The center of the twisted Heisenberg category, factorial Schur $Q$-functions, and transition functions on the Schur graph
Henry Kvinge Can Ozan Oğuz Michael Reeks
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The center of the twisted Heisenberg category]The center of the twisted Heisenberg category, factorial Schur Q-functions, and transition functions on the Schur graph









[
    Henry Kvinge, Can Ozan Oğuz, and Michael Reeks
    December 30, 2023
==================================================




We establish an isomorphism between the center of the twisted Heisenberg category and the subalgebra Γ of the symmetric functions generated by odd power sums. We give a graphical description of the factorial Schur Q-functions and inhomogeneous power sums as closed diagrams in the twisted Heisenberg category, and show that the bubble generators of the center correspond to two sets of generators of Γ which encode data related to up/down transition functions on the Schur graph. Finally, we describe an action of the trace of the twisted Heisenberg category, the W-algebra W^-⊂ W_1+∞, on Γ.







§ INTRODUCTION


In <cit.>, Khovanov describes a linear monoidal category ℋ which conjecturally categorifies the Heisenberg algebra. The morphisms of ℋ are governed by a graphical calculus of planar diagrams. This category has connections to many interesting areas of representation theory and combinatorics. The trace of ℋ, which can be defined diagrammatically as the algebra of diagrams on the annulus, was shown in <cit.> to be isomorphic to the W-algebra W_1+∞ at level one. 
 The center of ℋ, which is the algebra _ℋ() of endomorphisms of the monoidal identity, was shown in <cit.> to be isomorphic to the algebra of shifted symmetric functions Λ^* of Okounkov and Olshanski <cit.>. 

A twisted version of Khovanov's Heisenberg category was introduced by Cautis and Sussan in  <cit.>. The twisted Heisenberg category  is a ℂ-linear additive monoidal category, with an additional ℤ/2ℤ-grading. It conjecturally categorifies the twisted Heisenberg algebra. The center of , , was studied in <cit.> where it was shown that as a commutative C-algebra, 

    ≅C[d_0,d_2,d_4, … ] ≅C[d̅_2,d̅_4,d̅_6, …],

where d_2k and d̅_2k are certain clockwise and counterclockwise bubble generators respectively. While symmetric groups play a central role for ℋ in <cit.>, finite Sergeev superalgebras {n}_n ≥ 0 (also known as finite Hecke–Clifford algebras of type A) play the central role for . In particular, Cautis and Sussan construct a family of functors {n}_n ≥ 0 from  to bimodule categories of Sergeev algebras in order to categorify the Fock space representation of the twisted Heisenberg algebra. When restricted to , each functor gives a surjective algebra homomorphism n: ↠n where n is the even center of n. 

In this paper, we study the combinatorial and representation theoretic properties of . Our main result, Theorem <ref>, establishes an isomorphism 
: ∼⟶, 
where  is a subalgebra of the algebra of symmetric functions = C[p_1,p_3,p_5, …] ( is sometimes known as the algebra of supersymmetric <cit.> or doubly symmetric <cit.> functions). The construction of  relies on the fact that there are embeddings of both  and  into the algebra of functions on strict partitions, . In our proof of Theorem <ref> we identify the images of certain algebraically independent generators of these algebras in :  the closures of n-cycles from  and inhomogeneous analogues of odd power sums _n in . The latter were first investigated by Ivanov in his study of the asymptotic behavior of characters of projective representations of symmetric groups <cit.>. We go on to identify the closure of idempotents of n with scalar multiples of Ivanov's factorial Schur Q-functions. Intriguingly, the coefficients that appear on the image of idempotent closures when written in terms of factorial Schur Q-functions count the number of paths between specific vertices in the graph of all strict partitions (also known as the Schur graph). A similar phenomenon was observed in <cit.>. A dictionary between  and  is found in Table <ref>.

In parallel to the surjective homomorphisms {n}_n ≥ 0 from  to {n}_n ≥ 0, for all n ≥ 0 one can also construct surjective homomorphisms n: ↠n <cit.>. Our isomorphism  is canonical in the sense that it intertwines the pair n and n for each n ≥ 0.




at (-3,0) ;

at (3,0) ;

at (0,-2) n;

[<->] (-1.9,0) –(2.5,0);
[->>] (-2.1,-.3) – (-.5,-1.7);
[->>] (2.5,-.3) – (.5,-1.7);

at (0,.3) ;
at (-1.8,-1.2) n;
at (1.8,-1.2) n;





One interesting feature of the center of the non-twisted Heisenberg category ℋ is that, as shifted symmetric functions, the curl generators are best understood in terms of moments of Kerov's transition and co-transition measures on Young diagrams; fundamental tools used to answer probabilistic questions related to the asymptotic representation theory of symmetric groups <cit.>. In this paper we show that this connection to asymptotic representation theory extends to the twisted Heisenberg category. Specifically, we identify the clockwise bubble generators {d_2k}_k ≥ 0 and counterclockwise bubble generators {d̅_2k}_k ≥ 1 with two sets of algebraically independent generators for  discovered by Petrov <cit.>, {k}_k ≥ 0 and {k}_k ≥ 0 respectively. The functions {k}_k ≥ 0 (respectively {k}_k ≥ 0) encode down (resp. up) Markov transition kernels on the Schur graph. In particular, the difference between up and down transition functions manifests itself graphically in  as a difference in orientation of diagrams. This seems to be yet another indication of the “planar nature” of structures arising from noncommutative probability theory.

There is a natural action of the trace of a category on its center, which can be diagrammatically defined as gluing annular diagrams (elements of the trace) around planar ones (elements of the center). In the case of Khovanov's Heisenberg category, the results of <cit.> and <cit.> give rise to an action of  on the algebra of shifted symmetric functions. This representation of  was described in terms of symmetric group representation theory by Lascoux and Thibon in <cit.>. The trace of  was shown in <cit.> to be isomorphic to a classical-type subalgebra of  discovered by Kac, Wang, and Yan called  <cit.>. This result along with Theorem <ref> gives a representation of  on . In this paper we describe this representation which is a twisted version of the representation described in <cit.>. 

The paper is structured as follows. In Section <ref> we describe necessary background material on Schur's graph and the representation theory of Sergeev algebras. In Section <ref> we describe the subalgebra  of the symmetric functions and several of its bases. In Sections <ref> we recall the definition of the twisted Heisenberg category ℋ_tw and review the functors {n}_n ≥ 0. In Section <ref> we first establish the isomorphism between  and , and then describe the W-algebra  and its induced action on .

Acknowledgements: The authors would like to thank Weiqiang Wang and Nick Ercolani for many helpful discussions. The second author was partially supported by grants DMS-1664240 and DMS-1255334.





§ THE SCHUR GRAPH AND SERGEEV ALGEBRAS
 
  


 §.§ Transition functions on the Schur graph

Let n be the set of all partitions of n and set

    := ⋃_n ≥ 0n.

We freely identify a partition ρ with its corresponding Young diagram. If ρ∈n then we write |ρ| = n. If ρ = (ρ_1,ρ_2,…,ρ_r) and η = (η_1, η_2, …, η_t) ∈ then we write η⊂ρ when η_i ≤ρ_i for all i ≥ 1. We denote the number of parts (or length) of a partition ρ by ρ. A partition μ = (μ_1, …, μ_r) ∈n is called an odd partition if μ_i is odd for all 1 ≤ i ≤ r. We denote the collection of odd partitions of n by [n] and set := ⋃_n ≥ 0[n]. 

We call a partition λ∈n strict if all its nonzero parts are distinct. Let [n] be the set of all strict partitions of n and set := ⋃_n ≥ 0[n]. To a strict partition λ we can associate its shifted Young diagram λ which is obtained from the Young diagram (using English notation) by shifting all rows so that the ith row is shifted rightward by (i-1) cells. 


Let λ = (6,5,2,1) ∈[14], then 








at (-1,1.25) λ   =;


(2.5,1.5) rectangle (3,2);
(2,1.5) rectangle (2.5,2);
(1.5,1.5) rectangle (2,2);
(1,1.5) rectangle (1.5,2);
(.5,1.5) rectangle (1,2);
(0,1.5) rectangle (.5,2);

(2,1) rectangle (2.5,1.5);
(1.5,1) rectangle (2,1.5);
(1,1) rectangle (1.5,1.5);
(.5,1) rectangle (1,1.5);
(0,1) rectangle (.5,1.5);

(.5,.5) rectangle (1,1);
(0,.5) rectangle (.5,1);

(0,0) rectangle (.5,.5);

at (3.1,.2) ,;

at (6,1.25) S(λ)    =;


(9.5,1.5) rectangle (10,2);
(9,1.5) rectangle (9.5,2);
(8.5,1.5) rectangle (9,2);
(8,1.5) rectangle (8.5,2);
(7.5,1.5) rectangle (8,2);
(7,1.5) rectangle (7.5,2);

(9.5,1) rectangle (10,1.5);
(9,1) rectangle (9.5,1.5);
(8.5,1) rectangle (9,1.5);
(8,1) rectangle (8.5,1.5);
(7.5,1) rectangle (8,1.5);

(8.5,.5) rectangle (9,1);
(8,.5) rectangle (8.5,1);

(8.5,0) rectangle (9,.5);

at (10.1,.2) .;









Henceforth we reserve the variables λ and ν for strict partitions and the variables μ and γ for odd partitions.

For ν, λ∈, we write ν↗λ (respectively ν↘λ) when we can obtain λ from ν by adding (resp. removing)  a single cell . Set 

    νλ := 
    2    if ν↗λ, λ = ν,
    
    1    if ν↗λ and λ = ν + 1,
    
    0    otherwise.



The Schur graph  is the graded graph such that:



  * the vertex set of  corresponds to  and the nth graded component is [n],

  * the number of edges from ν to λ is given by νλ.




The version of  that we consider here is the same as that studied in <cit.>. Another version of the Schur graph without edge multiplicity was investigated in <cit.>. Both graphs have the same down transition functions (see (<ref>) below) so in principle we could have chosen to use either.

A standard shifted Young tableau of shape λ∈[n] is a bijective labeling of the cells of λ by the integers {1, …, n} such that entries increase from left to right across rows and down columns. Let λ be the number of standard shifted Young tableaux of shape λ. We can compute λ explicitly as

    λ = n!/λ_1!λ_2! …λ_r!∏_1 ≤ i < j ≤λλ_i - λ_j/λ_i + λ_j.

Following <cit.> we denote the number of paths from ∅ to λ in  by λ. Then 

    λ = 2^|λ| - λλ.


In <cit.> Borodin and Olshanski used coherent families of measures on partitions to construct infinite-dimensional diffusion processes. Petrov studied analogous processes on the Schur graph <cit.>. We review some basic definitions related to the latter of these works below. 

The down transition function p^↓: ×→Q  on  is defined so that for ν,λ∈, 

    λν := ν/λνλ.

In particular, when the first argument of p^↓ is restricted to [n] and the second to [n-1] the function p^↓ gives a Markov transition kernel from _n to _n-1. 

A coherent system on  with respect to down transition function p^↓ is a collection of probability measures {M_n}_n ≥ 0, with M_n a probability measure on _n, such that if ν∈[n-1], then

    M_n-1(ν) = ∑_λ↘νλν M_n(λ).

By abuse of notation we write M_n(λ) for M_n({λ}). One choice of coherent system with respect to the p^↓ defined by (<ref>) is the collection of Plancherel measures {n}_n ≥ 0 where for λ∈[n]

    n(λ) := 2^λ-nλ^2/n!.


Given the coherent system {n}_n ≥ 0 and the down transition function p^↓, the corresponding up transition function p^↑: ×→Q is defined as

    p^↑(ν,λ) := n+1(λ)/n(ν)λν = λ/ν(|ν| + 1)

(in <cit.> this is denoted by p^↑_∞).

In the next section we will make a connection between induction and restriction of simple Sergeev supermodules and p^↑, p^↓.


  
  
  

 §.§ The Sergeev algebra and the twisted hyperoctahedral group
 
  
Let n be the symmetric group on n elements with s_1, s_2, …, s_n-1 the Coxeter generators of n. We will work with the 
Clifford algebra  which is the unital associative algebra with n generators: 

    :=ℂ⟨  c_1,…,c_n  |   c_i^2=-1, c_ic_j=-c_jc_i  for  i≠ j ⟩.



There is another common presentation of  in which  is generated by c_1', …, c_n' subject to the relations c_i'^2 = 1 and c_i'c_j' = -c_j'c_i' for i ≠ j (see <cit.> and <cit.> for example). An equivalence between this presentation and (<ref>) can be obtained by setting c_i = √(-1)c_i'. 



The finite Sergeev algebra (also known as the finite Hecke-Clifford algebra of type A) is

    ≃⋊ℂ[S_n]

where the action of S_n on the Clifford generators is by permuting indices, i.e. 

    s_ic_i = c_i+1s_i,    s_ic_i+1 = c_is_i,   and   s_ic_j = c_js_i    for j ≠ i, i+1.


