Some extension algebras for standard modules over KLR algebras of type $A$
Doeke Buursma Alexander Kleshchev David J. Steinberg


Ext algebras for standard modules]Some extension algebras for standard modules over KLR algebras of type A


Department of Mathematics
University of Oregon
Eugene
OR 97403, USA
dbuursma@uoregon.edu


Department of Mathematics
University of Oregon
Eugene
OR 97403, USA
klesh@uoregon.edu


Department of Mathematics
University of Oregon
Eugene
OR 97403, USA
dsteinbe@uoregon.edu

[2010]16G99, 16E05, 17B37

The second author was supported by the NSF grants DMS-1161094, DMS-1700905, the Max-Planck-Institut, the Fulbright Foundation, and the DFG Mercator program through the University of Stuttgart.


	Khovanov-Lauda-Rouquier algebras R_θ of finite Lie type are affine quasihereditary with standard modules Δ(π) labeled by Kostant partitions of θ.
	Let Δ be the direct sum of all standard modules.
	It is known that the Yoneda algebra _θ:=_R_θ^*(Δ, Δ) carries a structure of an A_∞-algebra which can be used to reconstruct the category of standardly filtered R_θ-modules.
	In this paper, we explicitly describe _θ in two special cases: (1) when θ is a positive root in type , and (2) when θ is of Lie type [2].
	In these cases, _θ turns out to be torsion free and intrinsically formal.
	We provide an example to show that the A_∞-algebra _θ is non-formal in general.


[
    David J. Steinberg
    December 30, 2023
======================







§ INTRODUCTION


Let R_θ, be a Khovanov-Lauda-Rouquier (KLR) algebra of finite Lie type over a field  corresponding to θ∈ Q_+ <cit.>.
It is known that R_θ, is affine quasihereditary <cit.>, and in particular it comes with a family of standard modules {Δ(π)_|π∈(θ)}, where (θ) is the set of Kostant partitions of θ.
KLR algebras are defined over , so we have a -algebra R_θ with R_θ, ≅ R_θ⊗_.
The standard modules have natural integral forms Δ(π) with Δ(π)_≅Δ(π) ⊗_.
All modules and algebras are explicitly graded, and we refer to these gradings as KLR gradings.

Let Δ := ⊕_π∈(θ)Δ(π).
The Yoneda algebra _θ:=_R_θ^*(Δ, Δ) carries a structure of an A_∞-algebra <cit.>, which can be used to reconstruct the category ℱ(Δ) of modules which admit a finite filtration by standard modules <cit.>.
In view of <cit.> and <cit.>, understanding ℱ(Δ) is relevant for computing formal characters of the simple modules L(π)_ of R_θ,.
The smallest known example where the formal characters depend on the characteristic of  occurs in Lie type [5], see <cit.> (cf. <cit.>).

We now assume that the Lie type is [∞] with simple roots {α_i| i∈} so that the set of positive roots Φ_+ is {α_i+α_i+1+…+α_j| i≤ j}.
There is a natural lexicographic total order > on Φ_+.
Let Q_+ be the positive root lattice, and fix θ∈ Q_+.
If θ=∑ k_iα_i, we define the height of θ as (θ):=∑ k_i.
A Kostant partition of θ is a sequence π = (β_1^m_1, …, β_t^m_t) where m_1,…, m_t∈_>0, β_1 > ⋯ > β_t are positive roots, and m_1β_1 + ⋯ + m_tβ_t = θ.

We consider the Yoneda algebra _θ as the -linear category whose objects are (θ), and the set of morphisms from ρ∈(θ) to σ∈(θ) is

    _θ(ρ, σ) := ^*_R_θ(Δ(ρ), Δ(σ)).

The composition gf of g∈_θ(σ, τ) and f∈_θ(ρ, σ) is obtained using the composition of lifts of g in _R_θ(P_∙^σ, P_∙^τ) and f in _R_θ(P_∙^ρ, P_∙^σ), where P_∙^π is a projective resolution of Δ(π) for π∈(θ).
The category _θ has a homological grading for which the homogeneous components are _θ^m(ρ, σ) := _R_θ^m(Δ(ρ), Δ(σ)), and a KLR grading which is inherited from the KLR grading on the standard modules.
We use q to denote the KLR degree shift functor.
Theorems <ref> and <ref> describe the category _θ (as a bigraded category) in two special cases: (1) when θ is an arbitrary positive root, and (2) when θ is of type [2], i.e. θ is of the form c_1α_1 + c_2α_2.




 §.§ The case where θ is a positive root


Let θ = α_a + α_a+1 + ⋯ + α_b+1∈Φ_+.
Set l := b+2-a=(θ) and consider the polynomial algebra := [x_1, …, x_l].
We consider  to be graded with x_r = 2.
Note that (θ) is in bijection with the set of subsets of [1,l-1]: the subset associated to ρ = (β_1, …, β_u) ∈(θ) is D_ρ := { d_1, …, d_u-1} where d_t := (β_1)+⋯+(β_t).
For such D_ρ, set d_0 := 0 and d_u := l, and let J^ρ be the ideal of  generated by all x_r-x_s such that there is 1≤ t≤ u with d_t-1 < r,s ≤ d_t.
Define ^ρ := /J^ρ.
If D_ρ⊆ D_σ, then J^σ⊆ J^ρ so we have a natural projection _ρ^σ : ^σ^ρ.
We use the notation C⊆_m D to indicate that C⊆ D with |D∖ C| = m.


	Let θ = α_a + α_a+1 + ⋯ + α_b+1∈Φ_+ be a positive root.
	We have
	
    _θ^m(ρ,σ) ≅
    		q^-m^ρ   if  D_ρ⊆_m D_σ, 
    
    		0    otherwise.

	If D_ρ⊆_m D_σ⊆_n D_τ with f∈ q^-m^ρ≅_θ(ρ, σ) and g∈ q^-n^σ≅_θ(σ, τ), then the composition of g with f is given by _ρ^σ(g)f ∈ q^-(m+n)^ρ≅_θ(ρ, τ).





 §.§ The [2] case


For a nonnegative integer k, let Λ_k be the algebra of symmetric polynomials in k variables.
We impose a grading on Λ_k where linear symmetric polynomials have degree 2.
The space Λ_k is a free -module with basis { s_λ|λ∈(k)}, where (k) is the set of partitions with at most k parts, and s_λ is the Schur polynomial corresponding to λ <cit.>.
Letting V be the free graded -module with basis { v_0, v_1, v_2, …} such that v_i := 2i, there is an isomorphism of graded -modules

    γ_k: Λ_k  q^-k(k-1)^k V,  s_(λ_1, …, λ_k)↦ v_λ_k∧ v_λ_k-1+1∧⋯∧ v_λ_1+k-1

where ^k V is the kth exterior power of V.
Define

    ⋆ : Λ_a ⊗Λ_b → q^2abΛ_a+b,  f⊗ g ↦γ_a+b^-1(γ_a(f)∧γ_b(g)).


Considering Λ_a+b to be a subalgebra of Λ_a⊗Λ_b in the obvious way, we have that Λ_a ⊗Λ_b is free as a Λ_a+b-module with basis { s_λ⊗ 1 |λ∈(a,b)}, where (a,b) is the set of partitions with at most a nonzero parts, the first part being at most b, see <cit.> and <cit.>.
Moreover, <cit.> provides an explicit algorithm for writing any element of Λ_a⊗Λ_b as a Λ_a+b-linear combination of the basis elements s_λ⊗ 1.

Let c_1, c_2 ∈_≥ 0 and θ = c_1 α_1 + c_2 α_2.
Note that there is a bijection

    [0, min{ c_1, c_2} ] ∼⟷(θ),  r↦ (α_2^c_2-r, (α_1+α_2)^r, α_1^c_1-r),  r_ρρ.

For ρ, σ∈(θ) with r_ρ≥ r_σ, let

    ω(ρ, σ)    := -(r_ρ-r_σ)(1+(c_1-r_ρ)+(c_2-r_ρ)), 
    Λ(ρ,σ)    := q^ω(ρ,σ)Λ_c_2-r_ρ⊗Λ_r_ρ-r_σ⊗Λ_r_σ⊗Λ_c_1-r_ρ, 
    _ρ,σ   :=(r_ρ-r_σ,r_σ).

If f∈Λ_r_ρ-r_σ, we write

    f^ρ,σ:=1_Λ_c_2-r_ρ⊗ f⊗ 1_Λ_r_σ⊗ 1_Λ_c_1-r_ρ∈Λ(ρ,σ).

Then note that Λ(ρ,σ) is a free right Λ(ρ,ρ)-module with basis { s_λ^ρ,σ|λ∈_ρ,σ}.
We make Λ(ρ, σ) into a left Λ(σ, σ)-module via the composition of algebra homomorphisms:

    ξ: Λ(σ,σ) Λ_c_2-r_ρ⊗Λ_r_ρ-r_σ⊗Λ_r_σ⊗Λ_r_ρ-r_σ⊗Λ_c_1-r_ρ q^-ω(ρ,σ)Λ(ρ,σ);

the first map uses the embeddings Λ_c_2-r_σΛ_c_2-r_ρ⊗Λ_r_ρ-r_σ and Λ_c_1-r_σΛ_r_ρ-r_σ⊗Λ_c_1-r_ρ, and the second map is a⊗ b⊗ c⊗ d⊗ e↦ a⊗ bd⊗ c⊗ e (which we think of as identifying the two factors of Λ_r_ρ-r_σ).

If ρ, σ, τ∈(θ) with r_ρ≥ r_σ≥ r_τ, the tensor product Λ(σ,τ)⊗_Λ(σ,σ)Λ(ρ,σ) is now a free right Λ(ρ,ρ)-module with basis

    { s_μ^σ,τ⊗ s_λ^ρ,σ|μ∈_σ,τ, λ∈_ρ,σ}

and we define a map of right Λ(ρ,ρ)-modules

    Θ: Λ(σ,τ)⊗_Λ(σ,σ)Λ(ρ,σ) →Λ(ρ,τ),  s_μ^σ,τ⊗ s_λ^ρ,σ↦ (s_μ⋆ s_λ)^ρ,τ.

Let

    -♢- : Λ(σ,τ)⊗_Λ(ρ,σ) →Λ(ρ,τ),  g⊗ f ↦Θ(g⊗ f).

Thus, to compute g♢ f for some g∈Λ(σ, τ) and f ∈Λ(ρ, σ), the following steps must be performed: (1) write g = ∑_μ∈_σ,τ  s_μ^σ,τg_μ with g_μ∈Λ(σ, σ), (2) for each μ∈_σ,τ, write ξ(g_μ)f = ∑_λ∈_ρ,σ s_λ^ρ,σh_μ,λ with h_μ,λ∈Λ(ρ, ρ), (3) we have

    g♢ f = ∑_μ∈_σ,τ, λ∈_ρ,σ(s_μ⋆ s_λ)^ρ,τh_μ, λ.



	Let c_1, c_2 ∈_≥ 0 and θ = c_1 α_1 + c_2 α_2.
	We have
	
    _θ^m(ρ, σ) =
    	Λ(ρ, σ)    if  m = r_ρ-r_σ≥ 0, 
    
    		0    otherwise.

	If r_ρ≥ r_σ≥ r_τ with f∈Λ(ρ,σ) ≅_θ(ρ, σ) and g∈Λ(σ,τ) ≅_θ(σ, τ), then the composition of g with f is given by g♢ f ∈Λ(ρ,τ) ≅_θ(ρ, τ).





 §.§ Formality


For θ as in Theorem <ref> or <ref>, note that _R_θ(Δ(ρ), Δ(σ)) is torsion-free as a -module.
We do not know if this is true in general.

Also note that because _R_θ(Δ(ρ), Δ(σ)) is concentrated in homological degree |X_σ∖ X_ρ| (in the case of Theorem <ref>) or r_ρ-r_σ (in the case of Theorem <ref>), the A_∞-category structure of _θ must have m_n = 0 unless n = 2, so that _θ is intrinsically formal, see <cit.>.
In Section <ref>, we show that intrinsic formality and even formality does not occur in general:


	If θ = α_1 + 2α_2 + α_3, then the A_∞-category _θ is non-formal.





 §.§ The structure of the paper


The proofs of Theorems <ref> and <ref> occupy Sections <ref> and <ref>, respectively.
In the preliminary Section <ref>, we review the definition of the KLR algebra R_θ and the standard modules Δ(π).

In <ref> and <ref>, we record the relevant special cases of the projective resolution P_∙^ρ of Δ(ρ) constructed in <cit.>.
This resolution is finite and has the form P_∙^ρ = ⋯d_1→ P_1^ρd_0→ P_0^ρ_ρ→Δ(ρ) with P_n^ρ = ⊕_x∈ X_nq^s_xR_θ 1_x for some explicit index set X_n, integers s_x, and idempotents 1_x.
The map d_n: P_n+1^ρ→ P_n^ρ can be described as right multiplication by an X_n+1× X_n matrix (d_n^y,x) for some d_n^y,x∈ 1_y R_θ 1_x.

In <ref> and <ref>, we use the isomorphism _R_θ(R_θ 1_x, Δ(σ)) ≅ 1_x Δ(σ) to describe the complex _R_θ^∙(P_∙^ρ, Δ(σ)) in terms of familiar objects from commutative algebra; in the case of Theorem <ref>, these objects are polynomial rings and in the case of Theorem <ref>, they are rings of symmetric polynomials.
It turns out that in both cases, the complex _R_θ^∙(P_∙^ρ, Δ(σ)) is isomorphic to a Koszul complex corresponding to an explicit regular sequence, and we can therefore compute its homology H(_R_θ^∙(P_∙^ρ, Δ(σ))) =: _θ(ρ, σ) as a bigraded -module.

It remains to describe the composition in the category _θ.
This is done in <ref> and <ref>, where we explicitly lift elements of _R_θ^∙(P_∙^ρ, Δ(σ)) to _R_θ^∙(P_∙^ρ, P_∙^σ).
The function composition map _R_θ^∙(P_∙^σ, P_∙^τ) ⊗_R_θ^∙(P_∙^ρ, P_∙^σ) →_R_θ^∙(P_∙^ρ, P_∙^τ) induces a map on homology _θ(σ, τ) ⊗_θ(ρ, σ) →_θ(ρ, τ) which is the composition in the category _θ.

In Section <ref>, we review the A_∞-category structure on _θ following <cit.> and provide details for Example <ref>.





§ PRELIMINARIES





 §.§ Basic notation


Throughout, we work over an arbitrary principal ideal domain $̨ (since everything is defined over, one could just consider the case=̨).

Forr,s∈, we use the segment notation[r,s]:={ t∈|r≤t≤s},[r,s):={ t∈|r≤t< s}, etc.

Letqbe a variable, and((q))be the ring of Laurent series. Forn ∈_≥0, we define[n] := q^n-q^-n/q-q^-1,    [n]_± := q^±(n-1)[n],    [n]_(±)^!:=[1]_(±)[2]_(±)⋯ [n]_(±),and if0 ≤m ≤n,nm_(±) := [n]_(±)^!/[m]_(±)^![n-m]_(±)^!.We denote by_dthe symmetric group ondletters considered as a Coxeter group with generators{s_r:=(r,r+1)|1≤r<d}and the corresponding length functionℓ.
The longest element of_dis denotedw_0orw_0,d.
By definition,_dacts on[1,d]on the left.
For a setIthed-tuples fromI^dare written as words=i_1⋯i_d.
The group_dacts onI^dvia place permutations:w·=i_w^-1(1)⋯i_w^-1(d).

Given a compositionμ=(μ_1, …, μ_k)ofd, we have the corresponding standard parabolic subgroup_μ:=_μ_1 ×⋯×_μ_k ≤_d.
We denote by^μthe set of the shortest coset representatives for_d/_μ.




 §.§ Symmetric polynomials and the nil-Hecke algebra


We impose a grading on the polynomial algebra_d:=[̨x_1,…,x_d]such that(x_r) = 2.
The symmetric group_dacts by automorphisms on_dvia(w·f)(x_1, …, x_d) := f(x_w(1), …, x_w(d)).
The symmetric polynomial algebraΛ_d := _d^_dhas a basis consisting of Schur polynomials{ s_λ|λ∈(d)},where(d)is the set of partitions with at mostdnonzero parts, see <cit.>.

For a compositionμ= (μ_1, …, μ_k)ofd, the algebra ofμ-partially symmetric polynomials isΛ_μ:= _d^_μ.
We often writeΛ_μ_1,…,μ_kforΛ_μand identify it withΛ_μ_1 ⊗⋯⊗Λ_μ_k.
Fora,b ∈_≥0, let(a, b)be the set of partitions with at mostanonzero parts, each part being at mostb.
The following is known:


	The algebra Λ_μ is free as a Λ_d-module, and in the case where μ = (a,b) with a+b=d, a basis is given by
	{ s_λ⊗ 1 |λ∈(a, b)}.
	The freeness assertion is <cit.>, which, when combined with <cit.>, gives the basis assertion.


For an integerrwith1≤r < dand a reduced decompositionw = s_r_1⋯s_r_k∈_d, the Demazure operators on_dare defined as follows:∂_r := 𝕀__d - s_r/x_r+1 - x_r  and  ∂_w := ∂_r_1⋯∂_r_k.Note that∂_wdoes not depend on the choice of reduced decomposition and is a degree-2ℓ(w)element of__̨d̨.

For integersr,i,j ≥0withr+i+j ≤d, define

    U_r;i,j∈_d

to be the permutation which maps the interval[r+1, r+i]increasingly onto[r+1+j, r+i+j], and the interval[r+i+1, r+i+j]increasingly onto[r+1, r+j], and fixes all other elements of[1,d].
For example, we haveU_r;1,1 = s_r+1 = (r+1  r+2).
Recalling (<ref>), we have:

 <cit.>
	Let a,b ∈_≥ 0 with f∈Λ_a and g∈Λ_b.
	Then ∂_U_0;a,b(f⊗ g) = f⋆ g.

	Since we use different conventions from <cit.>, we provide a  translation for the reader's convenience.
	For w∈_a+b, let ∂_w' := (-1)^ℓ(w)∂_w.
	If k∈_≥ 0, recalling (<ref>), let γ_k' := (-1)^k2γ_k and note that γ_k'(s_λ) = v_λ_1+k-1∧ v_λ_2+k-2∧⋯∧ v_λ_k∈^k V for any λ = (λ_1, …, λ_k)∈(k).
	By <cit.>, we have ∂_U_0;a,b'(f⊗ g) = (γ_a+b')^-1m(γ_a'⊗γ_b')(f⊗ g), so
	
    ∂_U_0;a,b(f⊗ g)    = (-1)^ℓ(w)∂_U_0;a,b'(f⊗ g) 
       = (-1)^ab(γ_a+b')^-1m(γ_a'⊗γ_b')(f⊗ g) 
       = (-1)^ab + a+b2 + a2 + b2(γ_a+b)^-1m(γ_a⊗γ_b)(f⊗ g) 
       = f ⋆ g.


Definex_0 = x_0,d := ∏_r=1^d x_r^r-1∈_d.The following is well-known and easy to check:


	We have ∂_w_0(fx_0)=f for all f∈Λ_d.


The nil-Hecke algebra_dis given by generatorsτ_1,…,τ_d-1,  x_1,…,x_dsubject only to the relations

    x_r x_t=x_t x_r, τ_r^2=0, τ_rτ_r+1τ_r=τ_r+1τ_rτ_r+1, τ_rτ_s=τ_sτ_r  (|r-s|>1), 
    τ_r x_r=x_r+1τ_r-1, τ_r x_r+1=x_rτ_r+1, τ_r x_s=x_sτ_r  (s≠ r,r+1).

For a reduced decompositionw = s_r_1⋯s_r_k∈_d, we have a well-defined elementτ_w := τ_r_1 ⋯τ_r_k.
It is well-known that{τ_w x_1^k_1…x_d^k_d|w∈_d,  k_1,…,k_d∈_≥0}is a basis of_d.
In particular, we identify_das a subalgebra of_d.

 <cit.>
	The center of _d is equal to Λ_d.


We consider_das an_d-module withτ_wacting by∂_wandx_racting by multiplication withx_r.
Then we have the following easy to check and well-known properties:


	Let d∈_≥ 0, f∈_d, and w∈_d.
	Then in _d, we have
	τ_w_0 f τ_w = τ_w_0∂_w^-1(f)   and  τ_w f τ_w_0 = ∂_w(f)τ_w_0.

Define the following elements of_d:

    e_d := x_0,dτ_w_0,d  and   e_d' := τ_w_0,dx_0,d.

Lemmas <ref> and <ref> yield:


	In _d, the elements e_d and e_d' are idempotents.
	Moreover,
	
  * τ_w_0fτ_w_0=0 for any f ∈_d with f < d(d-1), and
		
  * τ_w_0x_0τ_w_0=τ_w_0.
	

 §.§ KLR Algebras


From now on, we setI := .
Ifi,j ∈Iwith|i-j|=1we set_i,j:=j-i∈{ 1,-1}.
We identifyIwith the set of vertices of the Dynkin diagram of type[∞]and denote by(_i,j)_i,j∈Ithe corresponding Cartan matrix so that_i,j=2ifi=j,_i,j=-1if|i-j| = 1, and_i,j=0otherwise.
We use the notationQ_+,Φ_+,, etc. introduced in Section <ref>.

Forθ∈Q_+of heightd, we defineI^θ:={=i_1⋯i_d∈I^d|α_i_1+…+α_i_d=θ}.
Let((q))·I^θ:=⊕_∈I^θ((q))·.
For1 ≤k ≤t, supposeθ_k∈Q_+are such thatθ_1+⋯+ θ_t = θ, and setd_k := (θ_k).
If^k ∈I^θ_k, then the concatenation^1⋯^t is considered as an element ofI^θ. Set^1⋯^t =: i_1⋯i_d.
Then the quantum shuffle product is

    ^1 ∘⋯∘^t := ∑_w ∈^(d_1, …, d_t) q^-e(w)w·(^1⋯^t) ∈((q))· I^θ,

wheree(w) := ∑_n < m,   w(n) > w(m)_i_n,i_m.
Ifa_k∈((q))·I^θ_k, we definea_1 ∘⋯∘a_t ∈((q))·I^θextending (<ref>) by linearity.

The KLR algebra <cit.> corresponding toθas above is the unital$̨-algebra R_θ (with identity denoted 1_θ) with generators

    { 1_|∈ I^θ}∪{ y_1, …, y_d}∪{ψ_1, …, ψ_d-1}

and defining relations

    y_r y_s = y_s y_r;R1
        1_ 1_=δ_,1_  and  ∑_∈ I^θ1_=1_θ;R2
        y_r 1_ = 1_ y_r   and  ψ_r 1_ = 1_s_r·ψ_r;R3
        (ψ_r y_t - y_s_r(t)ψ_r)1_ = δ_i_r,i_r+1(δ_t,r+1-δ_t,r)1_;R4
       ψ_r^2 1_ =
    	
    		0    if  i_r=i_r+1, 
    _i_r,i_r+1(y_r-y_r+1)1_   if  |i_r-i_r+1|=1, 
    
    		1_   otherwise;
    	R5
       ψ_r ψ_s = ψ_s ψ_r  if  |r-s|>1;R6
        (ψ_r+1ψ_rψ_r+1 - ψ_rψ_r+1ψ_r) 1_ =
    	_i_r,i_r+11_   if  |i_r-i_r+1|=1  and  i_r = i_r+2, 
    
    		0    otherwise.
    	R7

The right-hand sides of relations (<ref>) and (<ref>), when they are nonzero, will be referred to as error terms.
The algebra R_θ is graded with 1_=0; (y_s)= 2; (ψ_r1_)=-_i_r,i_r+1.

We will use the Khovanov-Lauda <cit.> diagrammatic notation for elements of R_θ.
In particular, for =i_1⋯ i_d∈ I^θ, 1≤ r<d and 1≤ s≤ d, we denote

    1_=
    baseline=1em (0,0)–(0,2) node[above]i_1;
    	 (1,0)–(1,2) node[above]i_2;
    	 (2,1) node⋯;
    	 (2,2) node[above]⋯;
    	 (3,0)–(3,2) node[above]i_d;
    ,   
    1_ψ_r =
    baseline=1em (0,0)–(0,2) node[above]i_1;
    	 (1.25,1) node⋯;
    	 (1,2) node[above]⋯;
    	 (2.5,0)–(2.5,2) node[above]i_r-1;
    	 (4.25,0)–(6,2) node[above]i_r+1;
    	 (6,0)–(4.25,2) node[above]i_r;
    	 (7.75,0)–(7.75,2) node[above]i_r+2;
    	 (9,1) node⋯;
    	 (9.25,2) node[above]⋯;
    	 (10.25,0)–(10.25,2) node[above]i_d;
    ,   
    1_ y_s=
    baseline=1em (0,0)–(0,2) node[above]i_1;
    	 (1.25,1) node⋯;
    	 (1,2) node[above]⋯;
    	 (2.5,0)–(2.5,2) node[above]i_s-1;
    	 (4,0)–(4,2) node[above]i_s;
    	 (5.5,0)–(5.5,2) node[above]i_s+1;
    	 (6.75,1) node⋯;
    	 (7,2) node[above]⋯;
    	 (8,0)–(8,2) node[above]i_d;
    	 (4,1);
    .


For each element w ∈_n, fix a reduced expression w=s_r_1⋯ s_r_l which determines an element ψ_w = ψ_r_1⋯ψ_r_l.
This element depends on the reduced expression of w.

 <cit.>, <cit.>
	Let θ∈ Q_+ and d=(θ).
	Then the following sets are $̨-bases ofR_θ:
	
    {ψ_w y_1^k_1… y_d^k_d1_| w∈_d, k_1,…,k_d∈_≥ 0,  ∈ I^θ}, 
    {y_1^k_1… y_d^k_dψ_w 1_| w∈_d, k_1,…,k_d∈_≥ 0,  ∈ I^θ}.


We identify the polynomial algebra

    _d := [̨y_1, …, y_d]

with the subalgebra ofR_θgenerated by{ y_1, …, y_d}according to Theorem <ref>.

The following lemma often simplifies calculations inR_θ.


	Let i_1,…,i_l∈ I be distinct, θ := α_i_1+…+α_i_l, = i_1⋯ i_l∈ I^θ, and w, w' ∈_l.
	In R_θ, we have ψ_w'ψ_w 1_ = ψ_w'w 1_ unless there are r, s∈ [1, l] such that |i_r-i_s|=1, r < s, w(r) > w(s), and w'w(r) < w'w(s).

	As i_1,…,i_l are distinct, the braid relation (<ref>) holds without error term in R_θ.
	Moreover, so long as there is no pair r,s as in the statement, the only quadratic relations we need to use are of the form ψ_t^2 1_ = 1_.
	Therefore ψ_w ψ_w' 1_ simplifies directly to ψ_ww'1_ as claimed.


 §.§ Parabolic subalgebras and divided power idempotents


Letθ_1,…,θ_t∈Q_+and setθ:=θ_1+…+θ_t.
Let1_θ_1,…,θ_t := ∑_^1∈ I^θ_1,…,^t∈ I^θ_t1_^1⋯^t∈ R_θ.Then we have an algebra embedding

    ι_θ_1,…,θ_t : R_θ_1⊗…⊗ R_θ_t 1_θ_1,…,θ_tR_θ_1+…+θ_t1_θ_1,…,θ_t

obtained by horizontal concatenation of the Khovanov-Lauda diagrams.
Forr_1∈R_θ_1,…, r_t∈R_θ_twe often writer_1∘…∘ r_t := ι_θ_1,…,θ_t(r_1⊗…⊗ r_t).For example,

    1_^1∘…∘ 1_^t = 1_^1⋯^t    (^1∈ I^θ_1,…,^t∈ I^θ_t).


We fix for the momenti∈I,d∈_≥0and takeθ= dα_i.
Then we have an isomorphismφ: _d R_dα_i,  x_r↦y_r,  τ_s ↦ψ_s.
Recalling (<ref>) and Lemma <ref>, the following element is an idempotent inR_dα_i:1_i^(d) := φ(e_d').Now letθ∈Q_+be arbitrary.
We defineI^θ_to be the set of all expressions of the formi_1^(d_1) ⋯i_r^(d_r)withd_1,…,d_r∈_≥0,i_1,…,i_r∈Iandd_1 α_i_1 + ⋯+ d_r α_i_r = θ.
We refer to such expressions as divided power words.
We identifyI^θwith the subset ofI^θ_which consists of all divided power words as above with alld_k=1.
We use the same notation for concatenation of divided power words as for concatenation of words.
For=i_1^(d_1)⋯i_r^(d_r)∈I^θ_, we define

    _(±)^! := [d_1]_(±)^! ⋯ [d_r]_(±)^!,   and  := i_1^d_1⋯ i_r^d_r∈ I^θ,

and the corresponding divided power idempotent is1_=1_i_1^(d_1)⋯ i_r^(d_r):=1_i_1^(d_1)∘…∘ 1_i_r^(d_r)∈ R_θ.We have the following generalization of (<ref>):1_^1∘…∘  1_^t=1_^1⋯^t    (^1∈ I^θ_1_,…,^t∈ I^θ_t_).In the algebra R_dα_i, if r_1+…+r_t=d then 1_i^(r_1)⋯ i^(r_t)ψ_w_0,d=ψ_w_0,d and 1_i^(r_1)⋯ i^(r_t)1_i^(d)=1_i^(d).

	Write ψ_w_0,d=(ψ_w_0,r_1∘…∘ψ_w_0,r_t)ψ_u for some u∈_d and use Lemma <ref>.


To be used as part of the Khovanov-Lauda diagrammatics, we denoteψ_w_0,d =:
baseline=0.25em0201w_0,    y_0,d =:
baseline=0.25em0201y_0,    1_i^(d) =
baseline=1.5em (0,2)–(0,3) node[above]i;
	 (1,2.5) node⋯;
	 (1,3) node[above]⋯;
	 (2,2)–(2,3) node[above]i;
	0201y_00212w_0 =:
baseline=0.25em0201.1i^d.For example, ifd=3, we have1_i^3ψ_w_0=
baseline=1em(0,1)–(0,2) node[above]i;
	(1,1)–(1,2) node[above]i;
	(2,1)–(2,2) node[above]i;
	0201w_0 =
baseline=1em (0,0)–(2,2) node[above]i;
	 (1,0)–(0,1)–(1,2) node[above]i;
	 (2,0)–(0,2) node[above]i;
,    1_i^3y_0 =
baseline=1em (0,1)–(0,2) node[above]i;
	 (1,1)–(1,2) node[above]i;
	 (2,1)–(2,2) node[above]i;
	0201y_0 =
baseline=1em (0,0)–(0,2) node[above]i;
	 (1,0)–(1,2) node[above]i;
	 (2,0)–(2,2) node[above]i;
	 (1,1);
	 (2,0.7);
	 (2,1.3);
,    1_i^(3) =
baseline=1em (0,0)–(2,2) node[above]i;
	 (1,0)–(0,1)–(1,2) node[above]i;
	 (2,0)–(0,2) node[above]i;
	 (0.75,0.25);
	 (1.3,0.7);
	 (1.7,0.3);
.More generally, we denote1_i_1^(d_1)⋯  i_r^(d_r) =:
baseline=0.5em0301.5i_1^d_1 (4.5,0.75) node⋯;
	6901.5i_r^d_r.§.§ Modules over R_θ


Letθ∈Q_+.
We denote byR_θthe category of graded leftR_θ-modules.
The morphisms in this category are all homogeneous degree zeroR_θ-homomorphisms, which we denote_R_θ(-,-).
ForV∈R_θ, letq^d Vdenote its grading shift byd, so ifV_mis the degreemcomponent ofV, then(q^dV)_m= V_m-d.
More generally, for a Laurent seriesa = a(q) = ∑_da_dq^d∈((q))with non-negative coefficients, we setaV := ⊕_d(q^d V)^⊕a_d.
ForU,V∈R_θ, we set_R_θ(U, V) := ⊕_d ∈ _R_θ(U, V)_d, where_R_θ(U, V)_d := _R_θ(q^d U, V) = _R_θ(U, q^-dV).We define^m_R_θ(U,V)and_R_θ(U)similarly fromext^m_R_θ(U,V)andend_R_θ(U).

For a free$̨-module V of finite rank we denote the rank of V by V.
A graded $̨-moduleV = ⊕_m∈V_mis called Laurentian if the graded componentsV_mare free of finite rank for allm∈andV_m = 0form ≪0.
For exampleR_θitself is Laurentian by Theorem <ref>.
The graded rank of a Laurentian$̨-module V is

    V:=∑_m∈( V_m) q^m∈((q)).


Recall that the ground ring $̨ is assumed to be a PID.
The following standard result often allows us to reduce to the case where$̨ is a field.


	If φ: V→ W is a degree 0 homomorphism of Laurentian $̨-modules such that the induced mapφ:V/JV →W/JWis an isomorphism of/̨J-vector spaces for every maximal idealJ, thenφis an isomorphism.


We say that anR_θ-moduleVis Laurentian if it is so as a$̨-module.
Recalling (<ref>), for a Laurentian R_θ-module V and ∈ I_^θ, by <cit.>, we have

    (1_ V) = 1/_+^!(1_ V),

which explains the usage of the term “divided power word” for ∈ I_^θ.
If V is a Laurentian R_θ-module then each 1_ V is a Laurentian $̨-module, and so we can define the formal character ofVas follows:V := ∑_∈ I^θ( 1_ V)·∈((q))· I^θ.Note that(q^dV)=q^d(V).

Forθ_1,…,θ_t∈Q_+andθ:= θ_1+⋯+θ_t, recalling (<ref>), we have a functor_θ_1,…,θ_t = R_θ 1_θ_1,…,θ_t⊗_R_θ_1⊗⋯⊗ R_θ_t-:(R_θ_1⊗⋯⊗ R_θ_t)→R_θ.ForV_1∈R_θ_1, …, V_t∈R_θ_t, we denote byV_1⊠⋯⊠V_tthe$̨-module V_1⊗⋯⊗ V_t, considered naturally as an (R_θ_1⊗⋯⊗ R_θ_t)-module, and set

    V_1∘⋯∘ V_t := _θ_1,…,θ_t V_1⊠⋯⊠ V_t.

By Theorem <ref>, setting d_k:=(θ_k), we have

    V_1 ∘⋯∘ V_t = ⊕_w∈^(d_1,…,d_t)ψ_w⊗ V_1⊗⋯⊗ V_t.

If V_1, …, V_t are Laurentian, then by <cit.>, recalling (<ref>), we have

    (V_1∘⋯∘ V_t) = (V_1)∘⋯∘(V_t).

For v_1∈ V_1,…, v_t∈ V_t, we denote

    v_1∘…∘ v_t := 1_θ_1,…,θ_t⊗ v_1⊗…⊗ v_t∈ V_1∘…∘ V_t.


If ^1∈ I_^θ_1, …, ^t∈ I_^θ_t, it is easy to check that

    R_θ_11_^1∘⋯∘ R_θ_t 1_^t≅ R_θ1_^1⋯^t.

Since R_θ 1_θ_1,…,θ_t is a free right R_θ_1⊗…⊗ R_θ_t-module of finite rank by Theorem <ref>, we get the following well-known properties:


	The functor _θ_1,…,θ_t is exact and sends finitely generated projectives to finitely generated projectives.


Let again θ_1, …, θ_t ∈ Q_+ and θ = θ_1 + ⋯ + θ_t.
Suppose (C_∙^k, d_k) is a chain complexes of R_θ_k-modules for 1 ≤ k ≤ t.
Let (C_∙^1 ∘⋯∘ C_∙^t)_n := ⊕_p_1 + ⋯ + p_t = n C_p_1^1 ∘⋯∘ C_p_t^t, and for x_1 ∈ C_p_1^1, …, x_t ∈ C_p_t^t, define

    d(x_1∘⋯∘ x_t) := ∑_k=1^t (-1)^p_k+1+ ⋯ + p_t x_1 ∘⋯∘ x_k-1∘ d_k(x_k) ∘ x_k+1∘⋯∘ x_t.

Then (C_∙^1 ∘⋯∘ C_∙^t, d) is a chain complex of R_θ-modules.
Proposition <ref> and <cit.> immediately imply the following.


	If C_∙^k is a projective resolution of M_k ∈R_θ_k for 1 ≤ k ≤ t, then C_∙^1 ∘⋯∘ C_∙^t is a projective resolution of M_1∘⋯∘ M_t ∈R_θ.





 §.§ Standard modules


The algebra R_θ is affine quasihereditary in the sense of <cit.>.
In particular, it comes with an important class of standard modules, which we now describe explicitly following <cit.>.

Fix β = α_i + ⋯ + α_j ∈Φ_+ of height l := j-i+1, and set _β := i (i+1)⋯ j∈ I^β.
We define the R_β-module Δ(β) to be a cyclic R_β-module generated by a vector v_β of degree 0 with defining relations

	
  * 1_ v_β = δ_,_βv_β for all  ∈ I^β;
	
  * ψ_r v_β = 0 for all 1≤ r < l;
	
  * y_r v_β = y_s v_β for all 1≤ r, s ≤ l.

The module Δ(β) can be considered as an (R_β, [̨x])-bimodule with the right action given by v_β x := y_1 v_β.
Diagrammatically, we represent

    v_β =
    baseline=0.5em (1.5,0) node[below]_β–(1.5,1)–(0,2);
    	 (1.5,0)–(1.5,1)–(1,2);
    	 (1.5,0)–(1.5,1)–(3,2);
    	 (2,1.9) node⋯;
    ,


    v_β x =
    baseline=0.5em (1.5,0) node[below]_β–(1.5,1)–(0,2);
    	 (1.5,0)–(1.5,1)–(1,2);
    	 (1.5,0)–(1.5,1)–(3,2);
    	 (2,1.9) node⋯;
    	 (1.5,0.5);
     = y_1 v_β =
    baseline=0.5em (1.5,0) node[below]_β–(1.5,1)–(0,2);
    	 (1.5,0)–(1.5,1)–(1,2);
    	 (1.5,0)–(1.5,1)–(3,2);
    	 (2,1.9) node⋯;
    	 (0.75,1.5);
     = … = y_l v_β =
    baseline=0.5em (1.5,0) node[below]_β–(1.5,1)–(0,2);
    	 (1.5,0)–(1.5,1)–(1,2);
    	 (1.5,0)–(1.5,1)–(3,2);
    	 (2,1.9) node⋯;
    	 (2.25,1.5);
    ,


    ψ_1 v_β =
    baseline=1em (1.5,0) node[below]_β–(1.5,1)–(0,2)–(1,3);
    	 (1.5,0)–(1.5,1)–(1,2)–(0,3);
    	 (1.5,0)–(1.5,1)–(3,2)–(3,3);
    	 (2,2) node⋯;
     = ⋯ = ψ_l-1 v_β =
    baseline=1em (1.5,0) node[below]_β–(1.5,1)–(0,2)–(0,3);
    	 (1.5,0)–(1.5,1)–(2,2)–(3,3);
    	 (1.5,0)–(1.5,1)–(3,2)–(2,3);
    	 (1,2) node⋯;
    =0.


The following lemma is easy to check.


	Let β∈Φ_+.
	Then there is an isomorphism of right [̨x]-modules [̨x] →Δ(β),  1 ↦ v_β.


For m∈_≥ 0 and β∈Φ_+, the R_mβ-module Δ(β)^∘ m is cyclicly generated by v_β^∘ m.
As explained in <cit.>, _m acts on Δ(β)^∘ m on the right so that

    v_β^∘ mx_r = v_β^∘ (r-1)∘ (v_β x)∘ v_β^∘ (m-r),    v_β^∘ mτ_s = v_β^∘ (s-1)∘ (ψ_w_l,l (v_β∘ v_β))∘ v_β^∘ (m-s-1),

where w_l,l is the longest element of ^(l,l).
Diagrammatically, we represent

    v_β^∘ m =
    baseline=0.5em (1.5,0) node[below]_β–(1.5,1)–(0,2);
    	 (1.5,0)–(1.5,1)–(1,2);
    	 (1.5,0)–(1.5,1)–(3,2);
    	 (2,1.9) node⋯;
    
    	 (5.5,0) node[below]_β–(5.5,1)–(4,2);
    	 (5.5,0)–(5.5,1)–(5,2);
    	 (5.5,0)–(5.5,1)–(7,2);
    	 (6,1.9) node⋯;
    
    	 (8,1) node⋯;
    
    	 (10.5,0) node[below]_β–(10.5,1)–(9,2);
    	 (10.5,0)–(10.5,1)–(10,2);
    	 (10.5,0)–(10.5,1)–(12,2);
    	 (11,1.9) node⋯;
    ,


    (v_β∘ v_β)τ_1 =
    baseline=1.5em (1.5,0) node[below]_β–(5.5,2)–(5.5,3)–(4,4);
    	 (1.5,0)–(5.5,2)–(5.5,3)–(5,4);
    	 (1.5,0)–(5.5,2)–(5.5,3)–(7,4);
    	 (6,3.9) node⋯;
    
    	 (5.5,0) node[below]_β–(1.5,2)–(1.5,3)–(0,4);
    	 (5.5,0)–(1.5,2)–(1.5,3)–(1,4);
    	 (5.5,0)–(1.5,2)–(1.5,3)–(3,4);
    	 (2,3.9) node⋯;
     = ψ_w_l,l(v_β∘ v_β) =
    baseline=1.5em (1.5,0) node[below]_β–(1.5,1)–(0,2)–(4,4);
    	 (1.5,0)–(1.5,1)–(1,2)–(5,4);
    	 (1.5,0)–(1.5,1)–(3,2)–(7,4);
    	 (2,1.9) node⋯;
    	 (2.2,3.9) node⋯;
    
    	 (5.5,0) node[below]_β–(5.5,1)–(4,2)–(0,4);
    	 (5.5,0)–(5.5,1)–(5,2)–(1,4);
    	 (5.5,0)–(5.5,1)–(7,2)–(3,4);
    	 (6,1.9) node⋯;
    	 (5.8,3.9) node⋯;
    .


Let ≤ be the lexicographic total order on Φ_+, i.e. for β = α_i + ⋯ + α_j∈Φ_+ and β' = α_i' + ⋯ + α_j'∈Φ_+, we have β < β' if and only if either i < i' or i = i' and j < j'.
Given θ∈ Q_+, a Kostant partition of θ is a sequence π = (β_1^m_1, …, β_t^m_t) such that m_1, …, m_t∈_>0, β_1 > ⋯ > β_t are positive roots, and m_1β_1 + ⋯ + m_tβ_t = θ.
We denote by (θ) the set of all Kostant partitions of θ.
For π = (β_1^m_1, …, β_t^m_t)∈(θ),

    Δ̂(π) := q^m_12+⋯+m_t2Δ(β_1)^∘ m_1∘⋯∘Δ(β_t)^∘ m_t

can now be considered as an (R_θ, _m_1⊗⋯⊗_m_t)-bimodule.
Recalling (<ref>), we define the corresponding standard module as

    Δ(π) := Δ̂(π)(e_m_1⊗⋯⊗ e_m_t).