 
There is a Z/2Z-grading making n into a superalgebra in which the n generators {s_i, 1≤ i ≤ n-1} are even and the  generators {c_j, 1≤ j ≤ n} are odd. For homogeneous element x ∈n we write |x| for the degree of x. The Sergeev algebras form a tower of superalgebras via the embedding n-1↪n which sends s_i ↦ s_i for 1 ≤ i ≤ n-2 and c_i ↦ c_i for 1≤ i ≤ n-1. We call this the standard embedding and use it implicitly throughout this paper. We set 0 = C.

It will be convenient to realize  as the quotient of a group algebra. Let C_2 denote the cyclic group of order two. Define the group

    n :=⟨ z, a_1,…,a_n  |  a_i^2=z, a_ia_j=za_ja_i, za_i = a_iz, z^2 = 1⟩.

The group n is a double cover of C_2^n via the short exact sequence:

    1 ⟶ C_2 ⟶n⟶ C_2^n ⟶ 1

in which C_2 is mapped to the subgroup {1,z}⊂n and z ∈n is mapped to 1.


The twisted hyperoctahedral group is defined as B_n := n⋊n where n acts on {a_i} by permuting their indices, and acts trivially on z. 


The algebra C[B_n] is also a superalgebra via the Z/2Z grading which sets (a_j) = 1 for 1≤ j≤ n and (z) = (s_i) = 0 for 1 ≤ i ≤ n-1. Using (<ref>) one can show that n is a double cover of the hyperoctahedral group n = C_2^n⋊n (i.e. the type B Weyl group) via the short exact sequence:


    1 ⟶ C_2 ⟶nf⟶n⟶ 1

where f sends z to 1. On the other hand from a comparison of generators and relations it is clear that

    ≃ℂ[B_n]/⟨ z +1⟩.

We denote the corresponding projection by n: C[B_n] →n.

Since z is central and z^2 = 1, for any C[n]-supermodule L, we have that z must act by multiplication by either 1 or -1. Hence studying n-supermodules is equivalent to studying C[n]-supermodules in which z acts as multiplication by -1 (these are commonly referred to as spin representations of n). Furthermore, via the super Wedderburn Theorem it follows that 

    C[n]   ≅  C[n] /⟨ z-1 ⟩⊕C[n]/⟨ z+1 ⟩  ≅  C[n] ⊕n.


The group algebras C[n] also form a tower of algebras with the embedding C[n-1] ↪C[n] which sends s_i ↦ s_i, a_i ↦ a_i, and z ↦ z. Note that this maps the subalgebra C[n-1] into the subalgebra C[n] in the usual way, and projected down to n-1 and n this becomes the standard embedding described above. We set ℂ[0] to be the subalgebra generated by z.

 
For n ≥ 2, 

    {  s_i … s_n-1a_n^ϵ  |   1 ≤ i ≤ n, ϵ∈{0,1}}

is a collection of left coset representatives of n-1 in n.


Note that we follow the convention that the elements corresponding to i = n are a_n^ϵ for ϵ∈{0,1} in Lemma <ref>.


The set {  s_i … s_n-1  |   1 ≤ i ≤ n  } forms a collection of minimal length left coset representatives of n-1 in n. It follows from this and the fact that n := n⋊n that any element g ∈n can be written as g = s_i … s_n-1ω a_n^ϵ a_J z^β where 1 ≤ i ≤ n, ω∈n-1, a_J = a_j_1… a_j_t for some J = {j_1, …, j_t}⊆{1,2, …, n-1}, and ϵ, β∈{0,1}. Since a_n commutes with n-1 we have x = s_i … s_n-1 a_n^ϵω  a_J z^β. Since ω  a_J z^β∈n-1, the set (<ref>) contains a set of left coset representatives. The result then follows from the observation that the size of (<ref>) is 2n while |n| = 2^n+1n! and |n-1| = 2^n(n-1)!.



If g = s_i … s_n-1a_n^ϵ for ϵ∈{0,1} then g^-1 = a_n^ϵ z^ϵ s_n-1… s_i and consequently while n(g) = s_i … s_n-1c_n^ϵ, we have

    n(g^-1) = (-1)^ϵ c_n^ϵ s_n-1… s_i = (-1)^|g| c_n^ϵ s_n-1… s_i.



We use the inclusions 1⊂2⊂…⊂n-1⊂n⊂… to iterate Lemma <ref> to get that for all 1 ≤ k < n, 

    nk := {  (s_i_n… s_n-1 a_n^ϵ_n)( s_i_n-1… s_n-2 a_n-1^ϵ_n-1) … (s_i_k+1… s_k a_k+1^ϵ_k+1)}


    |   1 ≤ i_j ≤ j,  ϵ_j ∈{0,1} }

is a collection of left coset representatives of k in n. Note in particular that

    |nk| = |n|/|k|=n^↓ k2^n-k

where n^↓ k is the falling factorial

    n^↓ k :=n!/(n-k)! = n(n-1)…(n-k+1)

for 1 ≤ k < n. The projection n: C[n] →n sends the elements of nk to distinct non-zero elements of n and we set

    nk := n(nk).


The set of conjugacy classes of n is indexed by pairs of partitions (ρ_+,ρ_-) such that |ρ_+| + |ρ_-| = n, plus an additional parameter ϵ∈{0,1} when either (ρ_+,ρ_-) = (μ, ∅) with μ∈[n] or (ρ_+,ρ_-) = (∅,λ) with λ∈[n]. We denote this indexing set by . A detailed description of the conjugacy class structure of n can be obtained by analyzing the conjugacy class structure of n (which follows from the basic theory for the conjugacy class structure of wreath products <cit.>) and investigating how the inverse image of these sets under the map γ: n↠n split into new conjugacy classes <cit.>. The additional parameter ϵ∈{0,1} appears precisely when a conjugacy class in n splits into two conjugacy classes in n. Since we are ultimately interested in the center of n which can be described using information about the conjugacy classes indexed by (μ, ∅, ϵ) for μ∈[n] and ϵ∈{0,1}, we limit ourselves to considering these classes. We call this set of conjugacy classes ⊂. For β∈, we write (β) for the corresponding conjugacy class.

We introduce a family of elements of n which will be useful for constructing representatives for the conjugacy classes from . For  μ = (μ_1,…,μ_r) ∈[k], set π_μ = 1 if μ = (1^k) and otherwise

    π_μ := (s_k-1… s_k-μ_r+1)…(s_μ_1+μ_2-1… s_μ_1 + 1) (s_μ_1 -1… s_2s_1)


    = (k,k-1,…,k-μ_r+1)…(μ_1+μ_2,…,μ_1+1) (μ_1,…,2,1) ∈k.

For n ≥ k we define μn := τ_0 π_μτ_0^-1, where τ_0 is the longest element of n by Coxeter length. Notice that μn has cycle type (μ,1^n-k) ∈[n] and fixes 1,2,…,n-k pointwise. 
















 
The elements { μn,  zμn  |  μ∈[n]} form a complete set of conjugacy class representatives for the conjugacy classes  in n with μn corresponding to (μ,∅,0) ∈ and zμn corresponding to (μ,∅,1) ∈.



This follows from the description of the conjugacy classes of n and results on conjugacy class splitting in n <cit.> (see <cit.> for an overview).


Note that under the projection map n: C[n] →n, the two sets of conjugacy classes {μn}_μ∈[n] and {zμn}_μ∈[n] are identified since n(z)=-1.

The size of the conjugacy classes (μ,∅,ϵ) will be important to us later. For ρ∈n, we denote by z_ρ the size of the stabilizer of an element of n of cycle type ρ under the conjugation action. Recall that 

    z_ρ = ∏_i ∈Z_≥ 0 i^iρiρ!

where iρ is the number of parts of size i in ρ. 

 <cit.> 
For μ∈[n], ϵ∈{0,1}

    |(μ,∅,ϵ)| = n!/z_μ 2^n - μ.



n has analogues to the classical Jucys-Murphy elements of C[S_n]. These elements {i}^n_i =1, which we also call Jucys-Murphy elements, are defined by

    1 := 0,     k := ∑_j =1^k-1 (1+c_jc_k)(j,k).

They generate a commutative subalgebra of n and their spectra have a combinatorial interpretation analogous to that of the classical Jucys-Murphy elements (<cit.>, <cit.>, <cit.> <cit.>).





 §.§ The super representation theory of n and B_n


In this subsection we will review basic facts about the super representation theory of n. Recall that any n-supermodule is by definition a spin representation of n, so all statements about n-supermodules also hold for n spin representations. We refer the reader to <cit.> and <cit.> for thorough accounts of these topics as well as a review of super representation theory. 

Let : →{0,1} be defined by

    (λ) :=  0    λ is even
     1    λ is odd.

The function  will be useful for describing quantities related to the representation theory of n.

 
  <cit.> 
  The set of simple n-supermodules are indexed by [n], and
for any λ∈[n], the simple n-supermodule indexed by λ has dimension

    (λ) = 2^n-λ-(λ)/2λ.




The algebras {n}_n ≥ 0 are semisimple. When N and M are n-supermodules we write

    [M : N] := (_n(M,N)).


The theorem below describes the branching for {n}_n ≥ 0.

 <cit.> 
Let λ∈[n] and ν∈[n-1], then:


  * 
[ν:^n_n-1λ] = 
2^2 + ν - (ν) - λ + (λ)/2   λ↘ν

0    otherwise.

  * 
[^n_n-1ν:λ] = 
2^2 + ν - (ν) - λ + (λ)/2   ν↗λ, 

0    otherwise.



The next theorem describes how n as a left n-supermodule decomposes into a direct sum of simple n-supermodules.

 
n as a left n-supermodule decomposes as

    n≅⊕_λ∈[n] (λ)^⊕(λ)/2^(λ).




For λ∈[n], the multiplicity of λ in n is equal to the multiplicity of the corresponding spin representation λ of n in the left regular super representation. If λ is of type 𝖬 (i.e. (λ) = 0), then λ is simple as an ungraded n-module <cit.>.
When λ is of type 𝖰 (i.e. (λ) = 1) then considered as an ungraded C[n]-module, λ splits into a direct sum of two simple C[n]-modules, each with dimension (λ)/2. The result then follows from the ungraded representation theory of n. 


We can now relate ·· and ·· to the representation theory of the algebras {n}_n ≥ 0.

 
Let λ∈[n] and ν∈[n-1]. Then


  * 
λν = [ν:^n_n-1λ](ν)/(λ),

  * 
2^(λ) - (ν)νλ = [^n_n-1ν:λ](λ)/(^n_n-1ν).




Both <ref> and <ref> follow from the branching rules for Sergeev algebras in Theorem <ref> and the dimension formula in Theorem <ref>.


For λ∈[n], we denote by λ the character corresponding to simple C[n]-supermodule λ. This descends to a character λ for simple n-supermodule λ with

    λ(n(g)) = λ(g).

The normalized character λ is defined such that for x ∈n

    λ(x) := λ(x)/(λ) = λ(x)/λ(1).


 
For λ∈[n], the character λ is uniquely determined by its value on the elements {μn  |  μ∈[n]}.



This follows from a similar statement <cit.> where each element μn is replaced by an element of n that is conjugate to it. Since characters are constant across conjugacy classes, the result follows.


Given Proposition <ref>, for μ∈[k] with k ≤ n, it is convenient to write λ(μ∪ 1^n-k) := λ(μn).
 
 



 §.§ The centers of n and C[n]


As a superalgebra the center of n breaks up into even and odd components of super-commutative elements. In this paper we will focus on n, which corresponds to the ungraded center of n. Note that n is exactly those elements that act on all simple n-modules as multiplication by a scalar. Following <cit.> we will construct a basis for n via the surjection π:C[n] ↠n.

Recall that the set  indexes the conjugacy classes of n. For β∈ set

    C_β := ∑_g ∈(β) g.

It is clear that {C_β}_β∈ is a basis for the ungraded center of C[n]. In <cit.>, Ivanov uses the subset of this basis corresponding to elements of  of the form (μ,∅,0) to construct a basis for n. For μ∈[n] let

    C_μ := n(C_(μ,∅,0)).


 <cit.> 
The set {C_μ  |  μ∈[n]} is a linear basis for n.


We now define a scaled version of Ivanov's basis of n which naturally appears from the center of the twisted Heisenberg category.

 
For k ≤ n and  μ∈[k], define

    μn := ∑_g ∈nn-k gμn g^-1

and 

    μn := π(μn).




 
Let k≤ n and μ∈[k] then:
 

  * μn∈ Z(C[n]) and μn∈n.

  *  
μn = 2^k-n+μz_μ∪ 1^n-k/(n-k)!C_(μ∪ 1^n-k,∅,0).


  *  Let h ∈n be an element not belonging to the same conjugacy class as μn or zμn for some μ∈[n] (i.e. h does not belong to a conjugacy class indexed by (μ,∅,ϵ), ϵ∈{0,1}). Then

    n(∑_g ∈n g h g^-1) = 0.