Setting

    Λ_π := Λ_m_1, …, m_t,

 by Theorem <ref>, Δ(π) is naturally an (R_θ, Λ_π)-bimodule.
In fact, by <cit.>, the bimodule structure yields the isomorphism

    _R_θ(Δ(π))^≅Λ_π.


The module Δ(π) is cyclic as a left R_θ-module with standard generator

    v_π := (v_β^∘ m_1∘⋯∘ v_β^∘ m_t)(e_m_1⊗⋯⊗ e_m_t).

Noting that Δ(π) = Δ(β_1^m_1)∘⋯∘Δ(β_1^m_t), by <cit.>, we have an isomorphism of R_θ-modules

    Δ̂(π) ≅ [m_1]_+^! ⋯ [m_t]_+^! Δ(π).


If $̨ is a field, the modules{Δ(π)|π∈(θ)}are the standard modules for an affine quasihereditary structure on the algebraR_θ, see <cit.>.
If=̨or_p, they can be thought of as integral forms of the standard modules, see <cit.>.

Forρ∈(θ), suppose we have a projective resolution ofΔ(ρ)of the formP_∙^ρ= ⋯d_1→ P_1^ρd_0→ P_0^ρ_ρ→ Δ(ρ)withP_n^ρ= ⊕_x∈X_n q^s_xR_θ1_xfor some index setX_n, integerss_x, and idempotents1_x.
The mapd_n: P_n+1^ρ→P_n^ρcan be described as right multiplication by anX_n+1×X_nmatrixD_n = (d_n^y,x)for somed_n^y,x ∈1_y R_θ1_x.
Using the isomorphism_R_θ(q^n R_θ1_x, Δ(σ)) q^-n1_x Δ(σ)and recalling (<ref>), we obtain:


	There is an isomorphism of complexes of (right) Λ_σ-modules
	^∙_R_θ(P_∙^ρ, Δ(σ))≅ T_∙^P_∙^ρ(Δ(σ)),
	where T_∙^P_∙^ρ(σ) = ⋯d^1 T_1^P_∙^ρ(σ) d^0 T_0^P_∙^ρ(σ) with T_n^P_∙^ρ(σ)=⊕_x∈ X_nq^-s_x1_x Δ(σ) and d^n given by left multiplication with the X_n+1× X_n matrix D_n.


This yields an isomorphism_θ(ρ, σ) ≅H(T_∙^P_∙^ρ(σ))ofΛ_σ-modules.
One can also use the resolutionsP_∙^ρ,P_∙^σ, andP_∙^τto describe the composition map_θ(σ, τ) ⊗_θ(ρ, σ) →_θ(ρ, τ).
Indeed, let_R_θ^m(P_∙^ρ, P_∙^σ)denote the homological degreemhomomorphisms.
Then_R_θ^∙(P_∙^ρ, P_∙^σ)is a complex with respect to the differentialδgiven by

    δ(φ) := dφ - (-1)^mφ d

forφ∈_R_θ^m(P_∙^ρ, P_∙^σ).
We have an isomorphism

    H(_R_θ^∙(P_∙^ρ, P_∙^σ)) ≅_θ(ρ, σ)

induced by the maps

    _R_θ^m(P_∙^ρ, P_∙^σ) →_R_θ(P_m^ρ, Δ(σ)),  φ↦ (-1)^m(m+1)/2_σ(φ|_P_m^ρ).

Now the composition map_θ^n(σ, τ) ⊗_θ^m(ρ, σ) →_θ^n+m(ρ, τ)is induced from the composition of homomorphisms_R_θ^n(P_∙^σ, P_∙^τ) ⊗_R_θ^m(P_∙^ρ, P_∙^σ) →_R_θ^n+m(P_∙^ρ, P_∙^τ).





§ THE CASE WHERE Θ IS A ROOT


Throughout the section,θ=α_a+…+α_b+1(witha≤b+1) is a positive root of heightl=b+2-a.
There is a bijection from(θ)to the set of all subsets of[a,b]π = (π_1, …, π_u) ↦ C_π:={max(π_2), max(π_3), …, max(π_u)},where, for a rootα= α_i + …+ α_j, we letα:= [i,j].
Forπ, τ∈(θ), ifC_τ⊇C_π, we say thatτis a refinement ofπand writeτ⊇π.
If, in addition,|C_τ∖C_π| = n, we writeτ⊇_n πand say thatτis an n-refinement ofπ.
IfC_τ∖C_π= { i}for somei ∈[a,b], we write^i(π) := τ.
For example, we have^i((θ)) = (α_i+1 + ⋯+ α_b+1, α_a + ⋯+ α_i).

Ifτ= (τ_1, …, τ_t) ∈(θ), the elements of

    ^τ := ^((τ_1), …, (τ_t))

are called τ-shuffles.
Setd_v := (τ_1) + ⋯ + (τ_v)    (0 ≤ v ≤ t).We say that integersr,s ∈[1,l]are τ-equivalent if there is somev ∈[1,t]withd_v-1 < r,s ≤d_v.
Recalling (<ref>) and (<ref>), we haveΛ_τ=[̨x_1,…,x_t] ≅_R_θ(Δ(τ))^.We have a surjection

    _τ: _l Λ_τ,  y_r ↦ x_v  if  d_v-1 < r ≤ d_v,

so that_τ(y_r)=_τ(y_s)if and only ifrandsareτ-equivalent.
If, in addition,τ⊇π= (π_1, …, π_u), then_πfactors as_π= ^τ_π_τ, where the surjection^τ_πis defined as follows: for eachm∈[0,u], we have(π_1) + ⋯+ (π_m) = d_v_mfor somev_m ∈[0,t]; now

    _π^τ: Λ_τΛ_π,  x_r ↦ x_m  if  v_m-1 < r ≤ v_m.


 §.§ The resolution S_∙^ρ


In this subsection, we fixρ= (ρ_1, …, ρ_t)∈(θ).
Forα= α_i + α_i+1 + ⋯+ α_j∈Φ_+, let_α := i(i+1) ⋯ j∈ I^α,    e_α := 1__α∈ R_α.Then set

    _ρ := _ρ_1⋯_ρ_t∈ I^θ,    e_ρ := 1__ρ∈ R_θ.

Forπ,τ∈(θ), letw(τ,π)∈_lbe the unique permutation with

    w(τ,π)·_π=_τ,

so thate_τψ_w(τ,π) = ψ_w(τ,π)e_π.
Ifτ=^i(π), we sets(τ, π) := (-1)^|C_π∩ [a,i)|.Forn∈_≥0, we setS_n^ρ := ⊕_π⊇_n ρq^nR_θ e_π.The boundary mapS_n+1^ρ→S_n^ρis defined to be right multiplication with the matrix

    d_n := (d_n^τ, π)_τ   ⊇_n+1ρ
    π   ⊇_n ρ,  where  d_n^τ, π :=
    	
    		s(τ, π)e_τψ_w(τ, π)e_π   if τ⊇π, 
    
    		0    otherwise.

We define the augmentation map by_ρ:S_0^ρ=R_θ e_ρ→Δ(ρ),  he_ρ↦ hv_ρ,wherev_ρis the standard generator forΔ(ρ), see (<ref>).


	The following is a projective resolution of Δ(ρ):
	0 ⟶ S_l-t^ρ⟶⋯⟶ S_n+1^ρd_n⟶ S_n^ρ⟶⋯⟶ S_0^ρ_ρ⟶Δ(ρ) ⟶ 0.
	Given a complex C_∙, we denote by C_∙ the same complex but with all the boundary maps negated.
	If v∈[1,t], we let
	S_∙^(ρ_v) =
	S_∙^(ρ_v)   if t-v is odd, 

		S_∙^(ρ_v)   if t-v is even.
	
	Using (<ref>) and the fact that the resolutions S_∙^(ρ_v) and S_∙^(ρ_v) are isomorphic, it is easy to note that
	S_∙^ρ=S_∙^(ρ_1)∘…∘S_∙^(ρ_t)≅ S_∙^(ρ_1)∘…∘ S_∙^(ρ_t),
	so by Lemma <ref>, we have reduced to the case t = 1, i.e. ρ = (ρ_1) = (θ).

	To complete the proof, we show that S_∙^(θ)≅ P_∙, where P_∙ is a resolution of Δ(θ) constructed in <cit.> (see also <cit.>) and which we now recall.
	For π∈(θ), put
	^π := a^δ_a∉C_π(a+1)^δ_a+1∉C_π⋯ b^δ_b∉C_π(b+1)b^δ_b∈ C_π⋯ (a+1)^δ_a+1∈ C_πa^δ_a∈ C_π∈ I^θ.
	For n∈_≥ 0, set P_n := ⊕_π⊇_n (θ) q^n R_θ 1_^π.
	If π, τ∈(θ) with τ = ^i(π), let u(τ,π)∈_l be determined from u(τ,π)·^π=^τ, and define the matrix
	∂_n := (∂_n^τ,π)_τ   ⊇_n+1(θ) 
π   ⊇_n (θ),  where ∂_n^τ,π :=
	
		s(τ, π)1_^τψ_u(τ,π)1_^π   if τ⊇π, 

		0    otherwise.
	
	Right multiplication with ∂_n defines a map P_n+1→ P_n.
	By <cit.> (see also <cit.>), noting that P_0 = S_0^(θ), we have that
	0 ⟶ P_b+1-a⟶⋯⟶ P_n+1∂_n⟶ P_n ⟶⋯⟶ P_0 _(θ)⟶Δ(θ) ⟶ 0
	is a projective resolution of Δ(θ).

	For π∈(θ), let w(π)∈_l be the unique permutation with w(π)·^π=_π so that e_πψ_w(π) = ψ_w(π)1_^π.
	We have the map
	
    S^(θ)_n = ⊕_π⊇_n (θ)q^n R_θ e_π   →⊕_π⊇_n (θ)q^n R_θ 1_^π=P_n, 
    
    		(h_π e_π)_π⊇_n (θ)   ↦ (h_π e_πψ_w(π)1_^π)_π⊇_n (θ).

	To check that this yields an isomorphism of complexes, let τ := ^i(π) for some i∈ [a,b]∖ C_π, and check the following using Lemma <ref>:
	
  * e_τψ_w(τ)ψ_u(τ, π) = e_τψ_w(τ, π)ψ_w(π),
		
  * e_πψ_w(π)ψ_w(π)^-1 = e_π,
		
  * 1_^πψ_w(π)^-1ψ_w(π) = 1_^π. 

 §.§ The $̨-module_θ(ρ, σ)Throughout this subsection, we fixρ, σ∈(θ).
Recall the word_σ∈ I^θand the idempotente_σfrom (<ref>).
Writing_σ = j_1 j_2 ⋯ j_l, fori∈ [a, b+1], there exists a uniquer∈ [1, l]such thatj_r = i, and we denote_σ(i) := r.

We have the standard moduleΔ(σ)with generatorv_σ∈ e_σΔ(σ), see (<ref>).
As in <ref>, we considerΔ(σ)as an(R_θ,Λ_σ)-bimodule.
Recalling Lemma <ref>, we writeT_∙^ρ(σ) := T_∙^S_∙^ρ(Δ(σ)), so thatT_n^ρ(σ)=⊕_π⊇_nρq^-ne_πΔ(σ).
Recall (<ref>) and (<ref>).


	In Δ(σ), we have:
	
  * y_r v_σ = v_σ_σ(y_r); in particular, y_r v_σ = y_s v_σ if r and s are σ-equivalent;
		
  * ψ_w v_σ = 0 whenever w ∈_l is not a σ-shuffle.
	
	Moreover, Δ(σ) is free as a right Λ_σ-module with basis {ψ_w v_σ| w ∈^σ}.

	Use Lemma <ref> and (<ref>).


Recalling (<ref>), we now get:


	Let π,σ∈(θ).
	If σ⊉π, then e_πΔ(σ) = 0.
	If σ⊇_nπ, then there is an isomorphism of right Λ_σ-modules q^nΛ_σ e_πΔ(σ),  f ↦ψ_w(π,σ) v_σ f.
	If σ is a refinement of π, then every pair of σ-equivalent integers is also π-equivalent, so w(π,σ) is a σ-shuffle and the result follows from Lemma <ref> since in this case we have (ψ_w(π,σ)e_σ) = n.

	On the other hand, if σ⊉π, then there is some i∈ C_π with i∉ C_σ.
	It follows that _σ(i) and _σ(i)+1 are σ-equivalent but w(π, σ)(_σ(i)) > w(π, σ)(_σ(i)+1), so w(π, σ) is not a σ-shuffle, and e_πΔ(σ) = 0 by Lemma <ref>(<ref>).

	We have:
	
  *  If σ⊉ρ, then T_∙^ρ(σ) = 0.
		
  *  If σ⊇_m ρ, then as right Λ_σ-modules,
		T_n^ρ(σ)=⊕_σ⊇π⊇_n ρq^-nψ_w(π, σ) v_σ·Λ_σ≅⊕_σ⊇π⊇_n ρq^m-2nΛ_σ.
		In particular, T_n^ρ(σ) = 0 for n>m and
		T_m^ρ(σ) = q^-m e_σΔ(σ) = v_σ·Λ_σ≅ q^-mΛ_σ.
	If σ is not a refinement of ρ, then it cannot be a refinement of any π⊇ρ, which implies (<ref>).
	Let σ⊇_m ρ.
	Recall that T^ρ_n(σ)=⊕_π⊇_nρq^-ne_πΔ(σ).
	Let π⊇_nρ.
	If σ⊉π, then e_πΔ(σ)=0.
	Otherwise e_πΔ(σ)=ψ_w(π, σ) v_σ·Λ_σ.

	Let π∈(θ).
	Suppose that σ⊇^i(π) for some i∈ [a, b].
	Then
	ψ_w(^i(π), π)ψ_w(π, σ) e_σ = ψ_w(^i(π), σ)(y__σ(i)-y__σ(i+1))e_σ.
	Let τ = ^i(π).
	We have that i and i+1 are the only adjacent elements of [a, b+1] with _τ(i) > _τ(i+1) and _π(i) < _π(i+1).
	Since σ⊇τ = ^i(π), we have _σ(i) > _σ(i+1).
	Now we compute:
	
    ψ_w(τ, π)ψ_w(π, σ) e_σ   = ψ_w(τ, π)s__π(i)ψ__π(i)^2 ψ_s__π(i)w(π, σ) e_σ
       = ψ_w(τ, π)s__π(i) (y__π(i+1) - y__π(i)) ψ_s__π(i)w(π, σ) e_σ
       = ψ_w(τ, π)s__π(i)ψ_s__π(i)w(π, σ) (y__σ(i) - y__σ(i+1)) e_σ
       = ψ_w(τ, σ)(y__σ(i) - y__σ(i+1)) e_σ,

	where the first equality is obtained using the fact that the braid relations (<ref>) hold without error term in R_θ, the second comes by applying a non-trivial quadratic relation (<ref>) on strands colored i and i+1, the third is obtained using the relation (<ref>), and the last comes from Lemma <ref>.


The proof of Theorem <ref> amounts to showing thatT_∙^ρ(σ)is isomorphic to a certain Koszul complex (see <cit.>) which we now define.
Supposeσ⊇_m ρ, and writeC_σ∖ C_ρ = { i_1, …, i_m}withi_1 < ⋯ < i_m.
LetNbe the free rightΛ_σ-module of graded rankmq^-2, that is,N := q^-2Λ_σ^⊕ m.
Fork∈ [1, m], we denoteϵ_k := (0,…, 0, 1, 0, …, 0)∈ N(with “1” in thekthentry).
Recalling (<ref>), define

    z_k    := s(^i_k(ρ),ρ)_σ(y__σ(i_k)-y__σ(i_k+1))∈Λ_σ     (k=1,…,m), 
    
    	Z    := (z_1, …, z_m) = ϵ_1 z_1 + ⋯ + ϵ_m z_m∈ N.

Note thatZis a homogeneous degree0element ofN.
We consider the Koszul complexq^m^∙ Nassociated to the regular sequenceZfor the ringΛ_σ:

    0 ⟵ q^m^m N ⟵⋯⟵ q^m^n+1 N    ⟵ q^m^n N ⟵⋯⟵ q^m^0 N ⟵ 0 
    
    	Z∧ a     a

where^n Nis thenthexterior power of the freeΛ_σ-moduleN.
Note that^n Nhas aΛ_σ-basis{ϵ_k_1∧⋯∧ϵ_k_n| 1≤ k_1 < ⋯ < k_n ≤ m}.

Letσ⊇_m ρ.
By Corollary <ref>(<ref>), themthcomponent of_R_θ^∙(S_∙^ρ, Δ(σ))is the last nonzero component and it can be identified with_R_θ(q^m R_θ e_σ, Δ(σ)).
Thus, every element of themthcomponent is a cocycle, so there is a surjective map[-]: _R_θ(q^m R_θ e_σ, Δ(σ))  _θ^m(ρ, σ)=H^m(_R_θ^∙(S_∙^ρ, Δ(σ))),  φ↦ [φ],where[φ]is the cohomology class ofφ.
Moreover, by Lemma <ref>, we have an isomorphismξ:q^-mΛ_σ_R_θ(q^m R_θ e_σ, Δ(σ)),  f ↦ (e_σ↦ v_σ f).We considerΛ_ρto be aΛ_σ-module via the homomorphism_ρ^σ: Λ_σΛ_ρ, see (<ref>).


	Let ρ, σ∈(θ).
	If σ⊉ρ, then _θ(ρ, σ) = 0.
	If σ⊇_m ρ, then _θ(ρ, σ) = _θ^m(ρ, σ) and there is an isomorphism of Λ_σ-modules _θ^m(ρ, σ)  q^-mΛ_ρ which makes the following diagram of Λ_σ-modules commute:
	
		q^-mΛ_σ[r,"∼","ξ" below] [d,"_ρ^σ"]    _R_θ(q^m R_θ e_σ, Δ(σ)) [d,"[-]"] 

		q^-mΛ_ρ   _θ^m(ρ, σ) [l,dashed,"∼"].
	
	If σ⊉ρ, then _θ(ρ, σ) = 0 by Lemma <ref> and Corollary <ref>(<ref>).

	Assume now that σ⊇_m ρ and write C_σ∖ C_ρ = { i_1 < ⋯ < i_m}.
	If σ⊇π⊇_n ρ, we set B(π) := { k∈ [1, m] | i_k∈ C_π}.
	Note that |B(π)| = n.
	We define a map Θ_n: T_n^ρ(σ)→ q^m^n N of Λ_σ-modules by defining it on the Λ_σ-basis {ψ_w(π,σ)v_σ|σ⊇π⊇_n ρ} of T_n^ρ(σ), see Corollary <ref>.
	If B(π) = { k_1 < ⋯ < k_n}, then define
	Θ_n(ψ_w(π,σ)v_σ) := ϵ_k_1∧⋯∧ϵ_k_n.
	It is easy to see that Θ_n is an isomorphism of (graded) Λ_σ-modules.
	To show that Θ_n defines an isomorphism of complexes T_∙^ρ(σ) → q^m^∙ N, we must verify that the following square commutes:
	
		T_n+1^ρ(σ) [d, "Θ_n+1"]     T_n^ρ(σ) [l, "d_n· -" above] [d, "Θ_n"] 

		q^m^n+1N     q^m^n N, [l, "Z∧ -" above]
	
	where d_n is the matrix defined in (<ref>).
	We check this using an arbitrary basis element ψ_w(π,σ)v_σ∈ T_n^ρ(σ).
	We have
	
    Θ_n+1(d_n ψ_w(π,σ)v_σ)    = ∑_σ⊇τ⊇_1 πs(τ,π)Θ_n+1(ψ_w(τ,π)ψ_w(π,σ)v_σ) 
       = ∑_k∈ [1,m]∖ B(π)s(^i_k(π),π)Θ_n+1(ψ_w(^i_k(π),π)ψ_w(π,σ)v_σ) 
       = ∑_k∈ [1,m]∖ B(π)(-1)^B(π)∩ [1,k)Θ_n+1(ψ_w(^i_k(π),σ)v_σ)z_k 
       = ∑_k∈ [1,m]∖ B(π)(-1)^B(π)∩ [1,k) (ϵ_k_1∧⋯∧ϵ_k ∧⋯∧ϵ_k_n)z_k 
       = Z∧(ϵ_k_1∧⋯∧ϵ_k_n) 
       = Z∧Θ_n(ψ_w(π,σ)v_σ)

	where the second equality follows by noting that k↦^i_k(π) defines a bijection from [1,m]∖ B(π) to the set of 1-refinements of π which are refined by σ, the third by Lemma <ref> and the observation that s(^i_k(π),π) = (-1)^B(π)∩ [1,k)s(^i_k(ρ),ρ), and the remaining equalities follow from the definitions.

	Since q^m^∙ N is a Koszul complex corresponding to a regular sequence, we now have that _θ^n(ρ, σ) ≅ H^n(q^m^∙ N) = 0 unless n = m.
	The proof is complete in view of Lemma <ref> upon noting that the kernel of _ρ^σ: Λ_σΛ_ρ is the ideal generated by (z_1, …, z_m), so _ρ^σ induces an isomorphism
	
    H^m(q^m^∙ N) = q^-mΛ_σ/(z_1, …, z_m) q^-mΛ_ρ.


 §.§ The category _θ


Throughout this subsection, we use Theorem <ref> to identify_θ(ρ, σ) = _θ^m(ρ, σ)withq^-mΛ_ρwheneverσ⊇_m ρ∈(θ).

Letσ⊇_mρ∈(θ).
For anyf̂∈ q^-m_landπ⊇_k σ, setf̂_ρ,σ^π:= (-1)^m(m+1)/2+mk w(π, σ)·f̂.We define an element of_R_θ^m(S_∙^ρ, S_∙^σ)by

    φ_ρ,σ^f̂: S_m+k^ρ = ⊕_π⊇_m+kρ q^m+kR_θ e_π   →⊕_τ⊇_k σq^k R_θ e_τ = S_k^σ
    
    	(h_π e_π)_π⊇_m+kρ   ↦ (h_τf̂_ρ,σ^τ e_τ)_τ⊇_k σ.

Recalling the differential (<ref>) on_R_θ^∙(S_∙^ρ, S_∙^σ), we have:


	Let σ⊇_m ρ∈(θ).
	If f∈ q^-mΛ_ρ = _θ^m(ρ, σ) and f̂∈ q^-m_l are such that _ρ(f̂) = f, then
	
  * δ(φ_ρ,σ^f̂) = 0, and
		
  *  the isomorphism (<ref>) sends the cohomology class of φ_ρ,σ^f̂ to f.
	
	We prove (<ref>) by checking that the following diagram either commutes (if m is even) or anticommutes (if m is odd) whenever τ⊇_1 π⊇_k σ:
	[column sep = 4em]
		q^k+m+1R_θ e_τ[r,"-· d_k+m^τ,π"] [d,"-·f̂^τ_ρ,σ" left]     q^k+mR_θ e_π[d,"-·f̂^π_ρ,σ"] 

		q^k+1R_θ e_τ[r, "-· d_k^τ,π"]     q^k R_θ e_π.
	
	Modulo the signs, this is checked by the computation:
	
    ψ_w(τ,π)(w(π,σ)·f̂) e_π =
    		(w(τ, π)w(π,σ)·f̂)ψ_w(τ,π)  e_π
    		= (w(τ,σ)·f̂)ψ_w(τ,π) e_π,

	and the signs are taken care of by
	(-1)^m(m+1)/2+mks(τ,π) = (-1)^m(-1)^m(m+1)/2+m(k+1)s(τ, π).

	To prove (<ref>), first note that the restriction of φ_ρ,σ^f̂ to S_m^ρ has image in S_0^σ and can therefore be realized as
	
    φ_ρ,σ^f̂|_S_m^ρ: S_m^ρ = ⊕_π⊇_m ρq^m R_θ e_π   → R_θ e_σ = S_0^σ
    
    		(h_π e_π)_π⊇_m ρ   ↦ h_σf̂_ρ,σ^σ e_σ = (-1)^m(m+1)/2 h_σf̂ e_σ.

	By Corollary <ref>(<ref>), we identify _R_θ(S_m^ρ, Δ(σ)) with _R_θ(q^m R_θ e_σ, Δ(σ)).
	Then the image of φ_ρ,σ^f̂ under the map (<ref>) is
	(-1)^m(m+1)/2_σ(φ_ρ,σ^f̂|_S_m^ρ) = (e_σ↦ v_σ f) ∈_R_θ(q^m R_θ e_σ, Δ(σ)).
	An application of Theorem <ref> completes the proof.

	Let τ⊇σ⊇_m ρ∈(θ).
	If f∈ q^-mΛ_ρ and f̂∈ q^-m_l are such that _ρ(f̂) = f, then we also have _ρ(w(τ,σ)·f̂) = f.

	Since τ⊇σ⊇ρ, we have that r and w(τ, σ)(r) are ρ-equivalent for any r∈ [1, l], so _ρ(w(τ,σ)· y_r) = _ρ(y_r).
	This implies the result.

	Let τ⊇_n σ⊇_m ρ∈(θ).
	If f∈ q^-mΛ_ρ, and g∈ q^-nΛ_σ, then there exist f̂∈ q^-m_l, ĝ∈ q^-n_l, and _ρ^σ(g)f∈ q^-(m+n)_l with _ρ(f̂) = f, _σ(ĝ) = g, and _ρ(_ρ^σ(g)f) = _ρ^σ(g)f, such that
	φ_σ,τ^ĝφ_ρ,σ^f̂ = φ_ρ,τ^_ρ^σ(g)f.
	Choose any two lifts f̂∈ q^-m_l and ĝ∈ q^-n_l of f and g, respectively.
	By Lemma <ref>, since τ⊇σ, we have that w(τ, σ)·f̂ is also a lift of f, and so _ρ^σ(g)f := ĝ(w(τ, σ)·f̂)∈ q^-(m+n)_l is a lift of _ρ^σ(g)f.
	By (<ref>), it suffices to show that for any π⊇τ we have (_ρ^σ(g)f)_ρ,τ^π = ĝ_σ,τ^πf̂_ρ,σ^π.
	Modulo the signs, this holds by the following computation:
	
    w(π, τ) · (ĝ(w(τ,σ)·f̂))    =
    		(w(π,τ)·ĝ)(w(π,τ)w(τ, σ)·f̂) 
       = (w(π, τ)·ĝ)(w(π, σ)·f̂),

	and if π⊇_k τ, the signs are taken care of by
	
    (-1)^(m+n)(m+n+1)/2+(m+n)k = (-1)^m(m+1)/2+m(k+n)(-1)^n(n+1)/2+nk.


We combine Lemmas <ref> and <ref> to obtain the following theorem.


	Let τ⊇_n σ⊇_m ρ∈(θ).
	The composition in the category _θ is given by
	[row sep = -0.5em]
		_θ^n(σ, τ) ⊗_θ^m(ρ, σ) [r]    _θ^m+n(ρ, τ) 
90≅   90≅

		q^-nΛ_σ⊗ q^-mΛ_ρ[r]     q^-(m+n)Λ_ρ

		g ⊗ f [r, mapsto]    _ρ^σ(g)f.
	

§ THE [2] CASE


Throughout this section, we use a special notationα:=α_1, β:=α_2, γ:=α_1+α_2,so thatα,β,γare now the positive roots of the root system of type[2].
We fixθ := aα + bβwitha, b∈_≥ 0.
There is a bijectionσ: [0, min{ a, b} ](θ), s ↦σ(s) := (β^b-s, γ^s, α^a-s).The standardR_θ-modules areΔ(s) := Δ(σ(s))    (0≤ s≤min{ a, b}).We denote the standard generator ofΔ(s)byv_s:=v_σ(s), see (<ref>).
Recall that_R_θ(Δ(s))^≅Λ_σ(s) = Λ_b-s,s,a-s=Λ_b-s⊗Λ_s⊗Λ_a-s,see (<ref>).




 §.§ The resolution P_∙^r


Let0≤ r≤min{ a, b}.
We recall the resolution ofΔ(r)defined in <cit.>.
Forn∈ [0, r], recalling (<ref>), we define:

    _r,n   := 2^b-r1^r-n2^r1^n+a-r∈ I^θ, 
    _r,n   := 2^(b-r)1^(r-n)2^(r)1^(n)1^(a-r)∈ I^θ_, 
    
    	e_r,n   := 1__r,n∈ R_θ, 
    _r,n   := n(r-n+1) - a-r2 - b-r2 - r(r-1), 
    
    	x_r,n   := U_b-n-1;1,r+n=(b+r,b+r-1,…,b-n) ∈_a+b, 
    
    	d_r,n   := e_r,n+1ψ_x_r,n e_r,n∈ R_θ, 
    
    	P_n^r    := q^_r,nR_θ e_r,n.

Note that the right multiplication withd_r,nyields the degree zeroR_θ-homomorphism-· d_r,n:P^r_n+1→ P^r_n.
Defineu_r∈_2rby

    u_r(i) =  r-i-1/2   if i is odd, 
     2r-i-2/2   if i is even.

Let

    _r: P_0^r Δ(r),  x e_r,0↦ x(ψ_w_0,b-r∘ψ_u_r∘ψ_w_0,a-r) v_r.

By <cit.> and Lemma <ref>, we have:


	The following sequence is a projective resolution of Δ(r):
	0 ⟶ P_r^r ⟶⋯⟶ P_n+1^r -· d_r,n⟶ P_n^r ⟶⋯⟶ P_0^r _r⟶Δ(r) ⟶ 0.

 §.§ Weight spaces of standard modules


The following lemmas are useful for finding bases for certain weight spaces of the standard modulesΔ(s).
The first of them concerns the nil-Hecke algebra and is well-known and easy to check.
Recall the notation from <ref> and the(R_cα_i, _c)-bimodule structure onΔ̂(α_i^c)from <ref>.


	Let i∈ I and c∈_≥ 0.
	The map
	q^-c2_c →Δ̂(α_i^c),  f↦ v_α_i^∘ c f τ_w_0
	is an injective map of right Λ_c-modules with image Δ(α_i^c).


Recalling (<ref>), we have:


	The map
	q^-2s2Λ_s →Δ̂(γ^s),  f↦ψ_u_sv_γ^∘ s f
	is an injective map of right Λ_s-modules with image 1_1^(s)2^(s)Δ(γ^s).

Diagrammatically, the map in the lemma is given byf↦baseline=2em
	 (0.5,0) – (0.5,0.5);
	 (4.5,0) – (4.5,0.5);
	 (0.5,1.5) – (0.5,2) – (0,3) – (0,4);
	 (4.5,1.5) – (4.5,2) – (4,3) – (2,4);
	 (0.5,1.5) – (0.5,2) – (1,3) – (3,4);
	 (4.5,1.5) – (4.5,2) – (5,3) – (5,4);
	 (0,5) – (0,5.5);
	 (2,5) – (2,5.5);
	 (3,5) – (3,5.5);
	 (5,5) – (5,5.5);
	
	0.54.50.51.5f0245w_03545w_0
	 (2.5,2) node⋯;
	 (1,6) node1^s;
	 (4,6) node2^s;
.In view of Lemma <ref>, we may assume that $̨ is a field.
	LetMbe the free graded$̨-module with basis ^(2^s), where the degree of the basis element w∈^(2^s) is set to equal to the degree of ψ_w 1_(12)^s in R_sα+sβ.
	Lemma <ref> and (<ref>) show that the map
	ξ: q^s2M⊗__s →Δ̂(γ^s),  w⊗ f↦ψ_w v_γ^∘ s f
	is an isomorphism of right _s-modules.

	Let φ be the map in the statement.
	Then φ is the composition of the inclusion q^-2s2Λ_s  q^s2M⊗__s,  f ↦ u_s⊗ f with ξ, so φ is injective.
	It is now enough to show that φ(f) ∈ 1_1^(s)2^(s)Δ(γ^s) and that (1_1^(s)2^(s)Δ(γ^s)) = q^-2s2(Λ_s).
	For the first claim, note that ψ_u_s v_γ^∘ s is a nonzero element of smallest possible degree in 1_1^s2^sΔ̂(γ^s), so recalling (<ref>), we have
	
    1_1^(s)2^(s)ψ_u_sv_γ^∘ s f e_s    =
    		ψ_u_s v_γ^∘ s f x_0τ_w_0
       = ψ_u_s v_γ^∘ s f x_0τ_1τ_s_1^-1w_0
       = ψ_u_s v_γ^∘ sτ_1 f x_0 τ_s_1^-1w_0 + ψ_u_s v_γ^∘ s∂_1(f x_0) τ_s_1^-1w_0
       = ψ_u_s v_γ^∘ s∂_1(f x_0) τ_s_1^-1w_0
       =⋮
       = ψ_u_s v_γ^∘ s∂_w_0(fx_0) 
       = ψ_u_s v_γ^∘ s f,

	where the first equality comes from Lemma <ref> and (<ref>), the third comes from the relations in _s, the fourth follows because (ψ_u_s v_γ^∘ sτ_1) =(ψ_u_s v_γ^∘ s)-2< (ψ_u_s v_γ^∘ s), and the last holds by Lemma <ref>.

	As for graded dimension, we have
	
    (1_1^(s)2^(s)Δ(γ^s))
    		   = 1/([s]_+^!)^3(1_1^s2^sΔ̂(γ^s)) 
       = q^2s2([s]_-^!)^2/([s]_+^!)^3 (_1)^s 
       = q^-2s2Λ_s.

	where the first equality follows from (<ref>) and (<ref>), the second from (<ref>), (<ref>), and Lemma <ref>, and the last from an elementary computation.


 §.§ The $̨-module_θ(r, s)In this subsection we fixr,s ∈ [0, min{ a, b} ].
Recalling Lemma <ref>, we writeT_∙^r(s) := T_∙^P_∙^r(Δ(s)).
The terms of this complex are of the formT_n^r(s) = q^-_r,ne_r,nΔ(s)    (n =0,…,r).Ifr ≥ sand0≤ n≤ r-s, we define

    ω_n(r, s)    := -(r-s)(1+(a-r)+(b-r))+(r-s-n+1)(r-s-n), 
    
    	K_n    := q^ω_n(r,s)Λ_b-r, r-s, s, r-s-n, n, a-r, 
    
    	w_n    := U_b-r;s,r-s-nU_b-n;r-s,sU_b-r+s;r-s,r-s-nU_b-r;r-s,sU_b;s,r-s-n.

The diagram for1__r,nψ_w_n 1__s,0is the top part of the diagram below.
Observe thatΛ_σ(s)=Λ_b-s,s,a-s⊆ K_nin a natural way, so we may considerK_nas aΛ_σ(s)-module.
Recalling (<ref>), we have


	Suppose 0 ≤ n ≤ r.
	
  *  If n > r-s, then T_n^r(s) = 0.
		
  *  If n≤ r-s, then the map
		
    Ξ_n: K_n    → q^-_r,nΔ̂(s), 
    
    			f    ↦ψ_w_n(v_β^∘ b-s∘ψ_u_sv_γ^∘ s∘ v_α^∘ a-s)  f  (τ_w_0,b-s⊗ 1 ⊗τ_w_0,a-s)

		is an injective degree zero map of Λ_σ(s)-modules with image T_n^r(s).
	

Diagrammatically,Ξ_nis given byf↦baseline=4.5em
	 (0,0) – (0,1);
	 (2,0) – (2,1);
	 (3,0) – (3,1);
	 (5,0) – (5,1);
	 (6.5,0) – (6.5,2);
	 (10.5,0) – (10.5,2);
	 (12,0) – (12,1);
	 (14,0) – (14,1);
	 (15,0) – (15,1);
	 (17,0) – (17,1);
	 (18,0) – (18,1);
	 (20,0) – (20,1);
	 (0,3) – (0,10);
	 (2,3) – (2,10);
	 (3,3) – (3,7) – (12,10);
	 (5,3) – (5,7) – (14,10);
	 (6.5,3) – (6.5,4) – (6,5) – (6,6);
	 (6.5,3) – (6.5,4) – (7,5) – (9,6);
	 (10.5,3) – (10.5,4) – (10,5) – (8,6);
	 (10.5,3) – (10.5,4) – (11,5) – (11,6);
	 (12,3) – (12,7) – (3,10);
	 (14,3) – (14,7) – (5,10);
	 (15,3) – (15,10);
	 (17,3) – (17,10);
	 (18,3) – (18,10);
	 (20,3) – (20,10);
	 (6,7) – (3,8.5) – (6,10);
	 (8,7) – (5,8.5) – (8,10);
	 (9,7) – (12,8.5) – (9,10);
	 (11,7) – (14,8.5) – (11,10);
	
	0512w_0;
	122012w_0;
	02023;
	 (8.5,2.5) nodef;
	6867w_0;
	91167w_0;
	[dotted, color=gray] (2.5,2) – (2.5,3);
	[dotted, color=gray] (5.75,2) – (5.75,3);
	[dotted, color=gray] (11.25,2) – (11.25,3);
	[dotted, color=gray] (14.5,2) – (14.5,3);
	[dotted, color=gray] (17.5,2) – (17.5,3);
	
	 (1,10.5) node2^b-r;
	 (4,10.5) node1^r-s-n;
	 (7,10.5) node1^s;
	 (10,10.5) node2^s;
	 (13,10.5) node2^r-s;
	 (16,10.5) node1^n;
	 (19,10.5) node1^a-r;
	
	 (20,8.5) node[right]-90^90ψ_w_n;
	 (20,5) node[right]-90^90v_β^∘ b-s∘ψ_u_sv_γ^∘ s∘ v_α^∘ a-s;
	 (20,1) node[right]-90^90τ_w_0,b-s⊗ 1 ⊗τ_w_0,a-s;(<ref>) The condition n > r-s is equivalent to the condition a-r+n > a-s, which easily implies, by (<ref>), that 1__r,nΔ(s) = 0 hence e_r,nΔ(s)=0.

	(<ref>) It is straightforward to check that Ξ_n is homogeneous and Λ_σ(s)-equivariant.
	Define
	
    ξ_α   : q^-a-s2Λ_r-s-n,n,a-r q^-a-s2_a-sΔ(α^a-s) 
    ξ_β   : q^-b-s2Λ_b-r,r-s q^-b-s2_b-sΔ(β^b-s) 
    ξ_γ   : q^-2s2Λ_s  1_1^(s)2^(s)Δ(γ^s)

	where the isomorphisms are from Lemmas <ref> and <ref>.
	Let d = (r-s)(r-s-n)-s(r-s-n)-s(r-s) and define
	ξ : Δ(β^b-s) ⊗ 1_1^(s)2^(s)Δ(γ^s) ⊗Δ(α^a-s) → q^-d 1__r,nΔ(s),  x⊗ y ⊗ z ↦ψ_w_n(x∘ y ∘ z).
	Since w_n ∈^(b-s, 2s, a-s), (<ref>) shows that ξ is injective.
	Observing that Ξ_n = ξ(ξ_β⊗ξ_γ⊗ξ_α), we see that Ξ_n is injective with image in 1__r,nΔ(s).

	Now we prove that Ξ_n⊆ e_r,nΔ(s) by showing that e_r,nΞ_n(f) = Ξ_n(f).
	We have, using the relations in R_θ,
	
    Ξ_n(f)    =
    		baseline=4.5em
    			 (6.5,0) – (6.5,3);
    			 (10.5,0) – (10.5,3);
    			 (0,4) – (0,10);
    			 (2,4) – (2,10);
    			 (3,4) – (3,6) – (9,9) – (12,10);
    			 (5,4) – (5,6) – (11,9) – (14,10);
    			 (6.5,4) – (6.5,5) – (6,6) – (3,8);
    			 (6.5,4) – (6.5,5) – (7,6) – (12,8);
    			 (10.5,4) – (10.5,5) – (10,6) – (5,8);
    			 (10.5,4) – (10.5,5) – (11,6) – (14,8);
    			 (12,4) – (12,6) – (6,9) – (3,10);
    			 (14,4) – (14,6) – (8,9) – (5,10);
    			 (15,4) – (15,10);
    			 (17,4) – (17,10);
    			 (18,4) – (18,10);
    			 (20,4) – (20,10);
    			 (3,9) – (6,10);
    			 (5,9) – (8,10);
    			 (12,9) – (9,10);
    			 (14,9) – (11,10);
    			
    			0501w_0;
    			122001w_0;
    			0212x_0;
    			3512x_0;
    			121412x_0;
    			151712x_0;
    			182012x_0;
    			0223w_0;
    			3523w_0;
    			121423w_0;
    			151723w_0;
    			182023w_0;
    			02034;
    			 (8.5,3.5) nodef;
    			3589w_0;
    			121489w_0;
    			[dotted, color=gray] (2.5,3) – (2.5,4);
    			[dotted, color=gray] (5.75,3) – (5.75,4);
    			[dotted, color=gray] (11.25,3) – (11.25,4);
    			[dotted, color=gray] (14.5,3) – (14.5,4);
    			[dotted, color=gray] (17.5,3) – (17.5,4);
    			
    			 (1,10.5) node2^b-r;
    			 (4,10.5) node1^r-s-n;
    			 (7,10.5) node1^s;
    			 (10,10.5) node2^s;
    			 (13,10.5) node2^r-s;
    			 (16,10.5) node1^n;
    			 (19,10.5) node1^a-r;
    		
       =
    		baseline=4em
    			 (6.5,0) – (6.5,2);
    			 (10.5,0) – (10.5,2);
    			 (0,3) – (0,8);
    			 (2,3) – (2,8);
    			 (3,3) – (3,5) – (9,7);
    			 (5,3) – (5,5) – (11,7);
    			 (6.5,3) – (6.5,4) – (6,5) – (3,7);
    			 (6.5,3) – (6.5,4) – (7,5) – (12,7);
    			 (10.5,3) – (10.5,4) – (10,5) – (5,7);
    			 (10.5,3) – (10.5,4) – (11,5) – (14,7);
    			 (12,3) – (12,5) – (6,7);
    			 (14,3) – (14,5) – (8,7);
    			 (15,3) – (15,8);
    			 (17,3) – (17,8);
    			 (18,3) – (18,8);
    			 (20,3) – (20,8);
    			 (3,8) – (6,9);
    			 (5,8) – (8,9);
    			 (6,8) – (3,9);
    			 (8,8) – (5,9);
    			 (9,8) – (12,9);
    			 (11,8) – (14,9);
    			 (12,8) – (9,9);
    			 (14,8) – (11,9);
    			
    			0501w_0;
    			122001w_0;
    			0212x_0;
    			3512x_0;
    			121412x_0;
    			151712x_0;
    			182012x_0;
    			02023;
    			 (8.5,2.5) nodef;
    			3578w_0;
    			6878w_0;
    			91178w_0;
    			121478w_0;
    			0289w_0;
    			151789w_0;
    			182089w_0;
    			[dotted, color=gray] (2.5,2) – (2.5,3);
    			[dotted, color=gray] (5.75,2) – (5.75,3);
    			[dotted, color=gray] (11.25,2) – (11.25,3);
    			[dotted, color=gray] (14.5,2) – (14.5,3);
    			[dotted, color=gray] (17.5,2) – (17.5,3);
    			
    			 (1,9.5) node2^b-r;
    			 (4,9.5) node1^r-s-n;
    			 (7,9.5) node1^s;
    			 (10,9.5) node2^s;
    			 (13,9.5) node2^r-s;
    			 (16,9.5) node1^n;
    			 (19,9.5) node1^a-r;
    		
       =
    		baseline=3.5em
    			 (6.5,0) – (6.5,2);
    			 (10.5,0) – (10.5,2);
    			 (0,3) – (0,7);
    			 (2,3) – (2,7);
    			 (3,3) – (3,5) – (9,7);
    			 (5,3) – (5,5) – (11,7);
    			 (6.5,3) – (6.5,4) – (6,5) – (3,7);
    			 (6.5,3) – (6.5,4) – (7,5) – (12,7);
    			 (10.5,3) – (10.5,4) – (10,5) – (5,7);
    			 (10.5,3) – (10.5,4) – (11,5) – (14,7);
    			 (12,3) – (12,5) – (6,7);
    			 (14,3) – (14,5) – (8,7);
    			 (15,3) – (15,7);
    			 (17,3) – (17,7);
    			 (18,3) – (18,7);
    			 (20,3) – (20,7);
    			
    			0501w_0;
    			122001w_0;
    			0212x_0;
    			3512x_0;
    			121412x_0;
    			151712x_0;
    			182012x_0;
    			02023;
    			 (8.5,2.5) nodef;
    			3878w_0;
    			91478w_0;
    			0278w_0;
    			151778w_0;
    			182078w_0;
    			[dotted, color=gray] (2.5,2) – (2.5,3);
    			[dotted, color=gray] (5.75,2) – (5.75,3);
    			[dotted, color=gray] (11.25,2) – (11.25,3);
    			[dotted, color=gray] (14.5,2) – (14.5,3);
    			[dotted, color=gray] (17.5,2) – (17.5,3);
    			
    			 (1,8.5) node2^b-r;
    			 (5.5,8.5) node1^r-n;
    			 (11.5,8.5) node2^r;
    			 (16,8.5) node1^n;
    			 (19,8.5) node1^a-r;

	where the first equality follows from Lemma <ref>, the second follows because f is symmetric in the variables as indicated by the vertical dotted lines, hence, by Theorem <ref>, commutes with the parabolic subalgebra _b-r⊗_r-s⊗_s⊗_r-s-n⊗_n ⊗_a-r, and the last equality is straightforward.
	Now we have e_r,nΞ_n(f) = Ξ_n(f) by Lemma <ref>.