  * Recall that we defined μn as a distinguished element from the conjugacy class of n indexed by (μ∪1^n-k, ∅, 0). Since μn is by definition a product of s_n-1,…,s_n-k+1 it commutes with n-k. Since nn-k is a collection of left coset representatives of n-k in n any element g ∈n can be written uniquely as g = σ h for σ∈nn-k and h ∈n-k. Thus gμng^-1 = σ h μn h^-1σ^-1 =  σμnσ^-1 and hence gμng^-1 is completely determined by the left coset to which g belongs. It follows that 

    ∑_g ∈n g μ̃n g^-1 = |n-k|∑_g ∈nn-k gμn g^-1 = |n-k|μn

and μn∈ Z(C[n]) since μn is a scalar multiple of a central element. 
  
 Finally, note that n is a degree-preserving homomorphism and μn is even, so π(μn)=μn∈n.


  * It follows from Lemma <ref> and the orbit stabilizer theorem that

    ∑_g ∈n gμn g^-1 = 2^μ + 1z_μ∪ 1^n-kC_(μ∪ 1^n-k,∅,0).

Then (<ref>) implies that

    |n-k|μn = 2^μ + 1z_μ∪ 1^n-kC_(μ∪ 1^n-k,∅,0).

The result follows.


  * It follows from the proofs of  <cit.> and <cit.> that the image in n, n(h), of any such element is conjugate to its negative. Hence each term in the image of the conjugacy class sum of any such h appears in a pair with its negative. The result follows. 



It follows from Proposition <ref>.<ref> that {μn  |  μ∈[n]} is also a linear basis of n. 

For a spin representation  λ of n, the corresponding character λ is a homomorphism when restricted to Z(C[n])_0 and λ and λ are homomorphisms on n. 

 
Let λ∈[n] and μ∈[k]. Then 

    λ(μn) = 2^kn^↓ kλ(μn)/λ(1).




This follows from the fact that characters are invariant under conjugation and (<ref>).


Another basis for n is given by the set of central idempotents of n corresponding to the simple n-supermodules. We denote these central idempotents by {λ  |  λ∈[n]}. 

 
For λ∈[n], the central idempotent λ∈n corresponding to the simple n-supermodule λ can be written as

    λ = 2^-λ-(λ)/2λ/n!∑_μ∈[n]λ(μ)C_μ.




The definition of n implies that λ is the image of the corresponding central idempotent λ in n under the projection map n. If λ is of type 𝖬, then λ is simple as an ungraded C[n]-module; if λ is of type 𝖰, then λ viewed as an ungraded C[n]-module breaks into the direct sum of two simples of equal dimension, λ  = L^λ_0⊕L^λ_1. Thus the central idempotent corresponding to λ is given by

    λ = (λ)/2^δ(λ)|n|∑_β∈λ(g)C_β,

where, if λ is of type 𝖰, then λ(g) = λ_0(g) + λ_1(g).

Applying n to λ, Proposition <ref>.<ref> implies that most terms go to zero and we are left with

    λ =  2^- λ-(λ)/2-1λ/n!∑_β∈λ(β)n(C_β).

Recall that β∈ contains pairs β = (μ,∅,0) and β̅ = (μ, ∅,1) for μ∈[n] such that if x ∈(β) then zx ∈(β̅). It follows that n(C_β̅) = -n(C_β). At the same time, since z acts as multiplication by -1 on λ then λ(zμn) = -λ(μn). It follows that 

    2^- λ-(λ)/2-1λ/n!∑_β∈λ(β)n(C_β) = 2^-λ-(λ)/2λ/n!∑_μ∈[n]λ(μ)C_μ.









 §.§ Interlacing coordinates for strict partitions


In <cit.> Kerov developed a useful way to parametrize Young diagrams via their interlacing coordinates. Petrov showed that shifted strict Young diagrams can be similarly parametrized with only slight modification <cit.>. For λ∈, and ∈λ with coordinates (i,j), the content of  is defined to be 

    := i - j.

Note that when  comes from a shifted diagram,  is always nonnegative. Let


  * λ be the set of contents for cells that we can add to λ to get another shifted strict partition. 

  * λ be the set of contents for cells that we can remove from λ to get another shifted strict partition. 

The set (λ, λ) uniquely characterizes S(λ) and is called the Kerov coordinates of S(λ). We follow <cit.> and denote the shifted diagram obtained by adding a cell  with content x ∈λ to S(λ) by S(λ) + (x) and the shifted diagram obtained by removing a cell  from S(λ) with content y ∈λ by S(λ) - (y).                

                                   
In the case of λ = (6,5,2,1) ∈[14], λ = {0,4}, λ = {2,6}, and








at (6,1.25) S(λ)    =;


(9.5,1.5) rectangle (10,2);
(9,1.5) rectangle (9.5,2);
(8.5,1.5) rectangle (9,2);
(8,1.5) rectangle (8.5,2);
(7.5,1.5) rectangle (8,2);
(7,1.5) rectangle (7.5,2);

(9.5,1) rectangle (10,1.5);
(9,1) rectangle (9.5,1.5);
(8.5,1) rectangle (9,1.5);
(8,1) rectangle (8.5,1.5);
(7.5,1) rectangle (8,1.5);

(8.5,.5) rectangle (9,1);
(8,.5) rectangle (8.5,1);

(8.5,0) rectangle (9,.5);



at (8.75,.25) 0;
at (9.75,1.25) 4;



at (9.25,.75) 2;
at (10.25,1.75) 6;

at (10.1,.2) .;


































As in <cit.>, we define λ := λ∖{0}. It follows from <cit.> that for all λ∈, |λ| = |λ|.
 

We set

    i := i(i+1).






The following proposition connects λ, · and λ· with the interlacing coordinates for λ.

 <cit.> 
Let λ∈[n]. Then we have
 

  *  
    ∑_x ∈λλλ + (x)/z-x = ∏_y ∈λ (z - y)/ z ∏_x ∈λ (z - x),


  *  
    1 - ∑_y ∈λ2|λ|λλ - (y)/z - y = ∏_x ∈λ(z - x)/∏_y ∈λ(z - y).




The left sides of (<ref>) and (<ref>) in Proposition <ref> can be rewritten as

    ∑_x ∈λλλ + (x)/z-x = ∑_k = 0^∞∑_x ∈λλλ + (x)x^k z^-k-1,

and

    1 - ∑_y ∈λλλ - (y)/z - y = 1 - 2|λ|∑_k = 0^∞∑_y ∈λλλ - (y)y^k z^-k-1.

Petrov defined functions on  corresponding to the coefficients on the right side of (<ref>) and (<ref>) 
 



Petrov defined two families of functions k: →Q and k: →Q, such that

    k(λ) := ∑_x ∈λλλ + (x)x^k

and

    k+1(λ) := 2|λ|∑_y ∈λλλ - (y)y^k

and investigated their properties in <cit.>. Note that 0 = 1.


k(λ) and k(λ) are the strict partition analog to moments of Kerov's transition and co-transition measure <cit.>. They will play a similar role to the one they played in <cit.>.


 <cit.> 
For λ∈[n],

    k = k +  ∑_i,j > 0, 
    i + j = kij.



We now give algebraic interpretations of k(λ) and k(λ) analogous to those found by Biane for Kerov's transition and co-transition measure on Young diagrams <cit.>. Let n-1: n→n-1 be the linear map defined such that for x ∈n

    n-1(x) :=  
    x    if x ∈n-1
    
    0    otherwise.


 
Let λ∈[n] for n ≥ 1 and k ≥ 0, then


  * λ(n(n+1^2k)) =  k(λ).

  * λ( ∑_x ∈nn-1 xn^rx^-1) = k+1(λ)    if r = 2k is even

0    otherwise.







  * Consider the character _n: n→C corresponding to n acting on itself by left multiplication. For x ∈n,

    _n(x) := 
    2^nn!    if x =1
    
    0    otherwise.

It follows from (<ref>) that 

    2(n+1)_n(n(x)) = _n+1(x).

Also note that if y ∈n and x ∈n+1 then

    n(yx) = yn(x).

Recall that e_λ is the central idempotent of n corresponding to simple n-supermodule λ. Then by Lemma <ref> there are 2^-(λ)(λ) copies of λ in the n-supermodule n so that

    2(n+1)_n(n(e_λn+1^2k)) = 2(n+1)_n(e_λn(n+1^2k)) 
    = 2^1-(λ)(n+1)(λ)λ(n(n+1^2k)).

On the other hand, the weight space decomposition for the Jucys-Murphy operators on n-supermodules implies that

    2(n+1)_n(n(e_λn+1^2k)) = _n+1(e_λn^2k)


    = ∑_x ∈λ[λ:^n+1_nλ+(x)](λ)(λ+(x))/2^(λ+(x)) s(x)^k.

Thus, taking the normalized character gives

    λ(n-1(n^2k)) 
       = ∑_x ∈λ 2^(λ+(x)) - (λ)[λ:^n+1_nλ+(x)](λ + (x)) /2(n+1)(λ)s(x)^k.

As (^n+1_nλ) = 2(n+1)(λ) and by Frobenious reciprocity

    [λ:^n+1_nλ+(x)] = [^n+1_nλ:λ+(x)],

applying Lemma <ref>.<ref> gives the desired result.


  * The elements c_i and the Jucys-Murphy elements i satisfy ic_i = -c_ii and for x = s_i … s_n-1c_n^ϵ we have x^-1 = (-1)^ϵc_n s_n-1… s_i. Therefore

    ∑_x ∈nn-1 xn^rx^-1   = ∑_i = 1^n s_i … s_n-1n^rs_n-1… s_i   -    s_i … s_n-1c_nn^rc_ns_n-1… s_i 
    
       = ∑_i = 1^n s_i … s_n-1n^rs_n-1… s_i   -    (-1)^r+1s_i … s_n-1n^rs_n-1… s_i.

When r is odd, this is then equal to zero. When r = 2k, (<ref>) is equal to

    2∑_i = 1^n s_i … s_n-1n^2ks_n-1… s_i.

Since characters are invariant under conjugation, we have

    λ(2∑_i = 1^n s_i … s_n-1n^2ks_n-1… s_i) = 2nλ(n^2k).

Decomposing n into its weight spaces then gives

    2nλ(n^2k) 
       = 2n∑_y ∈λ[λ - (y):^n_n-1λ](λ-(y))y^k/(λ)
       = 2n∑_y ∈λλλ - (y)y^k

where the last equality uses Lemma <ref>.<ref>



 



§ THE SUBALGEBRA 
 

We recall relevant facts about the algebra  following <cit.>. Let p_k be the kth power sum symmetric function, 



and recall that, for ρ∈,

    p_ρ := ∏_k=1^ℓ(ρ) p_ρ_k.











 Γ can be described as the subalgebra of the symmetric functions generated by the odd power sums

    Γ = C[p_1,p_3,p_5,…].


Elements of Γ can be evaluated on partitions in the following way. Let f ∈ and ρ∈, and define 

    f(ρ) := f(ρ_1, ρ_2, …, ρ_ℓ(ρ), 0, …).

Let  denote the algebra of functions from  to C with pointwise multiplication. 

 <cit.>) 
The algebra  embeds into  via the evaluation map (<ref>).










 An important linear basis of  is the Schur Q-functions {Q_λ}, indexed by  (cf. <cit.>). 
 
 

 
 Define numbers X_μ^λ for λ∈[n], μ∈[n], via 
 
    p_μ = ∑_λ∈[n] 2^-λX_μ^λ Q_λ.

 
 There is a “factorial” version of the Schur Q-functions, defined in <cit.>. For λ∈, the factorial Schur Q-polynomial corresponding to λ is defined as:
 
    Q^*_λ|N(x_1,…, x_N) := 2^λ/(N-l)!∑_ω∈ S_Nω(x_1^↓λ_1 x_2^↓λ_2… x_l^↓λ_l∏_1≤ i ≤ l 
     i < j ≤ Nx_i+x_j/x_i-x_j).

 If ℓ(λ)>N, then Q^*_λ|N is defined to be 0. The collection (Q_λ|N^*)_N=1,2,… defines an element of Γ, the factorial Schur Q-function Q_λ^*. Factorial Schur Q-functions have the following useful properties.
 
 <cit.> Let λ, ν∈.  
  
 
  * There exists g ∈Γ of degree less than |λ| such that 
 
    Q_λ^* = Q_λ+g.

 
  * The collection {Q_λ^*}_λ∈ is a linear basis of Γ. 
 
  * If ν∈[k], λ∈[n] for k ≤ n and ν⊈λ, Q_λ^*(ν) = 0.  
 
 
 
Let ψ: → be the linear map that sends Q_λ↦ Q_λ^*. For any μ∈, define the inhomogeneous analogue of the power sum _μ := ψ(p_μ) ∈Γ. Applying ψ to both sides of (<ref>) gives 

    _μ = ∑_λ∈[k] 2^-λX_μ^λ Q_λ^*.

It also follows from the fact that X_μ^λ = 2^-μ + λ -(λ)/2λ(μ) <cit.> and 

    Q_λ = ∑_μ∈[n]2^μ/z_μ X^λ_μ p_μ

that

    Q_λ^* = 2^λ-(λ)/2∑_μ∈[n]λ(μ)/z_μ_μ.

The elements {_μ}_μ∈ were first studied in <cit.>, where Ivanov proves that they satisfy the following properties.
 