	To complete the proof, in view of Lemma <ref>, we assume that $̨ is a field and check that K_n =  q^-_r,ne_r,nΔ(s).
	For brevity, we denote[M] :=  M.
	Letz = (r-s)s + (r-s-n)s + (r-s)(r-s-n).
	We have
	
    [1__r,nΔ(s)]    = q^zr-ns_- rs_- [Δ(β^b-s)] [1_1^s 2^sΔ(γ^s)] [Δ(α^a-s)] 
       = q^z r-ns_- rs_- ([s]_+^!)^2 [Δ(α^a-s)] [1_1^(s) 2^(s)Δ(γ^s)][Δ(α^a-s)] 
       = q^ω_n(r,s)+_r,n[r-n]_+^![r]_+^!/[r-n-s]_+^![r-s]_+^! [_b-s] [Λ_s][_a-s],

	where the first equality follows from (<ref>), then second from (<ref>), and the last from Lemmas <ref> and <ref>.
	Thus,
	
    q^-_r,n[e_r,nΔ(s)]    = q^-_r,n/[b-r]_+^![r-n]_+^![r]_+^![n]_+^![a-r]_+^![1__r,nΔ(s)] 
       = q^ω_n(r,s)[_b-s]/[b-r]_+^![r-s]_+^![Λ_s][_a-s]/[r-s-n]_+^![n]_+^![a-r]_+^!
       = q^ω_n(r,s)[Λ_b-r][Λ_r-s][Λ_s][Λ_r-s-n][Λ_n][Λ_a-r],

	where the first equality is (<ref>), the second is by the above computation, and the last is by an elementary computation.


Because of Lemma <ref>(<ref>), we assume for the rest of the subsection thatr ≥ s.
We use Lemma <ref>(<ref>) to understand the complexT_∙^r(s)and compute its cohomology.
First, we re-express the coboundary map ofT_∙^r(s).
For0 ≤ n < r-s, setg_n := ∏_k=1^r-s(x_b+r-s-n - x_b-r+k) ∈_a+b-s.Iff ∈ K_n, observe thatfg_n ∈Λ_b-r, r-s, s, r-s-n-1, 1, n, a-rso by Proposition <ref>, we have a mapδ_n: K_n → K_n+1,  f ↦∂_U_b+r-s-n-1;1,n(fg_n).Recalling the isomorphismsΞ_n:K_n T_n^r(s)from  Lemma <ref>(<ref>), we have:


	If 0 ≤ n < r-s, then the following diagram commutes:
	
		K_n+1[d,"Ξ_n+1"]     K_n [l,"δ_n" above] [d, "Ξ_n"] 

		T_n+1^r(s)     T_n^r(s) [l, "d_r,n· -" above].
	
	In particular, the maps δ_n make K_∙ into a complex isomorphic to T_∙^r(s).

	There exist polynomials h_j ∈[̨y_b-r+1, …, y_b-s] and k_j ∈[̨y_b+r-n] such that ∏_i=1^r-s (y_b+r-n - y_b-r+i) = ∑_j h_j k_j.
	Let f∈ K_n and note that
	d_r,nΞ_n(f) = e_r,n+1ψ_x_r,n e_r,nΞ_n(f) = e_r,n+1ψ_x_r,nΞ_n(f)
	since Ξ_n⊆ T_n^r(s)=q^-_r,ne_r,nΔ(s) by Lemma <ref>.
	We compute e_r,n+1ψ_x_r,nΞ_n(f):
	
    baseline=5.5em
    			 (0,2) – (0,13);
    			 (2,2) – (2,13);
    			 (3,2) – (3,6) – (15,10) – (14,13);
    			 (5,2) – (5,6) – (17,10) – (16,13);
    			 (6.5,2) – (6.5,3) – (6,4) – (6,5);
    			 (12.5,2) – (12.5,3) – (12,4) – (9,5);
    			 (6.5,2) – (6.5,3) – (7,4) – (10,5);
    			 (12.5,2) – (12.5,3) – (13,4) – (13,5);
    			 (6,6) – (3,8) – (7,10) – (7,13);
    			 (8,6) – (5,8) – (9,10) – (9,13);
    			 (9,6) – (6,8) – (10,10) – (20,13);
    			 (10,6) – (14,8) – (11,10) – (10,13);
    			 (13,6) – (17,8) – (14,10) – (13,13);
    			 (14,2) – (14,6) – (3,10) – (3,13);
    			 (17,2) – (17,6) – (6,10) – (6,13);
    			 (18,2) – (18,10) – (17,13);
    			 (20,2) – (20,10) – (19,13);
    			 (21,2) – (21,13);
    			 (23,2) – (23,13);
    			
    			0501w_0142301w_002312f6956w_0101356w_0[0.5]0213142^b-r3913141^r-n-1101613142^r172013141^n+1[0.5]212313141^a-r
    			 (0,7) node[left]=;
    		   
       baseline=5.5em
    			 (0,2) – (0,14);
    			 (2,2) – (2,14);
    			 (3,2) – (3,6) – (15,10) – (14,14);
    			 (5,2) – (5,6) – (17,10) – (16,14);
    			 (6.5,2) – (6.5,3) – (6,4) – (6,5) – (9,6) – (6,8) – (10,10) – (20,14);
    			 (8.5,2) – (8.5,3) – (8,4) – (7,5) – (6,6) – (3,8) – (7,10) – (7,13);
    			 (12.5,2) – (12.5,3) – (12,4) – (9,5) – (8,6) – (5,8) – (9,10) – (9,13);
    			 (6.5,2) – (6.5,3) – (7,4) – (10,5) – (14,8) – (11,10) – (10,13);
    			 (8.5,2) – (8.5,3) – (9,4) – (11,5) – (15,8) – (12,10) – (11,13);
    			 (12.5,2) – (12.5,3) – (13,4) – (13,5) – (17,8) – (14,10) – (13,13);
    			 (14,2) – (14,6) – (3,10) – (3,14);
    			 (17,2) – (17,6) – (6,10) – (6,14);
    			 (18,2) – (18,10) – (17,14);
    			 (20,2) – (20,10) – (19,14);
    			 (21,2) – (21,14);
    			 (23,2) – (23,14);
    			
    			0501w_0142301w_002312f791314w_010131314w_0
    			 (1,14.5) node2^b-r;
    			 (4.5,14.5) node1^r-s-n;
    			 (8,14.5) node1^s-1;
    			 (11.5, 14.5) node2^s;
    			 (15,14.5) node2^r-s;
    			 (18,14.5) node1^n;
    			 (20,14.5) node1;
    			 (22,14.5) node1^a-r;
    			
    			 (0,7) node[left]=;
    		    by the relation (<ref>) and Lemma <ref>
       baseline=5.5em
    			 (0,2) – (0,13);
    			 (2,2) – (2,13);
    			 (3,2) – (3,5) – (11,8) – (11,10) – (14,12) – (14,13);
    			 (5,2) – (5,5) – (13,8) – (13,10) – (16,12) – (16,13);
    			 (6.5,2) – (6.5,3) – (6,4) – (6,5) – (12,7) – (10,8) – (20,12) – (20,13);
    			 (8.5,2) – (8.5,3) – (8,4) – (7,5) – (3,7) – (3,10) – (7,12);
    			 (12.5,2) – (12.5,3) – (12,4) – (9,5) – (5,7) – (5,10) – (9,12);
    			 (6.5,2) – (6.5,3) – (7,4) – (10,5) – (14,8) – (14,10) – (10,12);
    			 (8.5,2) – (8.5,3) – (9,4) – (11,5) – (15,8) – (15,10) – (11,12);
    			 (12.5,2) – (12.5,3) – (13,4) – (13,5) – (17,8) – (17,10) – (13,12);
    			 (14,2) – (14,5) – (6,7) – (6,10) – (3,12) – (3,13);
    			 (17,2) – (17,5) – (9,7) – (9,10) – (6,12) – (6,13);
    			 (18,2) – (18,11) – (17,12) – (17,13);
    			 (20,2) – (20,11) – (19,12) – (19,13);
    			 (21,2) – (21,13);
    			 (23,2) – (23,13);
    			
    			0501w_0142301w_002312f791213w_010131213w_0
    			 (1,13.5) node2^b-r;
    			 (4.5,13.5) node1^r-s-n;
    			 (8,13.5) node1^s-1;
    			 (11.5, 13.5) node2^s;
    			 (15,13.5) node2^r-s;
    			 (18,13.5) node1^n;
    			 (20,13.5) node1;
    			 (22,13.5) node1^a-r;
    			
    			 (0,6.5) node[left]=;
    		    by the relation (<ref>) 
       baseline=4.5em
    			 (0,2) – (0,11);
    			 (2,2) – (2,11);
    			 (3,3) – (3,5) – (10,8) – (14,10) – (14,11);
    			 (5,3) – (5,5) – (12,8) – (16,10) – (16,11);
    			 (6.5,2) – (6.5,3) – (6,4) – (6,5) – (17,9);
    			 (8.5,2) – (8.5,3) – (8,4) – (7,5) – (3,8) – (7,10);
    			 (12.5,2) – (12.5,3) – (12,4) – (9,5) – (5,8) – (9,10);
    			 (6.5,2) – (6.5,3) coordinate (x);
    			 [dashed] (x) – (7,4) – (10,5) – (14,7) – (13,8) coordinate (x);
    			 (x) – (10,10);
    			 (8.5,2) – (8.5,3) – (9,4) – (11,5) – (15,7) – (14,8) – (11,10);
    			 (12.5,2) – (12.5,3) – (13,4) – (13,5) – (17,7) – (16,8) – (13,10);
    			 (14,2) – (14,5) – (6,8) – (3,10) – (3,11);
    			 (17,2) – (17,5) – (9,8) – (6,10) – (6,11);
    			 (18,2) – (18,10) – (17,11);
    			 (20,2) – (20,10) – (19,11);
    			 (21,2) – (21,11);
    			 (23,2) – (23,11);
    			 (17,10) – (20,11);
    			
    			0501w_0142301w_002312f3523h_i791011w_010131011w_016.617.4910k_i
    			 (1,11.5) node2^b-r;
    			 (4.5,11.5) node1^r-s-n;
    			 (8,11.5) node1^s-1;
    			 (11.5, 11.5) node2^s;
    			 (15,11.5) node2^r-s;
    			 (18,11.5) node1^n;
    			 (20,11.5) node1;
    			 (22,11.5) node1^a-r;
    			
    			 (0,5.5) node[left]= ∑_j;
    		    by several applications of the relation (<ref>) 
       baseline=4.5em
    			 (0,2) – (0,11);
    			 (2,2) – (2,11);
    			 (3,3) – (3,5) – (10,8) – (14,10) – (14,11);
    			 (5,3) – (5,5) – (12,8) – (16,10) – (16,11);
    			 (6.5,2) – (6.5,3) – (6,4) – (6,5) – (3,8) – (7,10);
    			 (8.5,2) – (8.5,3) – (8,4) – (8,5) – (17,9);
    			 (12.5,2) – (12.5,3) – (12,4) – (5,8) – (9,10);
    			 (6.5,2) – (6.5,3) – (7,4) – (7,5) – (13,8) – (10,10);
    			 (8.5,2) – (8.5,3) coordinate (x);
    			 [dashed] (x) – (9,4) – (15,7) – (14,8) coordinate (x);
    			 (x) – (11,10);
    			 (12.5,2) – (12.5,3) – (13,4) – (13,5) – (17,7) – (16,8) – (13,10);
    			 (14,2) – (14,5) – (6,8) – (3,10) – (3,11);
    			 (17,2) – (17,5) – (9,8) – (6,10) – (6,11);
    			 (18,2) – (18,10) – (17,11);
    			 (20,2) – (20,10) – (19,11);
    			 (21,2) – (21,11);
    			 (23,2) – (23,11);
    			 (17,10) – (20,11);
    			
    			0501w_0142301w_002312f3523h_j791011w_010131011w_016.617.4910k_j
    			 (1,11.5) node2^b-r;
    			 (4.5,11.5) node1^r-s-n;
    			 (8,11.5) node1^s-1;
    			 (11.5, 11.5) node2^s;
    			 (15,11.5) node2^r-s;
    			 (18,11.5) node1^n;
    			 (20,11.5) node1;
    			 (22,11.5) node1^a-r;
    			
    			 (0,5.5) node[left]= ∑_j;
    		    (see (*) below) 
       baseline=4.5em
    			 (0,2) – (0,11);
    			 (2,2) – (2,11);
    			 (3,3) – (3,5) – (10,8) – (14,10) – (14,11);
    			 (5,3) – (5,5) – (12,8) – (16,10) – (16,11);
    			 (6.5,2) – (6.5,3) – (6,4) – (6,5) – (3,8) – (7,10);
    			 (10.5,2) – (10.5,3) – (10,4) – (8,5) – (5,8) – (9,10);
    			 (12.5,2) – (12.5,3) – (12,4) – (12,5) – (17,9);
    			 (6.5,2) – (6.5,3) – (7,4) – (9,5) – (13,8) – (10,10);
    			 (10.5,2) – (10.5,3) – (11,4) – (11,5) – (15,8) – (12,10);
    			 (12.5,2) – (12.5,3) coordinate (x);
    			 [dashed] (x) – (13,4) – (13,5) – (16,7) – (16,8) – (14.5,9) coordinate (x);
    			 (x) – (13,10);
    			 (14,2) – (14,5) – (6,8) – (3,10) – (3,11);
    			 (17,2) – (17,5) – (9,8) – (6,10) – (6,11);
    			 (18,2) – (18,10) – (17,11);
    			 (20,2) – (20,10) – (19,11);
    			 (21,2) – (21,11);
    			 (23,2) – (23,11);
    			 (17,10) – (20,11);
    			
    			0501w_0142301w_002312f3523h_j791011w_010131011w_016.617.4910k_j
    			 (1,11.5) node2^b-r;
    			 (4.5,11.5) node1^r-s-n;
    			 (8,11.5) node1^s-1;
    			 (11.5, 11.5) node2^s;
    			 (15,11.5) node2^r-s;
    			 (18,11.5) node1^n;
    			 (20,11.5) node1;
    			 (22,11.5) node1^a-r;
    			
    			 (0,5.5) node[left]= ∑_j;
    		    repeat the argument from the previous step several times 
       baseline=4.5em
    			 (0,2) – (0,11);
    			 (2,2) – (2,11);
    			 (3,3) – (3,5) – (10,8) – (14,10) – (14,11);
    			 (5,3) – (5,5) – (12,8) – (16,10) – (16,11);
    			 (6.5,2) – (6.5,3) – (6,4) – (6,5) – (3,8) – (7,10);
    			 (10.5,2) – (10.5,3) – (10,4) – (8,5) – (5,8) – (9,10);
    			 (12.5,2) – (12.5,3) – (12,4) – (12,5) – (15,6) – (9,8) – (6,10) – (6,11);
    			 (6.5,2) – (6.5,3) – (7,4) – (9,5) – (13,8) – (10,10);
    			 (10.5,2) – (10.5,3) – (11,4) – (11,5) – (15,8) – (12,10);
    			 (12.5,2) – (12.5,3) – (13,4) – (13,5) – (16,6) – (16,8) – (13,10);
    			 (14,2) – (14,5) – (6,8) – (3,10) – (3,11);
    			 (16,2) – (16,5) – (8,8) – (5,10) – (5,11);
    			 (17,2) – (17,9);
    			 (18,2) – (18,10) – (17,11);
    			 (20,2) – (20,10) – (19,11);
    			 (21,2) – (21,11);
    			 (23,2) – (23,11);
    			 (17,10) – (20,11);
    			
    			0501w_0142301w_002312f3523h_j791011w_010131011w_016.617.4910k_j
    			 (1,11.5) node2^b-r;
    			 (4.5,11.5) node1^r-s-n;
    			 (8,11.5) node1^s-1;
    			 (11.5, 11.5) node2^s;
    			 (15,11.5) node2^r-s;
    			 (18,11.5) node1^n;
    			 (20,11.5) node1;
    			 (22,11.5) node1^a-r;
    			
    			 (0,5.5) node[left]= ∑_j;
    		    (see (**) below) 
       baseline=4.5em
    			 (0,2) – (0,11);
    			 (2,2) – (2,11);
    			 (3,2) – (3,5) – (10,8) – (14,10) – (14,11);
    			 (5,2) – (5,5) – (12,8) – (16,10) – (16,11);
    			 (6.5,2) – (6.5,3) – (6,4) – (6,5) – (3,8) – (6,10);
    			 (12.5,2) – (12.5,3) – (12,4) – (9,5) – (6,8) – (9,10);
    			 (6.5,2) – (6.5,3) – (7,4) – (10,5) – (13,8) – (10,10);
    			 (12.5,2) – (12.5,3) – (13,4) – (13,5) – (16,8) – (13,10);
    			 (14,2) – (14,5) – (7,8) – (3,10) – (3,11);
    			 (16,2) – (16,5) – (9,8) – (5,10) – (5,11);
    			 (17,2) – (20,5) – (20,11);
    			 (18,2) – (17,5) – (17,11);
    			 (20,2) – (19,5) – (19,11);
    			 (21,2) – (21,11);
    			 (23,2) – (23,11);
    			
    			0501w_0142301w_002312fg_n
    			
    			691011w_010131011w_0
    			 (1,11.5) node2^b-r;
    			 (4.5,11.5) node1^r-s-n-1;
    			 (7.5,11.5) node1^s;
    			 (11.5, 11.5) node2^s;
    			 (15,11.5) node2^r-s;
    			 (18,11.5) node1^n;
    			 (20,11.5) node1;
    			 (22,11.5) node1^a-r;
    			
    			 (0,5.5) node[left]=;
    		    by the relation (<ref>) 
       baseline=4.5em
    			 (0,2) – (0,11);
    			 (2,2) – (2,11);
    			 (3,2) – (3,6) – (10,9) – (14,11);
    			 (5,2) – (5,6) – (12,9) – (16,11);
    			 (6.5,2) – (6.5,3) – (6,4) – (6,5);
    			 (12.5,2) – (12.5,3) – (12,4) – (9,5);
    			 (6.5,2) – (6.5,3) – (7,4) – (10,5);
    			 (12.5,2) – (12.5,3) – (13,4) – (13,5);
    			 (14,2) – (14,6) – (7,9) – (3,11);
    			 (16,2) – (16,6) – (9,9) – (5,11);
    			 (17,2) – (17,11);
    			 (20,2) – (20,11);
    			 (21,2) – (21,11);
    			 (23,2) – (23,11);
    			 (6,6) – (3,9) – (6,11);
    			 (9,6) – (6,9) – (9,11);
    			 (10,6) – (13,9) – (10,11);
    			 (13,6) – (16,9) – (13,11);
    			
    			05-0.250.75w_01423-0.250.75w_00230.752∂_U_b+r-s-n-1;1,n(fg_n)6956w_0101356w_0
    			 (1,11.5) node2^b-r;
    			 (4.5,11.5) node1^r-s-n-1;
    			 (7.5,11.5) node1^s;
    			 (11.5, 11.5) node2^s;
    			 (15,11.5) node2^r-s;
    			 (18.5,11.5) node1^n+1;
    			 (22,11.5) node1^a-r;
    			
    			 (0,5.5) node[left]=;
    		    by Lemma <ref> and the relation (<ref>).
(*) To obtain this equality, we attempt to move the portion of dashed strand in the previous diagram to the left by applying a special case of relation (<ref>):
	
    ψ_t+1ψ_tψ_t+11_ = ψ_tψ_t+1ψ_t 1_ + 1_ (if  i_t+1 = i_t+1  and  i_t = i_t+2)

	several times.
	In all except the last application, the error term 1_ causes the rest of the diagram to become 0, so we only keep the term ψ_tψ_t+1ψ_t 1_.
	In the last application, because of the defining relations in Δ(s), the term ψ_tψ_t+1ψ_t 1_ causes the rest of the diagram to become 0, so we only keep the error term 1_, which yields the desired diagram.

	(**) To obtain this equality, we again attempt to move the portion of dashed strand in the previous diagram to the left by applying the relation (<ref>) several times.
	In the first application, the error term 1_ yields the desired diagram, so we wish to show that the term ψ_tψ_t+1ψ_t 1_ causes the rest of the diagram to become 0.
	Since f is symmetric in the variables x_b+1, …, x_b+r-s-n, the error term in any application of the relation (<ref>) other than the first causes the rest of the diagram to become 0.
	However, after the last application of the relation, the term ψ_tψ_t+1ψ_t 1_ also causes the rest of the diagram to become 0 because of the defining relations in Δ(s).

	The expression represented by the last diagram above is Ξ_n+1(δ_n(f)), which completes the proof of the lemma.


Given an interval(c,d]and a polynomial ind-cvariables, we denotef(x_(c,d]):=f(x_c+1,…,x_d).For example, if0≤ m≤ d-cthen we have themthelementary symmetric functionE_m(x_(c,d])=∑_c<i_1<…<i_m≤ dx_i_1⋯ x_i_m.Now, for0≤ k < r-s, we definez_k := (-1)^r-s-k(E_r-s-k(x_(b-r, b-s]) - E_r-s-k(x_(b, b+r-s])).These elements are considered as elements of the algebra

    Λ^r,s := Λ_b-r, r-s, s, r-s, a-r.

Note that sinceΛ^r,s⊆Λ_b-r,r-s,s,r-s-n,n,a-r, eachK_nis naturally a (right)Λ^r,s-module.
We use this to interpret the right-hand side of the lemma below as an element ofK_n+1.


	For 0≤ n < r-s and a symmetric polynomial f in n variables, we have
	δ_n(f(x_(b+r-s-n, b+r-s])) = ∑_k=0^r-s-1(x_b+r-s-n^k ⋆ f(x_(b+r-s-n, b+r-s]))z_k.
	For brevity, write f = f(x_(b+r-s-n, b+r-s]).
	We observe
	g_n = ∏_k=1^r-s(x_b+r-s-n - x_b-r+k) = ∑_k=0^r-s (-1)^r-s-kx_b+r-s-n^k E_r-s-k(x_(b-r, b-s]),
	so that
	
    δ_n(f) = ∑_k=0^r-s (-1)^r-s-k∂_U_b+r-s-n-1;1,n(x_b+r-s-n^k f)E_r-s-k(x_(b-r, b-s]).

	Next, using the identity
	x_b+r-s-n^r-s = -∑_k=0^r-s-1(-1)^r-s-kx_b+r-s-n^k E_r-s-k(x_(b, b+r-s]),
	(<ref>) becomes
	δ_n(f) = ∑_k=0^r-s-1∂_U_b+r-s-n-1;1,n(x_b+r-s-n^k f)z_k,
	and the result follows from Proposition <ref>.


The proof of Theorem <ref> below amounts to showing thatT_∙^ρ(σ) ≅ K_∙is isomorphic to a certain Koszul complex which we now define.
LetNbe the free rightΛ^r,s-module of graded rank∑_k=0^r-s-1q^2k-2(r-s).
Fork=0,…,r-s-1, we have the basis elementϵ_k := 1∈ q^2k-2(r-s)Λ^r,s⊆ N.
We setZ := (z_0, …, z_r-s-1) =  ϵ_0 z_0 + ⋯ + ϵ_r-s-1 z_r-s-1∈ N.Note thatZis a homogeneous degree0element ofN.
We consider the Koszul complexq^ω_0(r,s)^∙Nassociated to the regular sequenceZfor the algebraΛ^r,s(see <cit.>), which has the form

    ⋯⟵ q^ω_0(r,s)^n+1 N    ⟵ q^ω_0(r,s)^n N ⟵⋯
    
    	Z∧ a     a

where^n Nis thenthexterior power of the freeΛ^r,s-moduleN.
Note that^n Nhas basis{ϵ_i_1∧⋯∧ϵ_i_n| 0 ≤ i_1 < ⋯ < i_n < r-s}.
Recall (<ref>).

By Lemmas <ref> and <ref>(<ref>), the complex_R_θ^∙(P_∙^r, Δ(s))is zero in degrees larger thanr-s, so every element of ther-scomponent is a cocycle and there is a surjective map[-]: _R_θ(q^_r,r-s R_θ e_r,r-s, Δ(s)) _θ^r-s(r, s) = H^r-s(_R_θ^∙(P_∙^r, Δ(s))),  φ↦ [φ],where[φ]is the cohomology class ofφ.
Moreover, by Lemma <ref>(<ref>), we have an isomorphismξ: K_r-s_R_θ(q^_r,r-sR_θ e_r,r-s,Δ(s)),  f ↦ (e_r-s,s↦ v_s f).Forr, s ∈_≥ 0withmin{ a, b}≥ r ≥ s, we defineΛ(r,s) := q^ω_r-s(r,s)Λ_b-r, r-s, s, a-r.Note that there is a surjection

    _r,s: q^ω_r-s(r,s)Λ^r,sΛ(r,s), f_1⊗ f_2⊗ f_3⊗ f_4⊗ f_5↦ f_1⊗ f_2f_4⊗ f_3⊗ f_5,

obtained by identifying the twoΛ_r-scomponents.
SinceΛ_σ(s)=Λ_b-s,s,a-s⊆Λ^r,s, we considerΛ^r,sto be a rightΛ_σ(s)-module, and we considerΛ(r,s)to be a rightΛ_σ(s)-module via the composition of algebra homomorphismsΛ_σ(s)Λ^r,s_r,sΛ_b-r, r-s, s, a-r=q^-ω_r-s(r,s)Λ(r,s)and then degree shift.
Note that_r,sis aΛ_σ(s)-homomorphism.


	Let 0 ≤ r,s ≤min{ a, b}.
	If r < s, then _θ(r, s) = 0.
	If r ≥ s, then _θ(r,s) = _θ^r-s(r,s) and there is an isomorphism of right Λ_σ(s)-modules _θ^r-s(r,s)Λ(r,s) such that the following diagram of right Λ_σ(s)-modules commutes:
	
		q^ω_r,s(r-s)Λ^r,s[r, "∼", "ξ" below] [d, "_r,s"]    _R_θ(q^_r,r-sR_θ e_r,r-s, Δ(s)) [d, "[-]"] 
Λ(r,s)    _θ^r-s(r,s). [l, dashed, "∼"]
	
	If r < s, then _θ(r, s) = 0 by Lemmas <ref> and <ref>(<ref>), so assume r ≥ s.
	For 0≤ n≤ r-s and λ∈(n, r-s-n), let  s_λ := s_λ(x_(b+r-s-n, b+r-s]) ∈ K_n.
	By Proposition <ref>, { s_λ|λ∈(n, r-s-n)} is a basis of K_n as an Λ^r,s-module, so there exists an isomorphism Θ_n: K_n → q^ω_0(r,s)^n N of Λ^r,s-modules such that
	Θ_n(s_λ) = ϵ_λ_n∧ϵ_λ_n-1+1∧⋯∧ϵ_λ_1+n-1.

	We claim that the maps Θ_n define an isomorphism of complexes between K_∙ and q^ω_0(r,s)^∙ N.
	We must verify that the following square commutes:
	
		K_n+1[d, "Θ_n+1"]     K_n [l, "δ_n" above] [d, "Θ_n"] 

		q^ω_0(r,s)^n+1N     q^ω_0(r,s)^n N. [l, "Z∧ -" above]
	
	Fix some λ = (λ_1, …, λ_n)∈(n, r-s-n) and set X := {λ_n, λ_n-1+1, …, λ_1+n-1}.
	We then have by Lemma <ref> and (<ref>):
	
    Θ_n+1(δ_n(s_λ))    = ∑_k∈ [0,r-s)∖ X(-1)^|X ∩ [0,k)| (ϵ_λ_n∧⋯∧ϵ_k∧⋯∧ϵ_λ_1+n-1 )z_k
       = ∑_k∈ [0,r-s)∖ X (ϵ_k∧ϵ_λ_n∧⋯∧ϵ_λ_1+n-1)z_k 
       = Z ∧Θ_n(s_λ).


	Since Z is a regular sequence, we have _θ^n(r, s) ≅ H^n(q^ω_0(r,s)^∙ N) = 0 unless n = r-s.
	We complete the proof using Lemmas <ref> and <ref> and the observation that by the fundamental theorem of elementary symmetric polynomials, the kernel of _r,s: q^ω_r-s(r,s)Λ^r,sΛ(r,s) is the ideal generated by (z_0, …, z_r-s-1), so _r,s induces an isomorphism of Λ^r,s-modules (and therefore of Λ_σ(s)-modules)
	
    H^r-s(q^ω_0(r,s)^∙ N) = q^ω_r-s(r,s)Λ^r,s/(z_0, …, z_r-s-1) Λ(r,s).


 §.§ The category _θ


Throughout this subsection, we use Theorem <ref> to identify_θ^r-s(r, s)withΛ(r,s) = q^ω_r-s(r,s)Λ_b-r,r-s,s,a-rwhenevermin{ a, b}≥ r ≥ s ≥ 0.

In this subsection, we will need to consider not only partially symmetric polynomials in the variablesxbut also partially symmetric polynomials in the variablesy.
This will be important since elements of_a+bwill be considered as elements ofR_θ, cf. (<ref>).
For anydwe have an isomorphismι_y→ x:_d_d, y_r↦ x_r.We will use the notationΛ_m^:=_m^_mfor the symmetric polynomials iny_1,…,y_m.
More generally, given a compositionμ = (μ_1, …, μ_k)ofd, we have the algebra ofμ-partially symmetric polynomialsΛ_μ^ := _d^_μ.
We often writeΛ_μ_1,…,μ_k^forΛ_μ^and identify it withΛ_μ_1^⊗⋯⊗Λ_μ_k^.
The isomorphismι_y→ xrestricts to the isomorphismι_y→ x:Λ_μ^Λ_μ.

For integersr,s,twithmin{ a, b}≥ r≥ s ≥ t ≥ 0, define the following:

    Λ̂(r,s)    := q^ω_r-s(r,s)Λ^_b-r, r-s, s, s, r-s, a-r⊆_a+b, 
    Λ̂(r,s,t)    := q^ω_r-t(r,t)-4(r-s)(s-t)Λ^_b-r, r-s, s-t, t, t, s-t, r-s, a-r⊆_a+b.

Recalling (<ref>), there is a surjection_r,s: Λ̂(r,s)  q^ω_r-s(r,s)Λ^r,s, f_1⊗ f_2⊗ f_3⊗ f_4⊗ f_5⊗ f_6↦ι_y→ x(f_1⊗ f_2⊗ f_3 f_4⊗ f_5⊗ f_6).Recalling (<ref>), let_r,s := _r,s_r,s: Λ̂(r,s) →Λ(r,s).Let againmin{ a, b}≥ r≥ s ≥ t ≥ 0, and0 ≤ n ≤ r-s.
Set

    D(r,s,n)    := ∏_i∈ (b+s-n, b+s] 
     j∈ (b-r, b-s](y_i-y_j) ∈_a+b, 
    
    	u(s,n)    := U_b-n; n, s∈_a+b, 
    
    	v(r,s,n)    := U_b-r; r-s, 2s-nU_b+s-n; n, r-s∈_a+b, 
    
    	w(s,t)    := U_b-s; t, s-tU_b; s-t, tU_b-s+t; s-t, s-t∈_a+b, 
    
    	x(r,s,t)    := U_b-r; r-s, s-tU_b+t; s-t, r-s∈_a+b.


Letmin{ a, b}≥ r≥ s ≥ n ≥ 0.
For anyf̂∈Λ̂(r,s), set

    f̂_r,s^n    := (-1)^r-s+12+n(r-s)e_r,r-s+nψ_v(r,s,n)D(r,s,n)(u(s,n)·f̂)e_s,n
       = ±baseline=3em
    		 (0,2) – (0,6);
    		 (2,2) – (2,6);
    		 (3,2) – (9,6);
    		 (5,2) – (11,6);
    		 (6,2) – (3,6);
    		 (8,2) – (5,6);
    		 (9,2) – (6,6);
    		 (11,2) – (8,6);
    		 (12,2) – (15,6);
    		 (14,2) – (17,6);
    		 (15,2) – (12,6);
    		 (17,2) – (14,6);
    		 (18,2) – (18,6);
    		 (20,2) – (20,6);
    		
    		05012^b-s68011^s-n911012^s1214011^n1520011^a-s02012D(r,s,n)(u(s,n)·f̂)
    		
    		[0.4]02672^b-r35671^s-n611672^r1217671^r-s+n[0.4]1820671^a-r
    		
    	.

Define

    φ_r,s^f̂: P^r := ⊕_m=0^r q^_r,mR_θ e_r,m   →⊕_n=0^s q^_s,nR_θ e_s,n =: P^s, 
    
    	(h_m e_r,m)_m=0^r    ↦ (h_r-s+nf̂_r,s^n e_s,n)_n=0^s.

We think ofφ_r,s^f̂as an element of_R_θ^r-s(P_∙^r, P_∙^s).
Recalling the differential (<ref>) on_R_θ^∙(P_∙^r, P_∙^s), we have:


	Suppose 0 ≤ s ≤ r ≤min{ a, b} and let f∈Λ(r,s) = _θ(r,s).
	If f̂∈Λ̂(r,s) is such that _r,s(f̂) = f, then
	
  * δ(φ_r,s^f̂) = 0, and
		
  *  the isomorphism (<ref>) sends the cohomology class of φ_r,s^f̂ to f.
	
	We prove (<ref>) by checking that the following diagram either commutes (if r-s is even) or anticommutes (if r-s is odd) whenever 0 ≤ n < r-s.
	[column sep=5em]
		q^_r,n+1+r-sR_θ e_r,n+1+r-s[r, "-· d_r,r-s+n"] [d, "-·f̂_n+1"]     q^_r,r-s+nR_θ e_r,r-s+n[d, "-·f̂_n"] 

		q^_s,n+1R_θ e_s,n+1[r, "-· d_s,n"]     q^_s,nR_θ e_s,n.
	
	We compute ± d_r,r-s+nf̂_n:
	
    baseline=4.5em
    			 (0,2) – (0,5);
    			 (2,2) – (2,5);
    			 (3,2) – (10,5);
    			 (5,2) – (12,5);
    			 (6,2) – (3,5);
    			 (9,2) – (6,5);
    			 (10,2) – (7,5);
    			 (12,2) – (9,5);
    			 (13,2) – (16,5);
    			 (15,2) – (18,5);
    			 (16,2) – (13,5);
    			 (18,2) – (15,5);
    			 (19,2) – (19,5);
    			 (21,2) – (21,5);
    			 (0,6) – (0,10);
    			 (2,6) – (2,10);
    			 (3,6) – (3,10);
    			 (5,6) – (5,10);
    			 (6,6) – (18,10);
    			 (7,6) – (6,10);
    			 (12,6) – (11,10);
    			 (13,6) – (12,10);
    			 (18,6) – (17,10);
    			 (19,6) – (19,10);
    			 (21,6) – (21,10);
    			
    			05012^b-s69011^s-n1012012^s1315011^n1621011^a-s02112D(r,s,n)(u(s,n)·f̂)[0.4]02562^b-r36561^s-n712562^r1318561^r-s+n[0.4]1921561^a-r[0.3]0210112^b-r[0.6]3510111^s-n-161110112^r121810111^r-s+n+1[0.4]192110111^a-r
    			
    			
    		   
    
    		   baseline=4em
    			 (0,2) – (0,5) – (0,9);
    			 (2,2) – (2,5) – (2,9);
    			 (3,2) – (10,5) – (9,9);
    			 (5,2) – (12,5) – (11,9);
    			 (6,2) – (3,5) – (3,9);
    			 (8,2) – (5,5) – (5,9);
    			 (9,2) – (6,5) – (18,9);
    			 (10,2) – (7,5) – (6,9);
    			 (12,2) – (9,5) – (8,9);
    			 (13,2) – (16,5) – (15,9);
    			 (15,2) – (18,5) – (17,9);
    			 (16,2) – (13,5) – (12,9);
    			 (18,2) – (15,5) – (14,9);
    			 (19,2) – (19,5) – (19,9);
    			 (21,2) – (21,5) – (21,9);
    			
    			05012^b-s69011^s-n1012012^s1315011^n1621011^a-s02112D(r,s,n)(u(s,n)·f̂)[0.3]029102^b-r[0.6]359101^s-n-16119102^r12189101^r-s+n+1[0.4]19219101^a-r
    			
    			 (0,5) node[left]=;
    		    by Lemma <ref>
       baseline=3em
    			 (0,2) – (0,4);
    			 (2,2) – (2,4);
    			 (3,2) – (3,4);
    			 (5,2) – (5,4);
    			 (6,2) – (6,4);
    			 (8,2) – (8,4);
    			 (9,2) – (15,4);
    			 (10,2) – (9,4);
    			 (12,2) – (11,4);
    			 (13,2) – (12,4);
    			 (15,2) – (14,4);
    			 (16,2) – (16,4);
    			 (18,2) – (18,4);
    			 (19,2) – (19,4);
    			 (21,2) – (21,4);
    			 (0,5) – (0,7);
    			 (2,5) – (2,7);
    			 (3,5) – (9,7);
    			 (5,5) – (11,7);
    			 (6,5) – (3,7);
    			 (8,5) – (5,7);
    			 (9,5) – (6,7);
    			 (11,5) – (8,7);
    			 (12,5) – (15,7);
    			 (15,5) – (18,7);
    			 (16,5) – (12,7);
    			 (18,5) – (14,7);
    			 (19,5) – (19,7);
    			 (21,5) – (21,7);
    			
    			05012^b-s69011^s-n1012012^s1315011^n1621011^a-s02112u(s,n)·f̂02145D(r,s,n+1)[0.3]02782^b-r[0.6]35781^s-n-1611782^r1218781^r-s+n+1[0.4]1921781^a-r
    			
    			 (0,4) node[left]=;
    		    by several applications of the relations (<ref>) and (<ref>), and by Lemma <ref>
       baseline=3em
    			 (0,1) – (0,3);
    			 (5,1) – (5,3);
    			 (6,1) – (6,3);
    			 (8,1) – (8,3);
    			 (9,1) – (15,3);
    			 (10,1) – (9,3);
    			 (12,1) – (11,3);
    			 (13,1) – (12,3);
    			 (15,1) – (14,3);
    			 (16,1) – (16,3);
    			 (21,1) – (21,3);
    			 (0,4) – (0,6);
    			 (2,4) – (2,6);
    			 (3,4) – (9,6);
    			 (5,4) – (11,6);
    			 (6,4) – (3,6);
    			 (8,4) – (5,6);
    			 (9,4) – (6,6);
    			 (11,4) – (8,6);
    			 (12,4) – (15,6);
    			 (15,4) – (18,6);
    			 (16,4) – (12,6);
    			 (18,4) – (14,6);
    			 (19,4) – (19,6);
    			 (21,4) – (21,6);
    			
    			05012^b-s69011^s-n1012012^s1315011^n1621011^a-s02134D(r,s,n+1)(u(s,n+1)·f̂)[0.3]02672^b-r[0.6]35671^s-n-1611672^r1218671^r-s+n+1[0.4]1921671^a-r
    			
    			 (0,3.5) node[left]=;
    		    by Lemma <ref> and the relation (<ref>) 
       baseline=3em
    			 (0,1) – (0,3);
    			 (5,1) – (5,3);
    			 (6,1) – (6,3);
    			 (8,1) – (8,3);
    			 (9,1) – (15,3);
    			 (10,1) – (9,3);
    			 (12,1) – (11,3);
    			 (13,1) – (12,3);
    			 (15,1) – (14,3);
    			 (16,1) – (16,3);
    			 (21,1) – (21,3);
    			 (0,5) – (0,7);
    			 (2,5) – (2,7);
    			 (3,5) – (9,7);
    			 (5,5) – (11,7);
    			 (6,5) – (3,7);
    			 (8,5) – (5,7);
    			 (9,5) – (6,7);
    			 (11,5) – (8,7);
    			 (12,5) – (15,7);
    			 (15,5) – (18,7);
    			 (16,5) – (12,7);
    			 (18,5) – (14,7);
    			 (19,5) – (19,7);
    			 (21,5) – (21,7);
    			
    			05012^b-s[0.6]69011^s-n1012012^s1315011^n1621011^a-s05342^b-s[0.6]68341^s-n-1911342^s1215341^n+11621341^a-s02145D(r,s,n+1)(u(s,n+1)·f̂)[0.3]02782^b-r[0.6]35781^s-n-1611782^r1218781^r-s+n+1[0.4]1921781^a-r
    			
    			 (0,4) node[left]=;
    		    by Lemma <ref>.