 <cit.>  
 Let μ∈[k] and λ∈[n]. 
 
  * There exists g ∈Γ of degree less than |μ| such that 
    
    _μ = p_μ + g.

     
  * The family (_μ)_μ∈ is a linear basis of Γ.
     
  * 
     
    _μ(λ) = {[ n^↓ k·X_μ∪ (1^n-k)^λ/g_λ            if  |λ|≥ |μ|,;                        0                otherwise ].

     where in particular g_λ = X_1^|λ|^λ. 
     
  * Let γ∈. Define μ∪γ to be the partition formed by taking the disjoint union of parts of μ and γ and rearranging them in decreasing order. Then there exists g ∈ of degree less than |μ∪γ| such that 
    _μ·_γ = _μ∪γ + g.
 
 
 
 
 As a Corollary to part <ref> of the above Proposition, we have another formula for the value of _ρ. 
 
  <cit.> 
 Let μ∈[k] and λ∈[n]. We have 
 
    _μ(λ) = 2^k-ℓ(μ) n^↓ kχ^λ (μ∪ 1^n-k)/χ^λ(1^n).

 
 
   
The elements {_2k+1}_k ≥ 0 are algebraically independent and generate .
  

It is shown in <cit.> that viewed as elements of , {k}_k ≥ 1 and {k}_k ≥ 1 belong to .

 <cit.>
The elements {k}_k ≥ 1 and {k}_k ≥ 1 are each sets of algebraically independent generators of ,



 and

    (k) = (k) = 2k -1.


 
 
 
 

§ THE TWISTED HEISENBERG CATEGORY
 
 
 

 §.§ The definition of 

The twisted Heisenberg category  was introduced by Cautis and Sussan in <cit.>. It is a ℤ/2ℤ-graded additive monoidal category whose morphisms are described diagrammatically as oriented compact 1-manifolds immersed in R× [0,1]. There is an injective algebra homomorphism from the twisted Heisenberg algebra into the split Grothendieck group of K_0(Kar()), where Kar denotes the Karoubi envelope (idempotent completion). As in the untwisted case, this map is conjecturally surjective.

Cautis and Sussan define their version of  to be idempotent complete; since the center of a category remains invariant under passage to the idempotent completion, we use the non-idempotent complete version here. All results that hold for the center of  also hold for the center of its idempotent completion. 
   


The objects of  are monoidally generated by P and Q, so that a generic object in  is a direct sum of sequences of P's and Q's. We denote the empty sequence, which is the unit object of , by . The morphisms of ℋ_tw are generated by oriented planar diagrams up to boundary fixing isotopies,
 with generators 

    [thick,->](-2,-0.25) to (-2,0.75);
     (-1.99,0.25) circle [radius=2pt];
     at (-1.6,-.25) ,;
    [thick,<-](-1,-0.25) to (-1,0.75);
     (-1,0.25) circle [radius=2pt];
     at (-.5,-.25) ,;
    [thick,->] (0,-0.25) to (1,0.75);
    [thick,->] (1,-0.25) to (0,0.75);
     at (1.25,-.25) ,;
    [thick,->] (1.5,0.5) arc (180:360:5mm);
     at (2.6,-.25) ,;
    [thick,<-] (3,0.5) arc (180:360:5mm);
     at (4.2,-.25) ,;
    [thick,->] (4.5,-.2) arc (180:0:5mm);
     at (5.8,-.25) ,;
    [thick,<-] (6.2,-.2) arc (180:0:5mm);

where the first diagram corresponds to a map P→ P{1} and the second diagram corresponds to a map Q→ Q{1}, where {1} denotes the ℤ/2ℤ-grading shift. These generators satisfy the following relations:


    [baseline=(current bounding box.center),scale=0.6]
    
    [thick,->] (0,0) .. controls (1,1) .. (0,2);
    [thick,->] (1,2) .. controls (0,1) .. (1,0);
     at (1.75,1) =;
    [thick,->] (2.5,0) – (2.5,2);
    [thick,->] (3.5,2) – (3.5,0);
     at (4,0) ,;
    
    [->,thick] (5.25,0) .. controls (6.25,1) .. (5.25,2);
    [->,thick] (6.25,0) .. controls (5.25,1) .. (6.25,2);
     at (7,1) =;
    [->,thick] (7.75,0) – (7.75,2);
    [->,thick] (8.75,0) – (8.75,2);
     at (9.25,0) ,;
    
    [->,thick] (10.25,0) – (12.25,2);
    [->,thick] (12.25,0) – (10.25,2);
    [->,thick] (11.25,0) .. controls (12.25,1) .. (11.25,2);
     at (13,1) =;
    [->,thick] (13.75,0) – (15.75,2);
    [->,thick] (15.75,0) – (13.75,2);
    [->,thick] (14.75,0) .. controls (13.75,1) .. (14.75,2);	
     at (16.1,0) ,;



    [baseline=(current bounding box.center),scale=0.6]]
    
    [thick,->] (6,2) .. controls (7,1) .. (6,0);
    [thick,->] (7,0) .. controls (6,1) .. (7,2);
     at (8,1) =;
    [thick,->] (9,2) – (9,0);
    [thick,->] (10,0) – (10,2);
     at (11,1) -;        	
    [thick,->] (12,2) arc (180:360:0.75);
    [thick,->] (13.5,0) arc (0:180:0.75);
     at (14.3,1) -;
    [thick,->] (15,2) arc (180:360:0.75);
    [thick,->] (16.5,0) arc (0:180:0.75);
     (15.12,1.6) circle [radius = 3pt];
     (16.4,.4) circle [radius = 3pt];
     at (17,0) ,;
    [baseline=(current bounding box.center),scale=0.6]
    [thick,->] (3,2) arc (-180:180:5mm);
    
    =1,
    [baseline=(current bounding box.center),scale=1.5]
    [thick](1,0) to [out=90, in=-75](0.95,0.5);
    [thick](0.95,0.5) arc (5:355:2mm);
    [thick,->] (0.95,0.46) to [out=75, in=270] (1,1);
    
    =0,

































    [baseline=(current bounding box.center)]
        [thick,<-] (3,2) arc (180:0:5mm);
         (3.1,2.3) circle [radius=1.8pt];
        
        =
        -[baseline=(current bounding box.center)]
        [thick,<-] (3,2) arc (180:0:5mm);
         (3.95,2.2) circle [radius=1.8pt];
         at (4.2,2) ,;
        [baseline=(current bounding box.center)]
        [thick,<-] (3,2) arc (180:360:5mm);
         (3.07,1.75) circle [radius=1.8pt];
        
        =
        -[baseline=(current bounding box.center)]
        [thick,<-] (3,2) arc (180:360:5mm);
         (3.95,1.8) circle [radius=1.8pt];



    [baseline=(current bounding box.center),scale=0.6]
        [thick,->] (0,0) to (0,2);
        
        =
        [baseline=(current bounding box.center),scale=0.6]
        [thick,->] (0,0) to (0,2);
         (0,0.6) circle [radius=3pt];
         (0,1.2) circle [radius=3pt];
         at (.5,0) ,;
        [baseline=(current bounding box.center),scale=0.6]
        [thick,<-] (0,0) to (0,2);
        
        =
        -[baseline=(current bounding box.center),scale=0.6]
        [thick,<-] (0,0) to (0,2);
         (0,0.6) circle [radius=3pt];
         (0,1.2) circle [radius=3pt];
         at (.5,0) ,;
        [baseline=(current bounding box.center),scale=0.6]
        [thick,->] (3,2) arc (-180:180:5mm);
         (3.95,2.2) circle [radius=3pt];
        
        =0,
    [baseline=(current bounding box.center),scale=0.6]
        [thick,->] (0,0) to (0,2);
         (0,1.6) circle [radius=3pt];
        [fill=] (0.2,1) circle [radius=0.3pt];
        [fill=] (0.4,1) circle [radius=0.3pt];
        [fill=] (0.6,1) circle [radius=0.3pt];
        [thick,->] (0.8,0) to (0.8,2);
         (0.8,0.3) circle [radius=3pt];
        
        =-[baseline=(current bounding box.center),scale=0.6]
        [thick,->] (0,0) to (0,2);
         (0,0.3) circle [radius=3pt];
        [fill=] (0.2,1) circle [radius=0.3pt];
        [fill=] (0.4,1) circle [radius=0.3pt];
        [fill=] (0.6,1) circle [radius=0.3pt];
        [thick,->] (0.8,0) to (0.8,2);
         (0.8,1.6) circle [radius=3pt];
         at (1,.2) .;


Generators commute in all other situations (for instance, hollow dots commute with crossings).

If we denote a right-twist curl by a dot   [baseline=(current bounding box.center),scale=1.5]
[thick] (1,0) to [out=90, in=85](1.05,0.5);
[thick] (1.05,0.5) arc (-175:175:2mm);
[thick,->] (1.05,0.46) to [out=95, in=270] (1,1);

    :=
    
    [baseline=(current bounding box.center),scale=0.6]
    [thick,->](0,0) to (0,2);
    [fill](0,1) circle[radius=3pt];
    
  then we have the following relations:


    [baseline=(current bounding box.center),scale=0.6]
        [thick,->](0,0) to (0,2);
        [fill](0,0.6) circle[radius=3pt];
        (0,1.2) circle[radius=3pt];
        
        =
        -[baseline=(current bounding box.center),scale=0.6]
        [thick,->](0,0) to (0,2);
        (0,0.6) circle[radius=3pt];
        [fill](0,1.2) circle[radius=3pt];
         at (.5,.15) ,;
        [baseline=(current bounding box.center),scale=0.6]
        [thick,->] (0,0) to (0,2);
        [fill] (0,1.6) circle [radius=3pt];
        [fill] (0.2,1) circle [radius=0.3pt];
        [fill] (0.4,1) circle [radius=0.3pt];
        [fill] (0.6,1) circle [radius=0.3pt];
        [thick,->] (0.8,0) to (0.8,2);
        [fill] (0.8,0.3) circle [radius=3pt];
        
        =
        [baseline=(current bounding box.center),scale=0.6]
        [thick,->] (0,0) to (0,2);
        [fill] (0,0.3) circle [radius=3pt];
        [fill] (0.2,1) circle [radius=0.3pt];
        [fill] (0.4,1) circle [radius=0.3pt];
        [fill] (0.6,1) circle [radius=0.3pt];
        [thick,->] (0.8,0) to (0.8,2);
        [fill] (0.8,1.6) circle [radius=3pt]; 
         at (1,.15) ,;  
        [baseline=(current bounding box.center),scale=0.6]
        [thick,->] (0,0) to (0,2);
         (0,1.6) circle [radius=3pt];
        [fill] (0.2,1) circle [radius=0.3pt];
        [fill] (0.4,1) circle [radius=0.3pt];
        [fill] (0.6,1) circle [radius=0.3pt];
        [thick,->] (0.8,0) to (0.8,2);
        [fill] (0.8,0.3) circle [radius=3pt];
        
        =
        [baseline=(current bounding box.center),scale=0.6]
        [thick,->] (0,0) to (0,2);
         (0,0.3) circle [radius=3pt];
        [fill] (0.2,1) circle [radius=0.3pt];
        [fill] (0.4,1) circle [radius=0.3pt];
        [fill] (0.6,1) circle [radius=0.3pt];
        [thick,->] (0.8,0) to (0.8,2);
        [fill] (0.8,1.6) circle [radius=3pt];
         at (1,.15) .;


From <cit.> we have the following “dot sliding” relations,


    [baseline=(current bounding box.center),scale=0.6]
        [thick,->](0,0) to (1,2);
        [thick,->](1,0) to (0,2);
        [fill](0.25,0.5) circle[radius=3pt];
    
    =
    [baseline=(current bounding box.center),scale=0.6]
        [thick,->](0,0) to (1,2);
        [thick,->](1,0) to (0,2);
        [fill](0.75,1.5) circle[radius=3pt];
    
    +[baseline=(current bounding box.center),scale=0.6]
        [thick,->](0,0) to (0,2);
        [thick,->](0.5,0) to (0.5,2);
    +[baseline=(current bounding box.center),scale=0.6]
        [thick,->](0,0) to (0,2);
        [thick,->](0.5,0) to (0.5,2);
        (0,1.2) circle[radius=3pt];
        (0.5,0.6) circle[radius=3pt];
         at (1,.2) ,;




    [baseline=(current bounding box.center),scale=0.6]
        [thick,->](0,0) to (1,2);
        [thick,->](1,0) to (0,2);
        [fill](0.25,1.5) circle[radius=3pt];
    
    =
    [baseline=(current bounding box.center),scale=0.6]
        [thick,->](0,0) to (1,2);
        [thick,->](1,0) to (0,2);
        [fill](0.75,0.5) circle[radius=3pt];
    
    +
    [baseline=(current bounding box.center),scale=0.6]
        [thick,->](0,0) to (0,2);
        [thick,->](0.5,0) to (0.5,2);
    
    -
    [baseline=(current bounding box.center),scale=0.6]
        [thick,->](0,0) to (0,2);
        [thick,->](0.5,0) to (0.5,2);
        (0,1.2) circle[radius=3pt];
        (0.5,0.6) circle[radius=3pt];
         at (1,.2) .;


We can also move clockwise “bubbles” with dots on them through strands.

 
Let n ≥ 0, then


[scale = .7]


[thick,->] (0,0) circle [radius = 12pt];
at (-.42,0) ;
[fill=black] (-.3,.27) circle [radius = 2pt]; 
at (-.7,.4) 2n;


[thick,->] (1,-1)–(1,1);


at (2,0) =;


[thick,->] (3,-1)–(3,1);


[thick,->] (4,0) circle [radius = 12pt];
at (3.58,0) ;
[fill=black] (3.7,.27) circle [radius = 2pt]; 
at (3.3,.4) 2n;


at (4.9,0) +;


at (5.9,0) (4n+2);
[thick,->] (6.7,-1)–(6.7,1);
[fill = black] (6.7,.4) circle [radius = 2pt];
at (7.1,.55) 2n;


at (7.7,0) -;


at (9.7,0) 2 ∑_a + b = 2n-1∑_k = 1^b;


[thick,->] (13,0) circle [radius = 12pt];
at (12.58,0) ;
[fill=black] (13.3,.27) circle [radius = 2pt]; 
[fill=black] (11.7,.4) circle [radius = 2pt];
at (13.4,.7) b-k;
at (12.25,.53) a+k;

[thick,->] (11.7,-1)–(11.7,1);

at (13.6,-.8) .;







This follows from the proof of <cit.> along with the dot sliding relation (<ref>).


As a consequence of relations (<ref>) and (<ref>), there are homomorphisms n: n^opp→_(P^n) which send




[scale = .6]


[->,thick] (-.3,0)–(1,0);
[thick] (-.3,.1)–(-.3,-.1);
at (.2,.3) n;
at (-1,0) s_k;


[thick,->] (2,-1) – (2,1);
[thick,->] (3.2,-1) – (3.2,1);
[thick,->] (3.7,-1) to [out =90,in = 270] (4.2,1);
[thick,->] (4.2,-1) to [out =90,in = 270] (3.7,1);
[thick,->] (4.7,-1) – (4.7,1);
[thick,->] (5.9,-1) – (5.9,1);
[decorate,decoration=brace,amplitude=3pt,xshift=-4pt,yshift=0pt]
(3.3,-1.2) – (2,-1.2) node [black,midway,xshift=0cm,yshift = -.3cm] 
 k-1 strands;
[decorate,decoration=brace,amplitude=3pt,xshift=-4pt,yshift=0pt]
(6.1,-1.2) – (4.9,-1.2) node [black,midway,xshift=0cm,yshift = -.3cm] 
 n-k-1 strands;


at (2.6,0) …;
at (5.3,0) …;


 

[scale = .6]
[->,thick] (-.3,0)–(1,0);
[thick] (-.3,.1)–(-.3,-.1);
at (.2,.3) n;
at (-1,0) c_k;


[thick,->] (2,-1) – (2,1);
[thick,->] (3.2,-1) – (3.2,1);
[thick,->] (3.95,-1) to [out =90,in = 270] (3.95,1);
(3.95,0) circle [radius=3.3pt];
[thick,->] (4.7,-1) – (4.7,1);
[thick,->] (5.9,-1) – (5.9,1);
at (6.2,-.6) .;
[decorate,decoration=brace,amplitude=3pt,xshift=-4pt,yshift=0pt]
(3.3,-1.2) – (2,-1.2) node [black,midway,xshift=0cm,yshift = -.3cm] 
 k-1 strands;
[decorate,decoration=brace,amplitude=3pt,xshift=-4pt,yshift=0pt]
(6.1,-1.2) – (4.9,-1.2) node [black,midway,xshift=0cm,yshift = -.3cm] 
 n-k strands;


at (2.6,0) …;
at (5.3,0) …;






In order to simplify our diagrams we write the image of x ∈n under n as




    [scale = .8]
    
     at (-4.4,0) n(x)    =:;
    
    [thick,->] (-1.7,-1)–(-1.7,1);
    [thick,->] (-.8,-1)–(-.8,1);
    [thick,->] (-2,-1)–(-2,1);
    
    
     at (-1.35,.6) ·;
     at (-1.2,.6) ·;
     at (-1.05,.6) ·;
    
    [fill=white,thick] (-2.3,-.3) rectangle (-.5,.3);
    
     at (-1.3,0) x;
    
     at (-.3,-.9) .;
    
     [decorate,decoration=brace,amplitude=3pt,xshift=-4pt,yshift=0pt]
    (-.6,-1.2) – (-2,-1.2) node [black,midway,xshift=0cm,yshift = -.3cm] 
    n strands;




 

 §.§ The center of 
 
 
The center of a -linear monoidal category C is defined to be the endomorphism algebra of the monoidal unit  of C, that is _C(). In a diagrammatic category such as ,  is then by definition the commutative algebra of closed diagrams where multiplication of two closed diagrams corresponds to placing them next to each other.

We define the following elements of :

    d_2n := 
    [baseline=(current bounding box.center)]
    [thick,<-] (3,2) arc (-180:180:5mm);
     (3.95,2.2) circle [radius=2pt];
    
     at (4.3,2.2) 2n;
    
        and    d̅_2n:=
    [baseline=(current bounding box.center)]
    [thick,->] (3,2) arc (-180:180:5mm);
     (3.95,2.2) circle [radius=2pt];
    
     at (4.3,2.2) 2n;
    
    .

It follows from the defining relations of ℋ_tw that bubbles with an even number of hollow dots are equivalent to bubbles with no hollow dots, and that bubbles with an odd number of hollow dots are zero. Additionally, bubbles with odd numbers of solid dots are zero. Hence it suffices to consider the bubbles with an even number of solid dots and no hollow dots.








<cit.> 
The elements {d_2n}_n ≥ 0 are algebraically independent generators of , i.e. there is an isomorphism

    ≅C[d_0,d_2,d_4,…].

 
 
 The elements {d_2n}_n ≥ 0 and {d̅_2n}_n ≥ 0 are related via a recursive relation.
 <cit.> 
For n ≥ 1,

    d̅_2n = ∑_2a+2b = 2n-2d̅_2ad_2b.




The elements {d̅_2n}_n ≥ 1 are another algebraically independent generating set of .

 
Another natural set of diagrams in  come from the closure of permutations. We define



[scale=0.6]

[thick] (-1.7,-1)–(-1.7,1);
[thick] (-.8,-1)–(-.8,1);

at (-1.7,.6) ;
at (-.8,.6) ;

at (-1.4,.7) ·;
at (-1.2,.7) ·;
at (-1.0,.7) ·;

[fill=white,thick] (-2,-.3) rectangle (-.5,.3);

at (-1.3,0) k;

at (.5,0) =;

[decorate,decoration=brace,amplitude=7pt,xshift=-4pt,yshift=0pt]
(1.1,1.2) – (2.6,1.2) node [black,midway,xshift=0cm,yshift = .5cm] 
k strands;

[thick] (2.3,-1)–(2.3,1);
[thick] (1.4,-1)–(1.4,1);
[thick] (2.6,-1) to [out =90,in = 270] (2.6,-.5) to [out = 90, in = 270] (1.1,.5) to [out = 90, in =270] (1.1,1);

at (2.3,.6) ;
at (1.4,.6) ;
at (1.1,.6) ;

at (2,.7) ·;
at (1.8,.7) ·;
at (1.6,.7) ·;

at (3,-1) .;




For ν = (ν_1,…, ν_r) ∈k, let


    [scale=0.6]
    
    [thick] (-1.7,-1)–(-1.7,1);
    [thick] (-.8,-1)–(-.8,1);
    
     at (-1.7,.6) ;
     at (-.8,.6) ;
    
     at (-1.4,.7) ·;
     at (-1.2,.7) ·;
     at (-1.0,.7) ·;
    
    [fill=white,thick] (-2,-.3) rectangle (-.5,.3);
    
     at (-1.3,0) ν;
    
     at (.5,0) :=;
    
    
    
    [thick] (2.3,-1)–(2.3,1);
    [thick] (1.4,-1)–(1.4,1);
    
    [fill=white,thick] (2.6,-.3) rectangle (1.1,.3);
    
     at (2.3,.6) ;
     at (1.4,.6) ;
    
    
     at (2,.7) ·;
     at (1.8,.7) ·;
     at (1.6,.7) ·;
    
     at (3.2,0) ·;
     at (3.4,0) ·;
     at (3.6,0) ·;
    
     at (4.7,.7) ·;
     at (4.9,.7) ·;
     at (5.1,.7) ·;
    
    [thick] (4.5,-1)–(4.5,1);
    [thick] (5.4,-1)–(5.4,1);
    
     at (4.5,.6) ;
     at (5.4,.6) ;
    
    [fill=white,thick] (4.2,-.3) rectangle (5.7,.3);
    
     at (5.05,0) ν_r;
     at (1.85,0) ν_1;

then we define



[scale = .8]

at (-4,0) α_ν   :=;

[thick] (0,0) circle (1.8 cm);
[thick] (0,0) circle (1.6 cm);
[thick] (0,0) circle (1 cm);

[fill=white,thick] (-2,-.3) rectangle (-.5,.3);

at (-1,.5) ·;
at (-1.15,.55) ·;
at (-1.3,.6) ·;

at (-1.3,0) ν;

[rotate = 180] at (1.8,0) ;
[rotate = 180] at (1.6,0) ;
[rotate = 180] at (1,0) ;







We write






at (-4,0) α_ρ   :=;

[thick] (0,0) circle (1.8 cm);
[thick] (0,0) circle (1.6 cm);
[thick] (0,0) circle (1 cm);

[fill=white,thick] (-2,-.3) rectangle (-.5,.3);

at (-1,.5) ·;
at (-1.15,.55) ·;
at (-1.3,.6) ·;

at (-1.3,0) σ_ρ;

[rotate = 180] at (1.8,0) ;
[rotate = 180] at (1.6,0) ;
[rotate = 180] at (1,0) ;

at (2,-1.2) .;






where ρ is a partition and σ_ρ is a permutation with cycle type ρ.


We set α_k := α_(k). 

 
By an argument given in <cit.>, this notation is consistent with (<ref>) in the sense that the closures of the diagrams n(ω_1) and n(ω_2), for ω_1, ω_2 ∈n and ω_1 and ω_2 having the same cycle type ν, are both equal to α_ν.



One can impose a grading on  by setting:

    (d_0) = 0     and    (d_2k) = 2k+1.


 
In terms of the grading defined by (<ref>), 

    α_2k+1 = d_2k +




We can reduce the diagram α_2k+1 to a polynomial in d_0, d_2, d_4, … via repeated application of the dot sliding moves (<ref>)-(<ref>) and clockwise bubble sliding move from Lemma <ref>. Specifically, the innermost crossing in α_2k+1 forms part of a right twist curl, i.e. a solid dot. Modulo terms with at least 2 fewer crossings (all of which also have exactly two connected components), this dot can be moved to the outside of the diagram via repeated applications of relations (<ref>)-(<ref>). The resulting diagram once again has a right twist curl in its center. Repeating this process 2k times yields d_2k, plus terms which have 2 fewer crossings (including those in right twists curls written as solid dots) and exactly two connected components. 

It remains to reduce the lower order terms to polynomials in d_0, d_2, d_4, …. Note that the dot sliding moves (<ref>)-(<ref>) and bubble sliding relation in Lemma <ref> never increase the number of crossings, and can only increase the number of connected components by at most 1 with a corresponding decrease in crossings by 2 (see the second and third term on the right side of (<ref>)-(<ref>)). It follows that while reducing each of these additional terms to monomials in bubbles, in each diagram

    #crossings   +   #connected components≤ (2k - 2j) + (j+1) = 2k -j +1

where 1 ≤ j ≤ k is half the number of crossings that have been resolved via a dot or bubble slide. Hence the degree of each resulting monomial in bubbles is strictly less than 2k+1. Since d_2k has degree 2k+1, the result follows.



 
 is generated by {α_2k+1}_k ≥ 0 and these elements are algebraically independent.






 §.§ Diagrams as bimodule homomorphisms
 

An action of  on the category  whose objects are compositions of induction and restriction functors between ℤ/2ℤ-graded finite dimensional n-supermodules, for all n≥0, is described in <cit.>. Because induction and restriction functors for the algebras n can be written as

    ^n+1_n ( -) =  _n+1n+1⊗_n -     and    ^n+1_n ( - ) =  _nn+1⊗_n+1 -

(where above _n+1n+1 is a (n+1,n)-bimodule and _nn+1 is a (n,n+1)-bimodule) the objects of  can alternatively be described as tensor products of certain (k_1,k_2)-bimodules for all k_1,k_2 ≥ 0. We will use this interpretation extensively below. Let k_1, k_2 ≤ n, then we write


	
  * (n) for n considered as a (n,n)-bimodule,
	
  * (n)_k_2 for n considered as a (n,k_2)-bimodule,
	
  * _k_1(n) for n considered as a (k_1,n)-bimodule,
	
  * _k_1(n)_k_2 for n considered as a (k_1,k_2)-bimodule.