	The last diagram represents ±f̂_n+1d_s,n.
	The signs are taken care of by
	(-1)^r-s+12+(n+1)(r-s) = (-1)^r-s(-1)^r-s+12+n(r-s).

	To prove (<ref>), recalling (<ref>), we compute (-1)^(r-s)(r-s+1)/2_s(e_r,r-sf̂_r,s^0):
	
    baseline=3em
    			 (0,0) – (0,3);
    			 (5,0) – (5,3);
    			 (6.5,0) – (6.5,1) – (6,2) – (6,3);
    			 (10.5,0) – (10.5,1) – (10,2) – (8,3);
    			 (6.5,0) – (6.5,1) – (7,2) – (9,3);
    			 (10.5,0) – (10.5,1) – (11,2) – (11,3);
    			 (12,0) – (12,3);
    			 (17,0) – (17,3);
    			 (0,6) – (0,8);
    			 (2,6) – (2,8);
    			 (3,6) – (9,8);
    			 (5,6) – (11,8);
    			 (6,6) – (3,8);
    			 (8,6) – (5,8);
    			 (9,6) – (6,8);
    			 (11,6) – (8,8);
    			 (12,6) – (12,8);
    			 (14,6) – (14,8);
    			 (15,6) – (15,8);
    			 (17,6) – (17,8);
    			
    			0534w_06834w_091134w_0121734w_005452^b-s68451^s911452^s1217451^a-s01756f̂[0.3]02892^b-r35891^s611892^r1214891^r-s[0.3]1517891^a-r
    			
    			
    		   
    
    		   baseline=3em
    			 (0,0) – (0,3);
    			 (5,0) – (5,3);
    			 (6.5,0) – (6.5,1) – (6,2) – (6,3);
    			 (10.5,0) – (10.5,1) – (10,2) – (8,3);
    			 (6.5,0) – (6.5,1) – (7,2) – (9,3);
    			 (10.5,0) – (10.5,1) – (11,2) – (11,3);
    			 (12,0) – (12,3);
    			 (17,0) – (17,3);
    			 (0,5) – (0,7);
    			 (2,5) – (2,7);
    			 (3,5) – (9,7);
    			 (5,5) – (11,7);
    			 (6,5) – (3,7);
    			 (8,5) – (5,7);
    			 (9,5) – (6,7);
    			 (11,5) – (8,7);
    			 (12,5) – (12,7);
    			 (14,5) – (14,7);
    			 (15,5) – (15,7);
    			 (17,5) – (17,7);
    			
    			0534w_06834w_091134w_0121734w_001745f̂
    			 (1,7.5) node2^b-r;
    			 (4,7.5) node1^s;
    			 (7,7.5) node2^s;
    			 (10,7.5) node2^r-s;
    			 (13,7.5) node1^r-s;
    			 (16,7.5) node1^a-r;
    			
    			 (0,3.5) node[left]=;
    		    by Lemma <ref>
       baseline=3.5em
    			 (6.5,0) – (6.5,1);
    			 (10.5,0) – (10.5,1);
    			 (0,2) – (0,8);
    			 (2,2) – (2,8);
    			 (3,2) – (3,6) – (9,8);
    			 (5,2) – (5,6) – (11,8);
    			 (6.5,2) – (6.5,3) – (6,4) – (6,5);
    			 (10.5,2) – (10.5,3) – (10,4) – (8,5);
    			 (6.5,2) – (6.5,3) – (7,4) – (9,5);
    			 (10.5,2) – (10.5,3) – (11,4) – (11,5);
    			 (12,2) – (12,8);
    			 (14,2) – (14,8);
    			 (15,2) – (15,8);
    			 (17,2) – (17,8);
    			 (6,6) – (3,8);
    			 (8,6) – (5,8);
    			 (9,6) – (6,8);
    			 (11,6) – (8,8);
    			
    			0501w_0121701w_001712_r,s(f̂)6856w_091156w_0
    			 (1,8.5) node2^b-r;
    			 (4,8.5) node1^s;
    			 (7,8.5) node2^s;
    			 (10,8.5) node2^r-s;
    			 (13,8.5) node1^r-s;
    			 (16,8.5) node1^a-r;
    			
    			 (0,4) node[left]=;
    		    by Lemma <ref>, the relation (<ref>), and the defining relations of Δ(s).

	The last diagram represents Ξ_r-s(_r,s(f̂)), where Ξ_r-s is as in Lemma <ref>(<ref>).
	Thus, the image of φ_r,s^f̂ under the map (<ref>) is
	(e_r,r-s↦Ξ_r-s(_r,s(f̂)))∈_R_θ(q^_r,r-sR_θ e_r,r-s, Δ(s)).
	The proof is complete upon an application of Theorem <ref>.


Forf̂∈Λ̂(r,s)andĝ∈Λ̂(s,t), define

    : Λ̂(s,t)⊗Λ̂(r,s) →Λ̂(r,t),  ĝ⊗f̂↦∂_x(r,s,t)(D(r,s,s-t)ĝ(w(s,t)·f̂)).

	Let 0 ≤ t ≤ s ≤ r ≤min{ a, b}.
	If f̂∈Λ̂(r,s) and ĝ∈Λ̂(s,t), then φ_s,t^ĝφ_r,s^f̂ = φ_r,t^ĝf̂.

	Let y(s,t,n) := U_b-s;t-n,s-t∈_d and note that
	
    u(t,n)^-1y(s,t,n)u(s,s-t+n) = w(s,t)

	and
	
    D(r,s,s-t+n)D(s,t,n) = D(r,t,n)D(r,s,s-t).

	We compute f̂_r,s^s-t+nĝ_s,t^n:
	
    baseline=3em
    			 (0,2) – (0,4);
    			 (5,2) – (5,4);
    			 (6,2) – (12,4);
    			 (8,2) – (14,4);
    			 (9,2) – (6,4);
    			 (11,2) – (8,4);
    			 (12,2) – (9,4);
    			 (14,2) – (11,4);
    			 (15,2) – (18,4);
    			 (17,2) – (20,4);
    			 (18,2) – (15,4);
    			 (20,2) – (17,4);
    			 (21,2) – (21,4);
    			 (26,2) – (26,4);
    			 (0,6) – (0,9);
    			 (2,6) – (2,9);
    			 (3,6) – (12,9);
    			 (5,6) – (14,9);
    			 (6,6) – (3,9);
    			 (8,6) – (5,9);
    			 (9,6) – (6,9);
    			 (14,6) – (11,9);
    			 (15,6) – (18,9);
    			 (20,6) – (23,9);
    			 (21,6) – (15,9);
    			 (23,6) – (17,9);
    			 (24,6) – (24,9);
    			 (26,6) – (26,9);
    			
    			08012^b-t911011^t-n1214012^t1517011^n1826011^a-t02612D(s,t,n)(u(t,n)·ĝ)05452^b-s68451^t-n914452^s1520451^s-t+n2126451^a-s02656D(r,s,s-t+n)(u(s,s-t+n)·f̂)029102^b-r359101^t-n6149102^r15239101^r-t+n24269101^a-r
    			
    			
    		   
    
    		   baseline=3em
    			 (0,2) – (0,7);
    			 (2,2) – (2,7);
    			 (3,2) – (3,4) – (12,7);
    			 (5,2) – (5,4) – (14,7);
    			 (6,2) – (12,4) – (9,7);
    			 (8,2) – (14,4) – (11,7);
    			 (9,2) – (6,4) – (3,7);
    			 (11,2) – (8,4) – (5,7);
    			 (12,2) – (9,4) – (6,7);
    			 (14,2) – (11,4) – (8,7);
    			 (15,2) – (18,4) – (21,7);
    			 (17,2) – (20,4) – (23,7);
    			 (18,2) – (15,4) – (18,7);
    			 (20,2) – (17,4) – (20,7);
    			 (21,2) – (21,4) – (15,7);
    			 (23,2) – (23,4) – (17,7);
    			 (24,2) – (24,7);
    			 (26,2) – (26,7);
    			
    			08012^b-t911011^t-n1214012^t1517011^n1826011^a-t02612D(r,s,s-t+n)D(s,t,n)(u(t,n)·ĝ)(y(s,t,n)u(s,s-t+n)·â)02782^b-r35781^t-n614782^r1523781^r-t+n2426781^a-r
    			
    			 (0,4) node[left]=;
    		    by (<ref>), Lemma <ref>, and Theorem <ref>
       baseline=3em
    			 (0,2) – (0,7);
    			 (2,2) – (2,7);
    			 (3,2) – (6,4) – (12,7);
    			 (5,2) – (8,4) – (14,7);
    			 (6,2) – (3,4) – (9,7);
    			 (8,2) – (5,4) – (11,7);
    			 (9,2) – (9,4) – (3,7);
    			 (11,2) – (11,4) – (5,7);
    			 (12,2) – (12,4) – (6,7);
    			 (14,2) – (14,4) – (8,7);
    			 (15,2) – (15,4) – (21,7);
    			 (17,2) – (17,4) – (23,7);
    			 (18,2) – (21,4) – (18,7);
    			 (20,2) – (23,4) – (20,7);
    			 (21,2) – (18,4) – (15,7);
    			 (23,2) – (20,4) – (17,7);
    			 (24,2) – (24,7);
    			 (26,2) – (26,7);
    			
    			08012^b-t911011^t-n1214012^t1517011^n1826011^a-t02612D(r,t,n)D(r,s,s-t)u(t,n)·(ĝ(w(s,t)·f̂))02782^b-r35781^t-n614782^r1523781^r-t+n2426781^a-r
    			
    			 (0,4) node[left]=;
    		    by (<ref>), (<ref>), and the relation (<ref>) 
       baseline=2.5em
    			 (0,2) – (0,5);
    			 (2,2) – (2,5);
    			 (3,2) – (9,5);
    			 (8,2) – (14,5);
    			 (9,2) – (3,5);
    			 (11,2) – (5,5);
    			 (12,2) – (6,5);
    			 (14,2) – (8,5);
    			 (15,2) – (21,5);
    			 (17,2) – (23,5);
    			 (18,2) – (15,5);
    			 (23,2) – (20,5);
    			 (24,2) – (24,5);
    			 (26,2) – (26,5);
    			
    			08012^b-t911011^t-n1214012^t1517011^n1826011^a-t02612D(r,t,n)(u(t,n)·∂_x(s,t)(D(r,s,s-t)ĝ(w(s,t)·f̂)))02562^b-r35561^t-n614562^r1523561^r-t+n2426561^a-r
    			
    			 (0,3) node[left]=;
    		    by Lemma <ref>.

	The last diagram represents (ĝf̂)^n.

	Let 0 ≤ t ≤ s ≤ r ≤min{ a, b}, λ∈(r-s,s), and μ∈(s-t,t), and set f̂ := 1⊗ s_λ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1 ∈Λ̂(r,s) and ĝ := 1⊗ s_μ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1 ∈Λ̂(s,t).
	Then
	ĝf̂ = 1⊗ (s_λ⋆ s_μ) ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1 ∈Λ̂(r,t).
	We use the notation λ^c for the conjugate of a partition λ.
	For a partition κ = (κ_1, …, κ_s-t)∈(s-t,r-s), let κ̂ := (r-s-κ_s-t, …, r-s-κ_1)^c ∈(r-s,s-t).
	We have f̂ = w(s,t)f̂ and
	D(r,s,s-t) = ∑_κ∈(s-t, r-s)(-1)^|κ̂| 1⊗ s_κ̂⊗ 1 ⊗ 1 ⊗ 1 ⊗ s_κ⊗ 1 ⊗ 1 ∈Λ(r,s,t)
	by <cit.>, so that
	
    ĝf̂   = ∑_κ∈(s-t, r-s)(-1)^|κ̂|∂_x(r,s,t)(1⊗ (s_κ̂s_λ)⊗ s_μ⊗ 1 ⊗ 1 ⊗ s_κ⊗ 1 ⊗ 1) 
       = ∑_κ∈(s-t, r-s)(-1)^|κ̂|1⊗ ((s_κ̂s_λ)⋆ s_μ) ⊗ 1 ⊗ 1 ⊗ (s_κ⋆ 1) ⊗ 1 
       = 1⊗ (s_λ⋆ s_μ) ⊗ 1 ⊗ 1 ⊗ (s_(r-s)^(s-t)⋆ 1) ⊗ 1 
       = 1⊗ (s_λ⋆ s_μ) ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1.


We considerΛ(r,s)as a rightΛ(r,r)module via the natural algebra embeddingΛ(r,r)=Λ_b-r,r,a-rΛ_b-r,r-s,s,a-r=q^-ω_r-s(r,s)Λ(r,s).Iff∈Λ_r-s, we writef^r,s := 1_Λ_b-r⊗ f ⊗ 1_Λ_r⊗ 1_Λ_a-r∈Λ(r,s).By Proposition <ref>,Λ(r,s)is a free rightΛ(r,r)-module with basis{ s_λ^r,s|λ∈(r-s,s)}.We also makeΛ(r, s)into a leftΛ(s, s)-module via the composition of algebra homomorphisms:

    Λ(s,s)=Λ_b-s,s,a-sΛ_b-r,r-s,s,r-s,a-r=Λ^r,s_r,s⟶ q^-ω_r-s(r,s)Λ(r,s),

the first map being the natural embedding.
(This is similar to the definition of the rightΛ_σ(s)-module structure onΛ(r,s)used in the previous subsection; in factΛ(s,s)can be  identified with_R_θ(Δ(s))=Λ_σ(s)^, see the proof of Theorem <ref>.)

If0 ≤ t ≤ s ≤ r ≤min{ a, b}, then the tensor productΛ(s,t)⊗_Λ(s,s)Λ(r,s)is now a free rightΛ(r,r)-module with basis

    { s_μ^s,t⊗ s_λ^r,s|μ∈(s-t,t), λ∈(r-s,s)}

and we define a map of rightΛ(r,r)-modulesΘ: Λ(s,t)⊗_Λ(s,s)Λ(r,s) →Λ(r,t),  s_μ^s,t⊗ s_λ^r,s↦ (s_μ⋆ s_λ)^r,t.Let♢ : Λ(s,t)⊗_Λ̨(r,s) →Λ(r,t),  g⊗ f ↦Θ(g⊗ f).Let 0 ≤ t ≤ s ≤ r ≤min{ a, b}.
	The composition in the category _θ is given by
	[row sep = -0.5em]
		_θ(s,t) ⊗_θ(r,s) [r]    _θ(r,t) 
90≅   90≅
Λ(s,t) ⊗Λ(r,s) [r]    Λ(r,t) 

		g ⊗ f [r, mapsto]     g♢ f.
	
	According to Lemmas <ref> and <ref>, the composition of g∈Λ(s,t) with f∈Λ(r,s) is given by _r,t(ĝf̂), where ĝ∈Λ̂(s,t) and f̂∈Λ̂(r,s) are such that _s,t(ĝ) = g and _r,s(f̂) = f.

	First suppose r = s.
	Since D(s,s,s-t) = 1 and x(s,s,t) = 1, (<ref>) shows that _s,t(ĝf̂) = gf ∈Λ(s,t).
	Thus, the composition map Λ(s,t)⊗Λ(s,s) →Λ(s,s) coincides with the right Λ(s,s)-module structure of Λ(s,t).

	Now suppose s = t.
	Since D(r,s,0) = 1 and w(s,s) = x(r,s,s) = 1, (<ref>) shows that ĝf̂ = ĝf̂, so that _r,s(ĝf̂) = φ(g)f ∈Λ(r,s) where φ is the composition in (<ref>).
	Thus, the composition map Λ(s,s)⊗Λ(r,s) →Λ(r,s) coincides with the left Λ(s,s)-module structure of Λ(r,s).

	Associativity in _θ implies that the composition map Λ(s,t) ⊗Λ(r,s) →Λ(r,t) is Λ(s,s)-balanced and Λ(r,r)-equivariant, and is therefore completely determined by the image of the basis elements in (<ref>).
	Lemma <ref> now completes the proof.


§ NON-FORMALITY OF THE A_∞ STRUCTURE


We provide an example to show that theA_∞-category structure on_θis, in general, non-formal.
First, we recall some basic definitions and an important theorem, see <cit.>.

A ($̨-linear) A_∞-category consists of a class of objects (), a -graded $̨-module^∙(ρ, σ)for every pair of objectsρ, σ∈(), and for eachn ∈_>0and objectsπ_0, …, π_n ∈(), a degree2-n$̨-linear map
m_n^π_0, …, π_n: ^∙(π_n-1,π_n) ⊗^∙(π_n-2, π_n-1) ⊗⋯⊗^∙(π_0, π_1) →^∙(π_0, π_n),
such that for each n ∈_>0, we have

    *_n∑_r+s+t=n(-1)^r+stm_r+1+t(1^⊗ r⊗ m_s ⊗ 1^⊗ t) = 0,

where m_n := ⊕_π_0, …, π_n ∈()m_n^π_0, …, π_n.
An A_∞-functorF from = (, M_1, M_2, …) to = (, m_1, m_2, …) consists of a function F: () →(), and for each n ∈_≥ 0 and objects π_0, …, π_n ∈(), a degree 1-n map
F_n^π_0, …, π_n: ^∙(π_n-1,π_n) ⊗^∙(π_n-2, π_n-1) ⊗⋯⊗^∙(π_0, π_1) →^∙(F(π_0), F(π_n)),
such that for each n ∈_>0, we have

    **_n∑_r+s+t = n(-1)^r+stF_r+1+t(1^⊗ r⊗ M_s ⊗ 1^⊗ t) = ∑_i_1+⋯+i_k = n(-1)^ m_k(F_i_1⊗⋯⊗ F_i_k),

where := ∑_j=1^k-1(k-j)(i_j-1) and F_n := ⊕_π_0, …, π_n ∈()F_n^π_0, …, π_n.
The compositionGF of two A_∞-functors F: → and G: → is given on objects by (FG)(π) = F(G(π)) for π∈() with
(FG)_n := ∑_i_1+⋯+i_k = n(-1)^ F_k(G_i_1⊗⋯ G_i_k),
where  is as above.
The identity functor 1_ on  is the A_∞-functor F:→ with F(π) = π for each π∈(), F_1^ρ,σ = 𝕀_^∙(ρ, σ) for each ρ,σ∈(), and F_n = 0 for n > 1.
An isomorphism of A_∞-categories is an A_∞-functor F: → such that there exists an A_∞-functor G: → such that GF = 1_ and FG = 1_.

When we speak of the homologyH of an A_∞-category = (, m_1, m_2, …), we mean the homology with respect to m_1.
It is a graded category with the same object class as .
In fact, we have:

 <cit.>
	Let (, m_1, m_2, …) be an A_∞ category and H its homology.
	If each morphism space in H is a free graded $̨-module, thenHcarries the structure of anA_∞-categoryH = (H, M_1, M_2, …)such that
	
  * M_1 = 0 and M_2 = [m_2],
		
  *  there exists an A_∞-functor F: H→ such that F(π) = π for each π∈() and [F_1] = 𝕀_H.
	
	Moreover, theA_∞-category structure onHsatisfying (<ref>) and (<ref>) is unique up to (non-unique) isomorphism ofA_∞-categories.


Such anA_∞-category structure onHis called a minimal model of.
AnA_∞-categoryis called formal if its minimal model can be chosen so thatM_n = 0forn≠ 2.
A graded categoryis intrinsically formal if everyA_∞-category = (, m_1, m_2, …)whose homology is isomorphic toas a graded category is formal.
For example, the graded category_θin the situation of either Theorems <ref> or <ref> is intrinsically formal because by homological degree consideration, there is no way to impose anA_∞-category structure on_θwithM_n ≠ 0for anyn ≠ 2.

Kadeishvili's original proof is constructive and yields an inductive algorithm for producing a minimal model in the special case whereis a differential-graded algebra, i.e.,m_n = 0forn > 2:

 <cit.>
	Let = (, m_1, m_2, …) be an A_∞-category with m_k = 0 for k > 2 and H its homology.
	The following algorithm produces an A_∞-category structure (H, M_1, M_2, …) and an A_∞-functor F:H→ which satisfies the conditions in Theorem <ref>.

  Step 1:
	Let M_1 = 0 and take F_1^μ,ν: _θ(μ,ν) →_θ(μ,ν) to be a cycle-choosing homomorphism of $̨-modules.
	Setn := 2.

	
  Step 2:
	Sincem_k = 0fork > 2, we may rewrite (<ref>) as
	
    m_1 F_n = F_1 M_n - U_n,

	whereU_n := m_2 ∑_i=1^n-1(-1)^i-1(F_i⊗ F_n-i) - ∑_s=2^n-1∑_t=0^n-2(-1)^n-s-t+stF_n-s+1(1^⊗ n-s-t⊗ M_s ⊗ 1^⊗ t).We will also use the restriction toH(π_n-1,π_n)⊗⋯⊗ H(π_0,π_1):U_n^π_0, …, π_n : H(π_n-1,π_n)⊗⋯⊗ H(π_0,π_1) →(π_0,π_n).One can check thatm_1 U_n = 0.
	Thus, sinceM_k, F_khave been defined fork < n, we takeM_nto be the (well-defined) homology class[U_n]ofU_n.