The morphisms in  are certain natural transformations of these compositions of induction/restriction functors (or, equivalently, certain bimodule homomorphisms). 
To describe the morphisms of , we use the diagrams of , but this time we also label the rightmost region with a non-negative integer n.  Since an upward strand denotes the identity endomorphism of induction, it increases the label by one as one reads from right to left. Similarly a downward strand decreases the label by one. 



		
			[->,thick] (1,0) to (1,1);
			at (1.5,0.5) n;
			at (0,0.5) n+1;
		
			
		[->,thick] (1,1) to (1,0);
		at (2,0.5) n+1;
		at (0.5,0.5) n;
		



 We also set any  diagram with a negative label to be zero. Descriptions of other morphisms in  are most easily given in terms of bimodules, so we henceforth use this language exclusively.

Below are the generating morphisms and the corresponding bimodule maps: 
 
    [baseline=(current bounding box).center, scale=0.75]
     at (-4,1.6)(n+1)_n → (n+1)_n;
     at (-3.7,.9)g ↦ (-1)^|g|gc_n+1;
    [<->] (-.7,1.3)–(1.5,1.3);
    [->,thick] (4,.7) to (4,2);
     (4,1.35) circle [radius = 3pt];
     at (4.7,1.5)n;
     at (3.2,1.5)n+1;


 
    [baseline=(current bounding box).center, scale=0.75]
     at (-4,1.6)_n-1(n) → _n-1(n);
     at (-3.7,.9)g ↦ c_n+1g;
    [<->] (-.7,1.3)–(1.5,1.3);
    [<-,thick] (4,.7) to (4,2);
     (4,1.35) circle [radius = 3pt];
     at (4.7,1.5)n;
     at (3.2,1.5)n-1;



 
    [baseline=(current bounding box).center, scale=0.75]
     at (-4,1.6)q_n: (n) → (n)_n-1(n);
    [<->] (-.7,1.3)–(1.5,1.3);
    [<-,thick] (2.5,1.8) arc(-180:0:9mm); 
     at (4.7,1.5)n;
     at (3.4,1.6)n-1;



    [baseline=(current bounding box).center, scale=0.75]
     at (-4,1.6)n:  _n(n+1)_n→ (n);
    [<->] (-.7,1.3)–(1.5,1.3);
    [<-,thick] (2.5,1) arc(180:0:9mm); 
     at (4.7,1.5)n;
     at (3.4,1.2)n+1;



    [baseline=(current bounding box).center, scale=0.75]
     at (-4,1.6)(n+1)_n(n+1) → (n+1);
     at (-3.7,.9)g_1 ⊗ g_2 ↦ g_1g_2;
    [<->] (-.7,1.3)–(1.5,1.3);
    [->,thick] (2.5,1) arc(180:0:9mm); 
     at (5,1.5)n+1;
     at (3.4,1.2)n;



    [baseline=(current bounding box).center, scale=0.75]
     at (-4,1.6)i_n: (n) → _n(n+1)_n;
     at (-3.7,.9)g↦ g;
    [<->] (-.7,1.3)–(1.5,1.3);
    [->,thick] (2.5,1.8) arc(-180:0:9mm); 
     at (4.7,1.5)n;
     at (3.4,1.6)n+1;



    [baseline=(current bounding box).center, scale=0.75]
     at (-4,1.6)(n+2)_n→ (n+2)_n;
     at (-3.7,.9)g↦ gs_n+1;
    [<->] (-.7,1.3)–(1.5,1.3);
    [->,thick] (2.5,0)–(4,2);
    [->,thick] (4,0)–(2.5,2);
     at (4,1)n;
     at (3.3,2.25)n+1;
     at (2.4,1)n+2;


where n:  _n(n+1)_n→ (n) is the projection map given by n(g)=g if g∈n, n(g)=0 if g∉n, and q_n: (n) → (n)_n-1(n) is the bimodule map determined by q_n(1)=∑_x ∈nn-1 x ⊗ x^-1. 



The action of  on  can be lifted to the idempotent closures of these categories. This then becomes a categorification of the Fock space representation <cit.>. 


Following Khovanov's approach from <cit.>, let [n] be the full subcategory of  whose objects are (k,n)-bimodules, for all k∈ℤ_≥0. For every n ∈ℤ_≥0 there is a functor n:→[n] which is defined on objects of  such that it sends the rightmost P (respectively Q) from a P,Q sequence to (n+1)_n  (resp. _n-1(n)) and all other P and Q in the sequence are determined by this choice. Hence n maps all objects of  into [n].  Under n, a morphism (or diagram) is mapped to a morphism in [n] by labeling the rightmost region by n which determines the labelings of all other regions. An upward strand increases the label by 1, and a downward strand decreases the label by 1 as we read from right to left.

Note that the image of a closed diagram D under n will be an (n,n)-bimodule endomorphim of n which we denote as f: n→n. f is fully determined by the value f(1) since for any x ∈n, f(x) = xf(1). Furthermore, f(1) is an element of n because xf(1) = f(x) = f(1)x. In this way we can identify the image of  under n with elements of n.

We next study the image of some of the elements of  from Section <ref> under  the functor n.

 



  *  The diagram





[scale = .6]


[thick] (.9,-.5) arc (180:360:.6cm);
[thick] (0,-.5) arc (180:360:1.5cm);
[thick] (-.85,-.5) arc (180:360:2.4cm);


at (.9,-.5) ;
at (0,-.5) ;
at (-.85,-.5) ;


at (1.5,-.7) n-k;
at (-.3,-1) n-1;
at (-1.1,-1.2) n;


[thick,black] (.7,-.8) circle (.2mm);
[thick,black] (.55,-.85) circle (.2mm);
[thick,black] (.4,-.9) circle (.2mm);






corresponds to the (n,n)-bimodule homomorphism (n) → (n)_n-k(n) which sends

    1   ↦∑_x ∈nn-k x ⊗ x^-1.



  *  For μ∈[k] with k ≤ n, the diagram




[scale = .8]

[thick] (2.3,-1)–(2.3,1);
[thick] (1.4,-1)–(1.4,1);

[fill=white,thick] (2.6,-.3) rectangle (1.1,.3);

at (2.3,.6) ;
at (1.4,.6) ;


[thick,black] (2,.7) circle (.2mm);
[thick,black] (1.8,.7) circle (.2mm);
[thick,black] (1.6,.7) circle (.2mm);

[thick,black] (4.8,.7) circle (.2mm);
[thick,black] (5,.7) circle (.2mm);
[thick,black] (5.2,.7) circle (.2mm);

[thick,black] (3.2,0) circle (.2mm);
[thick,black] (3.4,0) circle (.2mm);
[thick,black] (3.6,0) circle (.2mm);

[thick] (4.5,-1)–(4.5,1);
[thick] (5.4,-1)–(5.4,1);

[rotate = 180] at (4.5,.6) ;
[rotate = 180] at (5.4,.6) ;

at (1.85,0) μ;

at (3.4,.5) n-k;
at (0,.3) n;
at (6.8,.3) n;






corresponds to the (n,n)-bimodule homomorphism (n)_n-k(n) → (n)_n-k(n) which for x,y ∈n sends

    x ⊗ y   ↦   xμn⊗ y.






Both <ref> and <ref> follow from calculations using the definitions of cup (<ref>) and crossing (<ref>) maps.


 
For μ∈[k],

    n(α_μ) = μn   if k ≤ n
    
    0    otherwise.




The diagram for α_μ can be broken into three components


[scale = .8]

at (-4,0) α_μ   :=;

[thick] (0,0) circle (1.8 cm);
[thick] (0,0) circle (1.6 cm);
[thick] (0,0) circle (1 cm);

[fill=white,thick] (-2,-.3) rectangle (-.5,.3);

at (1.4,0) ·;
at (1.3,0) ·;
at (1.2,0) ·;

at (-1.3,0) μ;

[rotate = 180] at (1.8,0) ;
[rotate = 180] at (1.6,0) ;
[rotate = 180] at (1,0) ;

at (2,-1.2) .;

[red,thick,dashed] (-2.8,.5)–(2.8,.5);
[red,thick,dashed] (-2.8,-.5)–(2.8,-.5);




Reading from bottom to top, the first component corresponds to Lemma <ref>.<ref>, and the second corresponds to Lemma <ref>.<ref> The composition of these two maps sends

    1   ↦  ∑_x ∈nn-k xμn⊗ x^-1.

The top component of k nested caps is the multiplication map which sends

    ∑_x ∈nn-k xμn⊗ x^-1  ↦  ∑_x ∈nn-k xμnx^-1 = μn.
 



 <cit.> 
For n -1 ≥ 0, the right twist curl 




    	[scale=.7]
    		[->,thick] (0,0) to (0,1);
    		[->,thick] (0,1) to (1,2);
    		[->,thick] (1,1) to (0,2);
    		[<-,thick] (2,1) to (2,2);
    		[->,thick] (2,1) arc(0:-180:0.5);
    		[->,thick] (0,2) to (0,3);
    		[->,thick] (1,2) arc(180:0:0.5);
    		[dashed] (-1,0) to (3,0);
    		[dashed] (-1,1) to (3,1);
    		[dashed] (-1,2) to (3,2);
    		[dashed] (-1,3) to (3,3);
    		at (2.7,1.5) n-1;
    	


corresponds to the (n,n-1)-bimodule homomorphism, (n)_n-1→ (n)_n-1 which multiplies x ∈n on the right by the Jucys-Murphy element n

    x   ↦   xn.

 

 
Let k ≥ 0 and n ≥ 1, then


  * n(d̅_2k) = n(n+1^2k),

  * n(d_2k) = ∑_x ∈nn-1 xn^2kx^-1.




These follow from direct calculation using the definitions of the cup (<ref>) and cap (<ref>) maps and Lemma <ref>.






§ MAIN RESULTS




 §.§ An isomorphism between  and 
 
In this subsection we establish an isomorphism between  and . The key step in the construction of this map will be identifying the elements of  with functions on , i.e. as elements of . To do this, let λ∈[n] and x ∈. Then we evaluate x on λ by

    x(λ) := λ(n(x)).

Because n is a homomorphism on  which maps into n and λ is a homomorphism when restricted to n, this defines a homomorphism into . 

 
For μ∈[k] and λ∈[n] we have 

    α_μ(λ) = 
    2^kn^↓ kλ(μ∪ 1^n-k)/λ(1^n)   if k ≤ n
    
    0    otherwise




This follows from Proposition <ref> and Proposition <ref>.


 
There is an algebra isomorphism : → which for any μ∈, sends

    α_μ↦ 2^μ_μ.




It is clear from Proposition <ref> and Corollary <ref> that 2^-μα_μ and _μ map to the same function in . Furthermore the collection of functions which are the image of {_2k+1}_k ≥ 0 are algebraically independent by Proposition <ref> and Corollary <ref>. By Proposition <ref>  is generated by the algebraically independent elements {α_2k+1}_k ≥ 0. It then follows that the map that sends α_μ↦ 2^μ_μ is an isomorphism.


Let μ∈[n]. It follows from Lemma <ref>, Remark <ref>, and Theorem <ref> that

    at (0,0) [scale = .8]
    
    
    [thick] (0,0) circle (1.8 cm);
    [thick] (0,0) circle (1.6 cm);
    [thick] (0,0) circle (1 cm);
    
    
    [fill=white,thick] (-2,-.3) rectangle (-.5,.3);
    
     at (-1,.5) ·;
     at (-1.15,.55) ·;
     at (-1.3,.6) ·;
    
     at (-1.3,0) C_μ;
    
    [rotate = -30] at (-1.5,1) ;
    [rotate = -30] at (-1.3,.9) ;
    [rotate = -30] at (-.8,.6) ;
    ;
    
    
     at (2.1,0) =;
    
     at (3.35,0) n!/z_μ2^n - μ;
    
     at (5.8,0) [scale = .8]
    
    
    [thick] (0,0) circle (1.8 cm);
    [thick] (0,0) circle (1.6 cm);
    [thick] (0,0) circle (1 cm);
    
    
    [fill=white,thick] (-2,-.3) rectangle (-.5,.3);
    
     at (-1,.5) ·;
     at (-1.15,.55) ·;
     at (-1.3,.6) ·;
    
     at (-1.3,0) μ;
    
    [rotate = -30] at (-1.5,1) ;
    [rotate = -30] at (-1.3,.9) ;
    [rotate = -30] at (-.8,.6) ;
    ;
    
     at (8.2,.3) ;
    [thick, ->] (7.6,0)–(8.8,0);
    [thick] (7.6,.1)–(7.6,-.1);
    
     at (10,0) n!/z_μ2^n_μ.;



Let λ∈[n]. Under the isomorphism : →, the closure of the central idempotent λ of n maps to λ Q_λ^*.



Recall from Lemma <ref> that

    λ = 2^-λ-(λ) /2λ/n!∑_μ∈[n]λ(μ)C_μ

while by (<ref>)

    Q^*_λ =2^λ-(λ)/2∑_μ∈[n]λ(μ)/z_μ_μ.

Combining these facts with Theorem <ref> and (<ref>) it follows that the closure of λ is equal to 2^n-λλQ^*_λ = λQ^*_λ.



Recall that the Schur Q-functions are related to the Schur P-functions by P_λ = 2^-λQ_λ. Ivanov also studied factorial Schur P-functions {P^*_λ}_λ∈[n] where P^*_λ = 2^-λQ^*_λ <cit.>. Then one alternative description of the closure of λ in  is as 2^nλP^*_λ.

   
Moving in the opposite direction, we can also identify the elements of  corresponding to the generators {d_2k}_k ≥ 0 and {d̅_2k}_k ≥ 0.

 
For k ≥ 0, 

	
  * ψ(d̅_2k) = k(·),
	
  * ψ(d_2k) = k+1(·).




This follows from Proposition <ref> and Proposition <ref>.



In light of Theorem <ref>, Proposition <ref> can be seen as a diagrammatic manifestation of Proposition <ref>.










 §.§ An action of  on 
 

Aside from taking the Grothendieck group or center, another method for decategorifing a category C is taking the categorical trace of C, (C) (also known as the zeroth Hochschild homology of C). See <cit.> for a discussion of this method of decategorification. In <cit.>, it is shown that the even part of the trace of , , is isomorphic to the vertex algebra  at level one, a subalgebra of  defined by Kac, Wang, and Yan <cit.>. 

In a diagrammatic setting such as this, the trace can be realized as the algebra of closed diagrams on an annulus. There is a natural action of (C) on the center of the category C, _C(), where diagrammatically a closed diagram on an annulus acts on a closed diagram in a disk by plugging the annulus with the disk, resulting in a new diagram in the disk. The results of <cit.> along with Theorem <ref> imply that  acts on . This action is similar to the action of  on the centers of symmetric group algebras described in <cit.>. In this section we will first review  and then describe the action of the generators of  on basis elements of .

We first review the vertex algebra W^-, which appears in the trace of ℋ_tw. 

Let 𝒟̂^- be the Lie algebra over the vector space spanned by {C}∪{ t^2k-1 g(D+(2k-1)/2);  g even}∪{ t^2k f(D+k) ; f odd} where k∈ℤ and even and odd refer to even and odd polynomial functions. The Lie bracket in 𝒟̂^- is defined by


    [t^rf(D),t^sg(D)]=t^r+s(f(D+s)g(D)-f(D)g(D+r))+ψ(t^rf(D),t^sg(D))C,

where

    ψ(t^rf(D),t^sg(D))= ∑_-r≤ j≤ -1f(j)g(j+r)     r=-s≥ 0 
    
          0     r+s≠ 0 
       ,

and C is a central element. Denote by W^- the universal enveloping algebra of 𝒟̂^-. The trace of ℋ_tw was shown in <cit.> to be isomorphic to the quotient W^-/⟨ω_0,0, C-1 ⟩. 




A generating set for W^-/⟨ω_0,0, C-1⟩ is given by ω_1,0, ω_0,3, and ω_± 2,1±ω_± 2,0<cit.>. In order to explicitly write down an action of the algebra W^- on , we will work with different generating set.


The algebra W^-/⟨ω_0,0, C-1⟩ is also generated by ω_1,0, ω_-1,0 and ω_0,3.



    We will show that we can obtain the aforementioned generators ω_± 2,1±ω_± 2,0 via the elements ω_1,0, ω_-1,0 and ω_0,3. It is a straightforward computation that 
   
    ω_0,1=-1/20[[ω_0,3, ω_-1,0], ω_1,0] +1/5ω_-1,0ω_1,0

   and using ω_0,1, we can obtain ω_-1,2-ω_-1,1 as follows:
   
    ω_-1,2-ω_-1,1=1/6[ω_0,3, ω_-1,0] + 1/3ω_-1,0ω_0,1.

Then one of the elements we are looking for is given by
   
    ω_-2,1 - ω_-2,0=1/2[ω_-1,2-ω_-1,1, ω_-1,0].

    To obtain ω_2,1 + ω_2,0, we follow a very similar computation:
    
    ω_1,2+ω_1,1=-1/6[ω_0,3, ω_1,0] + 1/3ω_0,1ω_1,0,

    and finally
    
    ω_2,1 + ω_2,0=-1/2[ω_1,2+ω_1,1, ω_1,0].



The images of these generators under the isomorphism W^-/⟨ω_0,0, C-1⟩→ from <cit.> are given below. In the following diagrams, an asterisk denotes a puncture in the plane, forming the annulus around which the diagrams are closed in the trace:


    √(2)ω_-1,0⟼;

	
	
    √(2)ω_1,0⟼;

	
	
    -2ω_0,3⟼[baseline=(current bounding box).center]
    				[thick] (2,2) circle (.5cm);
    				 at (2,2) *;
    				 at (1.5,2) ;
    				[fill=black] (1.62,2.33) circle (.08cm);
    				 at (1.4,2.5) 2;
    				
    				=
    				
    				[baseline=(current bounding box.center), scale=.25]
    				
    				[thick] (2,0) .. controls (2,1.25) and (0,.25) .. (0,2);
    				[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    				[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    				
    				 at (0,2);
    				 at (1,2);
    				 at (2,2);
    				 at (4,1)*;
    				
    				[thick] (2,2) arc (150:-150:2cm);
    				[thick] (1,2) arc (160.5:-160.5:3cm);
    				[thick] (0,2) arc (165.5:-165.5:4cm);
    				
    				+
    				
    				[baseline=(current bounding box.center), scale=.25]
    				
    				[thick] (1,0) to (1,2);
    				[thick] (2,0) to (2,2); 
    				
    				 at (1,2);
    				 at (2,2);
    				
    				[thick] (2,2) arc (150:-150:2cm);
    				[thick] (1,2) arc (160.5:-160.5:3cm);
    				 at (4,1)*;
    				.

		
		We will also use the elements ω_-(2n+1),0 and their images in :
		
		
    √(2)ω_-(2n+1),0⟼[baseline=(current bounding box).center, scale = .8]
    			
    			
    			[thick] (0,0) circle (1.8 cm);
    			[thick] (0,0) circle (1.6 cm);
    			[thick] (0,0) circle (1 cm);
    			
    			 at (0,0)*;
    			
    			[fill=white,thick] (-2,-.3) rectangle (-.5,.3);
    			
    			 at (-1,.5) ·;
    			 at (-1.15,.55) ·;
    			 at (-1.3,.6) ·;
    			
    			 at (-1.3,0) τ;
    			
    			[rotate = -30] at (-1.5,1) ;
    			[rotate = -30] at (-1.3,.9) ;
    			[rotate = -30] at (-.8,.6) ;

			where τ is a (2n+1)-cycle.


   We now describe the action of the generating set {ω_1,0, ω_-1,0, ω_0,3} of W^- on the vector space basis {_μ}_μ∈ of . We achieve this by describing the action of the corresponding generators of  on the basis {α_μ}_μ∈ of . 

 We have
	
    α_(μ,1)= α_μα_1 -2|μ|α_μ.




	  This simply follows from the local bubble sliding relation 
	  
	  
    [baseline=(current bounding box.center),scale=.6]
    	  [thick] (0,0) circle(1cm);
    	   at (-1,0);
    	  [thick] (1.5,1)–(1.5,-1);
    	  [rotate=180] at (1.5,0);
    	  
    	  =
    	  [baseline=(current bounding box.center),scale=.6]
    	  [thick] (0,0) circle(1cm);
    	   at (-1,0);
    	  [thick] (-1.5,1)–(-1.5,-1);
    	  [rotate=180] at (-1.5,0);
    	  
    	  -2
    	  [baseline=(current bounding box.center),scale=.6]
    	  [thick] (-1.5,1)–(-1.5,-1);
    	  [rotate=180] at (-1.5,0);

	  applied |μ| times to the diagram α_(μ,1), as we pull the clockwise bubble α_1 from within α_μ.


 We have
	
    ω_1,0·α_μα_1 =(α_1+2)ω_1,0·α_μ .




	We compute: 
    [baseline=(current bounding box.center),scale=.6]
    	
    	[thick] (0,0) circle (2cm);
    	[rotate=180] at (-2,0);
    	
    	[thick] (-1.5,-.5) rectangle (-.5,.5);
    	 at (-1,0)α_μ;
    	[thick] (-1,.5) arc (150:0:.5cm);
    	[thick] (-.07,.3) – (-.07,-.3);
    	[thick] (-1,-.5) arc (-150:0:.5cm);
    	[rotate=-90] at (-.5,.75);
    	
    	[thick] (1,0) circle(.7cm);
    	 at (.3,0);
    	   =
    	[baseline=(current bounding box.center),scale=.6]
    	
    	[thick] (0,0) circle (2cm);
    	[rotate=180] at (-2,0);
    	 at (2,0);
    	
    	[thick] (-1.5,-.5) rectangle (-.5,.5);
    	 at (-1,0)α_μ;
    	[thick] (-1,.5) arc (150:0:.5cm);
    	[thick] (-.07,.3) – (-.07,-.3);
    	[thick] (-1,-.5) arc (-150:0:.5cm);
    	[rotate=-90] at (-.5,.75);
    	
    	[thick] (1.7,0) circle(.7cm);
    	 at (1,0);
    	
    	+
    	[baseline=(current bounding box.center),scale=.6]
    	
    	[thick] (-2,0) arc (180:20:2cm);
    	[thick] (-2,0) arc (-180:-20:2cm);
    	[rotate=180] at (-2,0);
    	
    	[thick] (-1.5,-.5) rectangle (-.5,.5);
    	 at (-1,0)α_μ;
    	[thick] (-1,.5) arc (150:0:.5cm);
    	[thick] (-.07,.3) – (-.07,-.3);
    	[thick] (-1,-.5) arc (-150:0:.5cm);
    	[rotate=-90] at (-.5,.75);
    	
    	 at (.33,0);
    	[thick,red] (1.87,.7) to[out=-60,in=-60] (1.45,.65);
    	[thick] (1.45,.65) arc (60:300:.75cm);
    	[thick,red] (1.45,-.65) to[out=60, in=60] (1.87,-.7);
    	
    	+
    	[baseline=(current bounding box.center),scale=.6]
    	
    	[thick] (-2,0) arc (180:20:2cm);
    	[thick] (-2,0) arc (-180:-20:2cm);
    	[rotate=180] at (-2,0);
    	
    	[thick] (-1.5,-.5) rectangle (-.5,.5);
    	 at (-1,0)α_μ;
    	[thick] (-1,.5) arc (150:0:.5cm);
    	[thick] (-.07,.3) – (-.07,-.3);
    	[thick] (-1,-.5) arc (-150:0:.5cm);
    	[rotate=-90] at (-.5,.75);
    	
    	 at (.33,0);
    	[thick,red] (1.87,.7) to[out=-60,in=-60] (1.45,.65);
    	[thick] (1.45,.65) arc (60:300:.75cm);
    	[thick,red] (1.45,-.65) to[out=60, in=60] (1.87,-.7);
    	[thick,red] (1.6,.57) circle (4pt);
    	[thick,red] (1.8,-.57) circle (4pt);
    	
       =
    	[baseline=(current bounding box.center),scale=.6]
    	
    	[thick] (0,0) circle (2cm);
    	[rotate=180] at (-2,0);
    	
    	[thick] (-.7,-.5) rectangle (0.3,.5);
    	 at (-.2,0)α_μ;
    	[thick] (-.2,.5) arc (150:0:.5cm);
    	[thick] (.73,.3) – (.73,-.3);
    	[thick] (-.2,-.5) arc (-150:0:.5cm);
    	[rotate=-90] at (.3,.75);
    	
    	[thick] (3,0) circle(.7cm);
    	 at (2.3,0);
    	
    	+2
    	[baseline=(current bounding box.center),scale=.6]
    	
    	[thick] (0,0) circle (2cm);
    	[rotate=180] at (-2,0);
    	
    	[thick] (-.7,-.5) rectangle (0.3,.5);
    	 at (-.2,0)α_μ;
    	[thick] (-.2,.5) arc (150:0:.5cm);
    	[thick] (.73,.3) – (.73,-.3);
    	[thick] (-.2,-.5) arc (-150:0:.5cm);
    	[rotate=-90] at (.3,.75);
    	,

	as desired.




The generators  act on the basis elements {_μ}_μ∈ of  as follows:
 

  * ω_-1,0·_μ = √(2)_(μ,1) 
 
  * ω_1,0·_μ = 
1/√(2)_μ+ k/√(2)_μ̂

  * ω_0,3·_μ = -_3 _μ -2 _(1,1)_μ.

where k is the number of parts of size 1 of μ and μ̂ stands for the partition obtained by removing one part of size 1 from μ if this is possible. When μ = (1) then _(1)=1.



	For the action of ω_-1,0, note that the action of  on α_μ is diagrammatically just enclosing the diagram of α_μ by a clockwise oriented strand:
	
    ·α_μ = 
    	[baseline=(current bounding box).center]
    	[thick] (2,2) circle (.5cm);
    	 at (1.5,2) ;
    	 at (2,2) α_μ;

	and the resulting diagram is the diagram of α_(μ,1). Replacing α_μ by 2^μ_μ and the clockwise bubble by √(2)ω_-1,0, we get 
ω_-1,0·_μ = √(2)_(μ,1).
	