	
  Step 3:
	Note that[F_1 M_n - U_n] = [F_1 M_n] - [U_n] = M_n - M_n = 0, soF_1 M_n - U_nis a boundary, and chooseF_nsuch thatm_1 F_n = F_1 M_n - U_n.
	Incrementnand return to Step 2.


Letθ∈ Q_+.
For eachπ∈(θ), fix a projective resolutionP_∙^πofΔ(π).
Consider the differential-graded category_θwhose objects are the Kostant partitions ofθwith morphism spaces_θ^∙(ρ, σ) := _R_θ^∙(P_∙^ρ, P_∙^σ).
We denote by

    m_1^ρ,σ   : _θ^∙(ρ,σ) →_θ^∙(ρ,σ) 
    
    	m_2^ρ,σ,τ   : _θ^∙(σ,τ) ⊗_θ^∙(ρ,σ) →_θ^∙(ρ,τ)

the differential and composition in_θ, respectively.
Note thatm_1^ρ,σis preciselyδfrom (<ref>).
Being a differential-graded category,_θis also anA_∞-category (see <cit.>), so its homology_θcarries a structure of anA_∞-category according to Theorem <ref>.

For the rest of this section, we letθ := α_1 + 2α_2 + α_3 ∈ Q_+and set

    π   := (α_2, α_1+α_2+α_3)    ρ   := (α_2+α_3, α_1+α_2) 
    σ   := (α_3, α_2, α_1+α_2)    τ   := (α_3, (α_2)^2, α_1).

Note that there is one other Kostant partition,(α_2+α_3, α_2, α_1), which will not play a role in our construction.
Recall the standard generators (<ref>) of the standard modules and the idempotents (<ref>) in the nil-Hecke algebra.
We define

    v̂_τ   := v_α_3∘ v_α_2∘ v_α_2∘ v_α_1∈Δ̂(τ), 
    
    	e_τ   := e_1⊗ e_2 ⊗ e_1 ∈_1 ⊗_2 ⊗_1,

so thatv_τ = v̂_τ e_τ.
We list the resolutions of the corresponding standard modules from <cit.> and Lemma <ref> below:[column sep=2em, row sep=1em]
	0 [r]     [-1em] q^2 R_θ 1_2321[r,"d_1^π"]     q R_θ 1_2231⊕ q R_θ 1_2132[r,"d_0^π"]     R_θ 1_2123[r,"ϵ_π"]    Δ(π) [r]     [-1em] 0, 

	0 [r]     q^2 R_θ 1_3221[r,"d_1^ρ"]     q R_θ 1_2321⊕ q R_θ 1_3212[r,"d_0^ρ"]     R_θ 1_2312[r,"ϵ_ρ"]    Δ(ρ) [r]     0, 
    0 [r]     q R_θ 1_3221[r," d_0^σ"]     R_θ 1_3212[r,"ϵ_σ"]    Δ(σ) [r]     0, 
       0 [r]     q^-1 R_θ 1_32^(2)1[r,"ϵ_τ"]    Δ(τ) [r]     0,where a matrix label stands for right multiplication with that matrix, and

    d_1^π   :=
    	[
    		-ψ_2 1_2231   ψ_3ψ_2 1_2132],    
    	d_0^π   :=
    	[ψ_3ψ_2 1_2123
    ψ_3 1_2123],    ϵ_π   :=
    	[
    		v_π], 
    
    	d_1^ρ   :=
    	[
    		-ψ_1 1_2321   ψ_3 1_3212],    
    	d_0^ρ   :=
    	[ψ_3 1_2312
    ψ_1 1_2312],    ϵ_ρ   :=
    	[
    		v_ρ], 
          
    	d_0^σ   :=
    	[ψ_3 1_3212],    ϵ_σ   :=
    	[
    		v_σ], 
                ϵ_τ   :=
    	[ψ_2 v_τ] =
    	[v̂_ττ_2
    	].


One can easily check, using (<ref>), that the complexesT_∙^π(ρ),T_∙^ρ(σ),T_∙^σ(τ), andT_∙^π(τ)are, respectively, the top complexes in the four diagrams below.
Moreover, the diagrams define isomorphisms of complexes.[column sep=2em, ampersand replacement=&]
	& [-1em] 0 & [-0.5em] q^-11_2132Δ(ρ) [l] & [3em] 1_2123Δ(ρ) [l,"[(d_0^π)_2,1]" above] & [-1em] 0 [l] 
& 0 & q^-1_2 [l] [u,"f ↦ψ_2 v_ρ f" right,"90∼" left] & q_2 [l,"[x_2-x_1]" above] [u,"f ↦ψ_3 ψ_2 v_ρ f" right,"90∼" left] & 0 [l], 
& 0 & q^-11_3212Δ(σ) [l] & 1_2312Δ(σ) [l,"[(d_0^ρ)_2,1]" above] & 0 [l] 
& 0 & q^-1_3 [l] [u,"f ↦ v_σ f" right,"90∼" left] & q_3 [l,"[x_2-x_1]" above] [u,"f ↦ψ_1 v_σ f" right,"90∼" left] & 0 [l], 
& 0 & 1_3221Δ(τ) [l] & q 1_3212Δ(τ) [l,"[(d_0^σ)_1,1]" above] & 0 [l] 
& 0 & q^-2_4 [l] [u,"f ↦v̂_τ f τ_2" right,"90∼" left] &_4 [l,"[x_4-x_3]" above] [u,"f ↦ψ_3 v̂_τ f τ_2" right,"90∼" left] & 0 [l], 

	0 & q^-11_2321Δ(τ) [l] & 1_2231Δ(τ) ⊕ 1_2132Δ(τ) [l,"d_1^π" above] & q 1_2123Δ(τ) [l,"d_0^π" above] & 0 [l] 

	0 & q^-2_4 [l] [u,"f↦ψ_1v̂_τ f τ_2" right,"90∼" left] &_4 ⊕_4 [l,"[ -(x_3-x_1)     x_4-x_3 ]" below=0.5em] [u,"(f,g) ↦ (ψ_2ψ_1 v̂_τ f τ_2, ψ_2 ψ_1 ψ_3 v̂_τ g τ_2)" right,"90∼" left] & q^2 _4 [l,"[ x_4-x_3 
 x_3-x_1 ]"] [u,"f ↦ψ_3 ψ_2 ψ_3 ψ_1 v̂_τ f τ_2" right,"90∼" left] & 0 [l]Thus, denoting_k := [̨z_1, …, z_k], we have

    _θ(π,ρ)    = _θ^1(π,ρ) ≅ q^-1_2 / (x_1 = x_2)  q^-1_1,  x̅_1 ↦ z_1 
    _θ(ρ,σ)    = _θ^1(ρ,σ) ≅ q^-1_3 / (x_1 = x_2)  q^-1_2,  x̅_1 ↦ z_1, x̅_3 ↦ z_2 
    _θ(σ,τ)    = _θ^1(σ,τ) ≅ q^-2_4 / (x_3 = x_4)  q^-2_3,  x̅_1 ↦ z_1, x̅_2 ↦ z_2, x̅_3 ↦ z_3 
    _θ(π,τ)    = _θ^2(π,τ) ≅ q^-2_4 / (x_1 = x_3 = x_4)  q^-2_2,  x̅_1 ↦ z_1, x̅_2 ↦ z_2.

	There is an A_∞-category structure _θ = (_θ, M_1, M_2, …) satisfying the conditions in Theorem <ref> such that
	[row sep = -0.5em]
		_θ^1(σ,τ) ⊗_θ^1(ρ,σ) ⊗_θ^1(π,ρ) [r,"M_3^π,ρ,σ,τ"]    _θ^2(π,τ) 
90≅   90≅

		q^-2_3 ⊗ q^-1_2 ⊗ q^-1_1 [r]     q^-2_2 

		z_1^a z_2^b z_3^c ⊗ z_1^m z_2^n ⊗ z_1^w [r,mapsto]     z_1^c+m+w z_2^b+nz_2^a - z_1^az_2-z_1.
	
	Moreover, there is no A_∞-category structure on _θ satisfying the conditions in Theorem <ref> with M_3 = 0.

	We apply Algorithm <ref>.
	We will need to examine the complexes T_∙^π(σ) and T_∙^ρ(τ).
	They are, respectively, the top complexes in the two diagrams below, and the diagrams define isomorphisms of complexes.
	[column sep=2em, ampersand replacement=&]
		& [-1em] 0 & q^-11_2132Δ(σ) [l] & [2em] 1_2123Δ(σ) [l,"[(d_0^π)_2,1]" above] & [-1em] 0 [l] 
& 0 &_3 [l] [u,"f ↦ψ_2 ψ_1 v_σ f" right,"90∼" left] & q^2 _3 [l,"[x_3-x_1]" above] [u,"f ↦ψ_3 ψ_2 ψ_1 v_σ f" right,"90∼" left] & 0 [l] 

		0 & q^-11_3221Δ(τ) [l] & 1_2321Δ(τ) ⊕ 1_3212Δ(τ) [l,"d_1^ρ" above] & q 1_2312Δ(τ) [l,"d_0^ρ" above] & 0 [l] 

		0 & q^-3_4 [l] [u,"f ↦v̂_τ f τ_2" right,"90∼" left] & q^-1_4 ⊕ q^-1_4 [l,"[ -(x_2-x_1)     x_4-x_3 ]" below=0.5em] [u,"(f,g) ↦ (ψ_1 v̂_τ f τ_2, ψ_3 v̂_τ g τ_2)" right,"90∼" left] & q_4 [l,"[ x_4-x_3 
 x_2-x_1 ]"] [u,"f ↦ψ_1 ψ_3 v̂_τ f τ_2" right,"90∼" left] & 0 [l]
	
	so that
	
    _θ(π,σ)    = _θ^1(π,σ) ≅_3 / (x_1=x_3) _2,  x̅_1 ↦ z_1, x̅_2 ↦ z_2 
    _θ(ρ,τ)    = _θ^2(ρ,τ) ≅ q^-3_4/(x_1=x_2,x_3=x_4)  q^-3_2, x̅_1 ↦ z_1, x̅_3 ↦ z_2.


	Instead of F_1, it will only be relevant to determine the restrictions F_1^π,ρ, F_1^ρ,σ, F_1^σ,τ, and F_1^ρ,τ.
	According to the algorithm, we are free to take for these any cycle-choosing homomorphisms.
	We define
	   F_1^π,ρ(z_1^a) :=    [column sep=2em, row sep=3em, ampersand replacement=&]
			0 [r] & q^2 R_θ 1_2321[r,"d_1^π"] [dr,"[ y_4^a 1_2321    0 ]" right=1.2em] & q R_θ 1_2231⊕ q R_θ 1_2132[r,"d_0^π"] [dr,"[ 0 
 -ψ_2 y_3^a 1_2312]" right=2em] & R_θ 1_2123[r] & 0, 

			0 [r] & q^2 R_θ 1_3221[r,"d_1^ρ" below] & q R_θ 1_2321⊕ q R_θ 1_3212[r,"d_0^ρ" below] & R_θ 1_2312[r] & 0,
		
   F_1^ρ,σ(z_1^a z_2^b) :=    [column sep=2em, row sep=3em, ampersand replacement=&]
			0 [r] & q^2 R_θ 1_3221[r,"d_1^ρ"] [dr,"[ y_2^a y_4^b 1_3221]" right=1.2em] & q R_θ 1_2321⊕ q R_θ 1_3212[r,"d_0^ρ"] [dr,"[ 0 
 -y_2^a y_3^b 1_3212]" right=2em] & R_θ 1_2312[r] & 0, 
& 0 [r] & q R_θ 1_3221[r,"d_0^σ" below] & R_θ 1_3212[r] & 0,
		
   F_1^σ,τ(z_1^a z_2^b z_3^c) :=    [column sep=2em, row sep=3em, ampersand replacement=&]
			0 [r] & q R_θ 1_3221[r,"d_0^σ"] [dr,"[ -y_1^a y_2^b y_4^c 1_32^(2)1]" right=1em] & R_θ 1_3212[r] & 0, 
& 0 [r] & q^-1 R_θ 1_32^(2)1[r] & 0,
		
   F_1^ρ,τ(z_1^a z_2^b) :=    [column sep=2em, row sep=3em, ampersand replacement=&]
			0 [r] & [-1em] q^2 R_θ 1_3221[r,"d_1^ρ"] [drr,"[ -y_2^a y_4^b 1_32^(2)1]" right=3em] & q R_θ 1_2321⊕ q R_θ 1_3212[r,"d_0^ρ"] & [-1em] R_θ 1_2312[r] & [-1em] 0, 
&& 0 [r] & q^-1 R_θ 1_32^(2)1[r] & 0.
		

	Using the above, we now show that the following choices are in accordance with the algorithm:
	
    M_2^π,ρ,σ = 0,    F_2^π,ρ,σ = 0,    M_2^π,ρ,τ = 0,    F_2^π,ρ,τ = 0

	By our choices of F_1^π,ρ and F_1^ρ,σ, we have U_2^π,ρ,σ = 0, so M_2^π,ρ,σ = 0, and according to (<ref>), we may take F_2^π,ρ,σ = 0.
	Since _θ(π,ρ) is concentrated in homological degree 1 and _θ(ρ,τ) is concentrated in homological degree 2, the image of U_2^π,ρ,τ = m_2^π,ρ,τ(F_1^ρ,τ⊗ F_1^π,ρ) is in _θ^3(π,τ), which is zero since P_∙^π has length 2.
	Thus M_2^π,ρ,τ = 0 and according to (<ref>), we may take F_2^π,ρ,τ = 0.

	We now have
	
    U_3^π,ρ,σ,τ   = m_2^π,σ,τ(F_1^σ,τ⊗ F_2^π,ρ,σ) - m_2^π,ρ,τ(F_2^ρ,σ,τ⊗ F_1^π,ρ) + F_2^π,σ,τ(1^σ,τ⊗ M_2^π,ρ,σ) 
                - F_2^π,ρ,τ(M_2^ρ,σ,τ⊗ 1^π,ρ) 
       = -m_2^π,ρ,τ(F_2^ρ,σ,τ⊗ F_1^π,ρ),

	so we only need to make a choice for F_2^ρ,σ,τ.
	We have U_2^ρ,σ,τ = m_2^ρ,σ,τ(F_1^σ,τ⊗ F_1^ρ,σ) so that U_2^ρ,σ,τ(z_1^a z_2^b z_3^c ⊗ z_1^m z_2^n) is given by
	[column sep=2em, row sep=3em, ampersand replacement=&]
		0 [r] & q^2 R_θ 1_3221[r,"d_1^ρ"] [drr,"[ -y_1^a y_2^b+m y_4^c+n 1_32^(2)1]" right=3em] & q R_θ 1_2321⊕ q R_θ 1_3212[r,"d_0^ρ"] & R_θ 1_2312[r] & 0 
&& 0 [r] & q^-1 R_θ 1_32^(2)1[r] & 0.
	
	Thus
	
    M_2^ρ,σ,τ(z_1^a z_2^b z_3^c ⊗ z_1^m z_2^n) = z_1^a+b+mz_2^c+n,

	and (F_1^ρ,τM_2^ρ,σ,τ - U_2^ρ,σ,τ)(z_1^a z_2^b z_3^c ⊗ z_1^m z_2^n) is given by the above diagram, except the diagonal arrow is right multiplication with
	[
		-y_2^b+m y_4^c+n (y_2^a - y_1^a) 1_32^(2)1].
	We may now take F_2^ρ,σ,τ(z_1^a z_2^b z_3^c ⊗ z_1^m z_2^n) to be:
	[column sep=2em, row sep=4em, ampersand replacement=&]
		0 [r] & q^2 R_θ 1_3221[r,"d_1^ρ"] & q R_θ 1_2321⊕ q R_θ 1_3212[r,"d_0^ρ"] [dr,"[ψ_1 y_2^b+my_4^c+ny_2^a-y_1^a/y_2-y_1 1_32^(2)1
 0 ]" right=2em] & R_θ 1_2312[r] & 0 
&& 0 [r] & q^-1 R_θ 1_32^(2)1[r] & 0.
	

	Now U_3^π,ρ,σ,τ(z_1^a z_2^b z_3^c ⊗ z_1^m z_2^n ⊗ z_1^w) is given by:
	[column sep=2em, row sep=4em, ampersand replacement=&]
		0 [r] & q^2 R_θ 1_2321[r,"d_1^π"] [drr,"[ -ψ_1 y_2^b+my_4^c+n+wy_2^a-y_1^a/y_2-y_1 1_32^(2)1]" right=2em] & q R_θ 1_2231⊕ q R_θ 1_2132[r,"d_0^π"] & R_θ 1_2123[r] & 0, 
&& 0 [r] & q^-1 R_θ 1_32^(2)1[r] & 0,
	
	so that M_3^π,ρ,σ,τ = [U_3^π,ρ,σ,τ] is as in the theorem statement.

	For the second assertion, note that the existence of a second A_∞-category structure _θ = (_θ, N_1, N_2, …) satisfying the conditions of Theorem <ref> implies the existence of an isomorphism of A_∞-categories G: (_θ, M_1, M_2, …) → (_θ, N_1, N_2, …) with G_1 being the identity on each morphism space.
	Assume, toward a contradiction, that such an isomorphism exists, and that N_3 = 0.

	Recall that we take M_1 = N_1 = 0, so (**_2) applied to G reads G_1 M_2 = N_2(G_1⊗ G_1), and since G_1 is the identity, we have M_2 = N_2.
	Now since N_3 = 0 by assumption, (**_3) reads
	
    G_2(M_2⊗ 1 - 1⊗ M_2) + M_3 = M_2(1⊗ G_2 - G_2⊗ 1).

	The restriction of (<ref>) to _θ(σ,τ)⊗_θ(ρ,σ)⊗_θ(π,ρ) is
	
    G_2^π,ρ,τ(M_2^ρ,σ,τ⊗ 1^π,ρ) - G_2^π,σ,τ(1^σ,τ⊗ M_2^π,ρ,σ) + M_3^π,ρ,σ,τ
                = M_2^π,σ,τ(1^σ,τ⊗ G_2^π,ρ,σ) - M_2^π,ρ,τ(G_2^ρ,σ,τ⊗ 1^π,ρ),

	so according to (<ref>), (<ref>) becomes
	
    G_2^π,ρ,τ(M_2^ρ,σ,τ⊗ 1^π,ρ) + M_3^π,ρ,σ,τ = M_2^π,σ,τ(1^σ,τ⊗ G_2^π,ρ,σ).


	We have a formula for M_2^ρ,σ,τ given by (<ref>).
	We define
	F_1^π,σ(z_1^m z_2^n) :=
	[column sep=2em, row sep=3em, ampersand replacement=&]
		0 [r] & q^2 R_θ 1_2321[r,"d_1^π"] [dr,"[ψ_1 y_2^n y_4^m 1_3221]" right=1.2em]
		& q R_θ 1_2231⊕ q R_θ 1_2132[r,"d_0^π"] [dr,"[ 0 
 -ψ_2 ψ_1 y_2^n y_3^m 1_3212]" right=2em]
		& R_θ 1_2123[r] & 0, 
& 0 [r] & q R_θ 1_3221[r,"d_0^σ" below] & R_θ 1_3212[r] & 0,
	
	By our choices of F_1^π,σ and F_1^σ,τ, we have that U_2^π,σ,τ(z_1^a z_2^b z_3^c ⊗ z_1^m z_2^n) is given by
	[column sep=2em, row sep=3em, ampersand replacement=&]
		0 [r] & q^2 R_θ 1_2321[r,"d_1^π"] [drr,"[ -ψ_1 y_1^a y_2^b+ny_4^c+m 1_32^(2)1]" right=2em] & q R_θ 1_2231⊕ q R_θ 1_2132[r,"d_0^π"] & R_θ 1_2123[r] & 0, 
&& 0 [r] & q^-1 R_θ 1_32^(2)1[r] & 0
	
	so that M_2^π,σ,τ(z_1^a z_2^b z_3^c ⊗ z_1^m z_2^n) = z_1^a+c+m z_2^b+n.

	Denote the left- and right-hand sides of (<ref>) by L and R, respectively.
	We apply L and R to two elements: z_1⊗ 1 ⊗ 1 and z_2⊗ 1 ⊗ 1.
	Note that the map
	G_2^π,ρ,σ : q^-1_2⊗ q^-1_1 ≅_θ^1(ρ,σ) ⊗_θ^1(π,ρ) →_θ^1(π,σ) ≅_2,
	is a KLR degree 0 map, so we must have G_2^π,ρ,σ(1⊗ 1) = 0.
	Thus, R(z_1⊗ 1 ⊗ 1) = R(z_2⊗ 1 ⊗ 1) = 0.
	On the other hand, we have M_3^π,ρ,σ,τ(z_1⊗ 1⊗ 1) = 1, M_3^π,ρ,σ,τ(z_2⊗ 1⊗ 1) = 0, and M_2^ρ,σ,τ(z_1⊗ 1) = M_2^ρ,σ,τ(z_2⊗ 1) = z_1, so by (<ref>), we have
	
    0    = R(z_1 ⊗ 1 ⊗ 1) = L(z_1 ⊗ 1 ⊗ 1) = G_2(z_1⊗ 1) + 1, 
    
    		0    = R(z_2 ⊗ 1 ⊗ 1) = L(z_2⊗ 1 ⊗ 1) = G_2(z_1⊗ 1),

	a clear contradiction.
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