	
We also know that ω_-(2n+1),0·_μ= √(2)_(μ,2n+1) from (<ref>). Note that we have
[ω_-1,0, ω_1,0]=-1 and, for n≥ 0, we have
	[ω_-(2n+1),0, ω_1,0]=0.
	
	
	
	
    ·α_ρ = 
    	[baseline=(current bounding box).center]
    	[thick] (2,2) circle (.5cm);
    	[rotate=180] at (1.5,2) ;
    	 at (2,2) α_ρ;

	 
	
	To simplify the notation in the following computations, denote :=√(2)ω_1,0.	

		We start by showing that if the partition μ doesn't contain any parts of size one, then ·α_μ=α_μ by induction on μ. We provide a diagrammatic proof for the base case μ=1 (i.e. α_μ=α_k for k≠1 odd).
		
		In the diagram [baseline=(current bounding box).center]
		[thick] (2,2) circle (.5cm);
		[rotate=180] at (1.5,2) ;
		at (2,2) α_k;
		, we claim that we can pass α_k through the outer strand for free, meaning that all the resolution terms that appear as a result of relation <ref> are zero.
		
		We provide the computation for the case of α_k = α_5, and explain how the arguments generalize to any α_k. We have
		
    [baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		[thick] (-3,2) arc (172:-172:7cm);
    		   =
    		
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[rotate=-180] at (9.9,1);
    		 at (9.3,1);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		[thick] (-3,2) arc (172:29:7cm);
    		[thick] (-3,0) arc (-172:-29:7cm);
    		
    		[thick, red] (10,4.5) to [out=-50, in=90] (9.3,1);
    		[thick, red] (10,-2.5) to [out=50, in=-90] (9.3,1);
    		
    		+
    		
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		
    		[rotate=-180] at (-3,1.9);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:30:6cm);
    		[thick] (-2,0) arc (-171:-30:6cm);
    		[thick] (-3,2) arc (172:29:7cm);
    		[thick] (-3,0) arc (-172:-29:7cm);
    		
    		[thick,red] (10,4.5) .. controls (10.3,3.7) and (9.2,3.6) .. (9.1,4.1);
    		[thick,red] (9.1,-2.1) .. controls (9.1,-1.7) and (10.3,-1.6) .. (10,-2.5);
    		
       +
    		
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		
    		[rotate=-180] at (-3,1.9);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:30:6cm);
    		[thick] (-2,0) arc (-171:-30:6cm);
    		[thick] (-3,2) arc (172:29:7cm);
    		[thick] (-3,0) arc (-172:-29:7cm);
    		
    		[thick,red] (10,4.5) .. controls (10.3,3.7) and (9.2,3.6) .. (9.1,4.1);
    		[thick,red] (9.1,-2.1) .. controls (9.1,-1.7) and (10.3,-1.6) .. (10,-2.5);
    		
    		[thick] (9.4,3.8) circle (.3cm);
    		[thick] (10,-2) circle (.3cm);

		and the two hollow dots appearing in the last term cancel with each other if we slide them along the outermost strand. This observation will hold for the rest of the computation, so we will omit drawing the second resolution term and instead write the first resolution term with coefficient 2. We will show that all resolution terms coming from crossings on the: outermost strand, innermost strand, and intermediate strands are zero. 
		
		For the resolution term coming from the crossing of outermost strands, we have 
		
    [baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		
    		[rotate=-180] at (-3,1.9);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:30:6cm);
    		[thick] (-2,0) arc (-171:-30:6cm);
    		[thick] (-3,2) arc (172:29:7cm);
    		[thick] (-3,0) arc (-172:-29:7cm);
    		
    		[thick,red] (10,4.5) .. controls (10.3,3.7) and (9.2,3.6) .. (9.1,4.1);
    		[thick,red] (9.1,-2.1) .. controls (9.1,-1.7) and (10.3,-1.6) .. (10,-2.5);
    		
    		=
    		
    		[baseline=(current bounding box.center), scale=.3]
    		
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		
    		[thick,red] (-3,2) arc (180:3:.5cm);
    		[thick,red] (-3,0) arc (-180:0:.5cm);
    		
    		=0
  
		where the last equality follows from (<ref>).
		
		For the resolution term coming from the crossing of intermediate strands, consider a generic intermediate strand. We have

		
    [baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[rotate=-180] at (9.9,1);
    		[rotate=-180] at (8.9,1);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:50.5:4cm);
    		[thick] (0,0) arc (-165.5:-50:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		[thick] (-3,2) arc (172:35:7cm);
    		[thick] (-3,0) arc (-172:-35:7cm);
    		
    		[thick, red] (9.66,5.05) to [out=-70, in=-50] (6.4,4.1);
    		[thick, red] (9.66,-3.05) to [out=70, in=50] (6.4,-2.1);
    		   =
    		
    		
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[rotate=-180] at (9.9,1);
    		[rotate=-180] at (8.9,1);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		
    		[thick,red] (-3,2) arc (180:2:1.5cm);
    		[thick,red] (-3,0) arc (-180:0:1.5cm);
    		
    		=
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		[rotate=-20] at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[rotate=-180] at (9.9,1);
    		[rotate=-180] at (8.9,1);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		
    		[thick,red] (-3,2) arc (180:2:1.5cm);
    		[thick,red] (-3,0) to[out=-70,in=-130] (-.7,1.3);
    		[thick,red] (-.7,1.3) to[out=50, in=-100] (1,2);
    		
       =[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		[rotate=-10] at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[rotate=-180] at (9.9,1);
    		[rotate=-180] at (8.9,1);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		
    		[thick,red] (-3,2) arc (180:2:1.5cm);
    		[thick,red] (-3,0) to[out=-70,in=-130] (-1.7,1.3);
    		[thick,red] (-1.7,1.3) to[out=50, in=-100] (1,2);
    		
    		=
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		
    		[rotate=-180] at (9.9,1);
    		[rotate=-180] at (8.9,1);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		
    		[thick,red] (0,2) to[out=90, in=0] (-.35,2.7);
    		[thick,red] (-.35,2.7) to[out=180, in=130] (-.35,1.3);
    		[thick, red] (-.35,1.3) to[out=-50, in=-100] (1,2);
    			
    		= 0

where the second and third equalities follow from a Reidemeister III move, and the fourth is a result of relation (<ref>). Hence these resolution terms are zero as well. In general, for a resolution term coming from a crossing of intermediate strands, we can first pull the red string above the permutation using Reidemeister III moves, and then pull the red string into the permutation using relation (<ref>) to get a left twist curl.
		
		Finally, for the resolution term coming from the crossing of intermediate strands the situation is simpler:

		
    [baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[rotate=-180] at (9.9,1);
    		[rotate=-180] at (8.9,1);
    		[rotate=-180] at (7.9,1);
    		[rotate=-180] at (6.8,1);
    		
    		[thick] (2,2) arc (150:30:2cm);
    		[thick] (2,0) arc (-150:-30:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		[thick] (-3,2) arc (172:35:7cm);
    		[thick] (-3,0) arc (-172:-35:7cm);
    		
    		[thick, red] (9.66,5.05) to [out=-50, in=-50] (5.45,2);
    		[thick, red] (9.66,-3.05) to [out=50, in=50] (5.45,0);
    		
    		=
    		
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[rotate=-180] at (9.9,1);
    		[rotate=-180] at (8.9,1);
    		[rotate=-180] at (7.9,1);
    		[rotate=-180] at (6.8,1);
    		
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		
    		[thick,red] (-3,2) arc (180:0:2.5cm);
    		[thick,red] (-3,0) arc (-180:0:2.5cm);
    		
    		=
    		
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (-2.5,0) to[out=70, in=-90] (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[rotate=-180] at (9.9,1);
    		[rotate=-180] at (8.9,1);
    		[rotate=-180] at (7.9,1);
    		[rotate=-180] at (6.8,1);
    		
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		
    		[thick,red] (-3,2) arc (180:0:2.5cm);
    		[thick,red] (-3,0) arc (-180:0:0.25cm);
    		
    		=0.
	

		Hence all the resolution terms are zero. This leaves us with

		
    [baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		[thick] (-3,2) arc (172:-172:7cm);
    		   =
    		
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		[rotate=-180] at (9.9,1);
    		[rotate=-180] at (8.9,1);
    		[rotate=-180] at (7.9,1);
    		[rotate=-180] at (6.8,1);
    		[rotate=-180] at (5.7,1);
    		 at (5,1);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		[thick] (-3,2) arc (172:35:7cm);
    		[thick] (-3,0) arc (-172:-35:7cm);
    		
    		[thick, red] (9.66,5.05) to [out=-50, in=90] (5,1);
    		[thick, red] (9.66,-3.05) to [out=50, in=-90] (5,1);
    		
    		=
    		
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		 at (3,1);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		
    		[thick, red] (-3,2) to [out=90, in=90] (3,1);
    		[thick, red] (-3,0) to [out=-90, in=-90] (3,1);
    		
       =
    		
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		[thick] (-3,0)–(-3,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		[rotate=-180] at (-3,1.9);
    		
    		 at (-2.5,.7);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		
    		[thick, red] (-3,2) to [out=90, in=90] (3,2);
    		[thick, red] (-3,0) to [out=-90, in=-90] (3,-.7);
    		[thick,red] (3,2) to[out=-90, in=90] (-2.5,.7);
    		[thick,red] (-2.5,.7) to[out=-90, in=90] (3,-.7);
    		
    		=
    		
    		[baseline=(current bounding box.center), scale=.25]
    		
    		[thick] (-2,0) .. controls (-2,1) and (-1.2,.8) .. (-1,2);
    		[thick] (-1,0) .. controls (-1,1) and (-.2,.8) .. (0,2);
    		[thick] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
    		[thick] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
    		[thick] (2,0) .. controls (2,1.25) and (-2,.25) .. (-2,2);
    		
    		 at (2,1.9);
    		 at (1,1.9);
    		 at (0,1.9);
    		 at (-1,1.9);
    		 at (-2,1.9);
    		
    		[rotate=180] at (-5,1);
    		
    		[thick] (2,2) arc (150:-150:2cm);
    		[thick] (1,2) arc (160.5:-160.5:3cm);
    		[thick] (0,2) arc (165.5:-165.5:4cm);
    		[thick] (-1,2) arc (169:-169.5:5cm);
    		[thick] (-2,2) arc (171:-171:6cm);
    		
    		[thick,red] (-4,1) circle (1cm);

		and a counter-clockwise oriented bubble is equal to 1 by the defining relation (<ref>). These diagrammatic arguments clearly hold for arbitrary k > 1. Hence the action of ω_(1,0) on α_k for k≠1 is trivial.
		
		
		This concludes the proof of the base case μ=1. Now suppose ·α_μ=α_μ for some μ∈ such that μ=m-1, and let n be a positive integer. Then
		
    0   =[√(2)ω_-(2n+1),0, ]·α_μ = √(2)ω_-(2n+1),0·(·α_μ)- · (√(2)ω_-(2n+1),0·α_μ)
        = √(2)ω_-(2n+1),0·α_μ - · (√(2)ω_-(2n+1),0·α_μ) 
       = α_(μ,2n+1) - ·α_(μ,2n+1),

and the result follows by induction.
				
Hence if μ doesn't contain any parts of size 1, then 
·α_μ=α_μ.

Now suppose γ is an odd partition without parts of size 1. We will prove that  

    ·α_(γ,1^k) = α_(γ,1^k) + 2kα_(γ,1^k-1)

by induction on k. The base case k=0 was proved above. Suppose the formula holds for α_(γ,1^k). 
	
    ·α_(γ,1^k+1)   =·(α_(γ,1^k)α_1-2|(γ,1^k)|α_(γ,1^k))  by Lemma <ref>
       = α_1·α_(γ,1^k) +2·α_(γ,1^k) -2|(γ,1^k)|·α_(γ,1^k)
       = (α_1+2-2|(γ,1^k)|) ·α_(γ,1^k)
       = (α_1+2-2|(γ,1^k)|) (α_(γ,1^k)+2k α_(γ,1^k-1) )    by the inductive hypothesis
       = (α_1+2-2|(γ,1^k)|)α_(γ,1^k) +2k(α_1+2-2|(γ,1^k)|)α_(γ,1^k-1)
       = α_(γ,1^k+1) +4 α_(γ,1^k) + 2k α_(γ,1^k) by Lemma <ref>
       = α_(γ,1^k+1)+ 2(k+1) α_(γ,1^k),

and the result follows after the identification α_μ→ 2^μ_μ.
	
	
	
	
	
	
	
For the action of ω_0,3, note that this element acts on the center as multiplication by itself. Therefore 
    ω_0,3·α_μ=α_3α_μ+α_(1,1)α_μ,

		
    -2ω_0,3· 2^μ_μ= 2^μ+1_3_μ+2^μ+2_(1,1)_μ.



amsalpha


