The totally nonnegative part of G/P is a ball
Pavel Galashin Steven N. Karp Thomas Lam
math.RT math.AG math.CO 14M15, 15B48, 20Gxx





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Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue,
Cambridge, MA 02139, USA
mailto:galashin@mit.edugalashin@mit.edu

Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, USA
mailto:snkarp@umich.edusnkarp@umich.edu
mailto:tfylam@umich.edutfylam@umich.edu



T.L. acknowledges support from the NSF under agreement No. DMS-1464693.





We show that the totally nonnegative part of a partial flag variety (in the sense of Lusztig) is homeomorphic to a closed ball.



[2010]14M15, 
15B48, 
20Gxx



The totally nonnegative part of G/P is a ball
    Thomas Lam
    December 30, 2023
=============================================



equationsection


§ INTRODUCTION

Let G be a simply connected, semisimple algebraic group defined and split over , and let ^ be the (real) partial flag variety of G associated to a subset  of the set  of simple roots.  Lusztig <cit.> has defined a “remarkable polyhedral subspace” ^_≥ 0 of ^, called the totally nonnegative part.  In this paper we establish the following result.


The totally nonnegative part ^_≥ 0 is homeomorphic to a closed ball of dimension equal to that of ^.


Lusztig proved that ^_≥ 0 is contractible <cit.>. He also defined a partition of ^_≥ 0 8.15, which was shown to be a cell decomposition by Rietsch <cit.>. Williams conjectured that this cell decomposition forms a regular CW complex <cit.>. Rietsch and Williams showed that ^_≥ 0 forms a CW complex <cit.>, and have proved Williams's conjecture up to homotopy <cit.>. These results complement work by Hersh <cit.> in the unipotent group case.

Our proof of <ref> employs the vector field τ on ^ which is the sum of all Chevalley generators of , which Lusztig used to show that ^_≥ 0 is contractible. The flow defined by τ is a contractive flow on ^_≥ 0 in the sense of our paper <cit.>, so in particular it contracts all of ^_≥ 0 to a unique fixed point p_0∈^_≥ 0. The machinery of <cit.> (see <ref>) generates a homeomorphism from ^_≥ 0 to a closed ball B⊂^_≥0 centered at p_0, by mapping each trajectory in ^_≥ 0 to its intersection with B. This generalizes <cit.> from type A Grassmannians to all partial flag varieties. We remark that <ref> is new in all other cases, including for the complete flag variety and multi-step flag varieties in ^n.



§ PRELIMINARIES


In this section, we recall some background from Lusztig <cit.> and <cit.>.



 §.§ Pinnings


Let  denote the Lie algebra of G over . We fix Chevalley generators (e_i,f_i)_i∈ of , so that the elements h_i:=[e_i,f_i] (i∈) span the Lie algebra of a split real maximal torus T of G. For i∈ and t∈ℝ, we define the elements of G

    x_i(t) := exp(te_i),   y_i(t) := exp(tf_i).

We also let α_i^∨:ℝ^∗→ T be the homomorphism of algebraic groups whose tangent map takes 1∈ℝ to h_i. The x_i(t)'s (respectively, y_i(t)'s) generate the unipotent radical U^+ of a Borel subgroup B^+ (respectively, U^- and B^-) of G, with B^+∩ B^- = T. The data (T,B^+,B^-,x_i,y_i;i∈) is called a pinning for G.



 §.§ Total positivity


Let i_1,…,i_ℓ be a sequence of elements of  such that the product of simple reflections s_i_1⋯ s_i_ℓ is a reduced decomposition of the longest element of the Weyl group. We define the totally positive parts

    ^+:={x_i_1(t_1)⋯ x_i_ℓ(t_ℓ) : t_1,…,t_ℓ>0 },  ^-:={y_i_1(t_1)⋯ y_i_ℓ(t_ℓ) : t_1,…,t_ℓ>0 }.

Let :=^+ ^-=^- ^+, where  is generated by α_i^∨(t) for i∈ and t>0.

The complete flag variety  of G is the space of all Borel subgroups B of G. Its totally positive part is

    :={u B^+ u^-1 : u∈^-}={u B^- u^-1 : u∈^+},

and its totally nonnegative part  is the closure of  in .

We now fix a subset ⊂ and define P_^+ to be the subgroup of G generated by B^+ and {y_j(t) : j∈,  t∈}. Let ^:={g P_^+ g^-1 : g∈ G} be a partial flag variety of G. (In the case J=∅, we have ^ =.) For each B∈, there is a unique P∈^ such that B⊂ P; denote by π^:→^ the natural projection map that sends B∈ to the P∈^ that contains it. We define the totally positive and totally nonnegative parts

    ^:=π^(),  ^:=π^().



Let G:=_3(). We may take a pinning with I = {1, 2} and

    e_1 = 0.80[ 0 1 0; 0 0 0; 0 0 0 ], 
    f_1 = 0.80[ 0 0 0; 1 0 0; 0 0 0 ], 
    x_1(t) = 0.80[ 1 t 0; 0 1 0; 0 0 1 ], 
    y_1(t) = 0.80[ 1 0 0; t 1 0; 0 0 1 ], α_1^∨(t) = 0.80[    t    0    0;    0 t^-1    0;    0    0    1 ],
    
    e_2 = 0.80[ 0 0 0; 0 0 1; 0 0 0 ], 
    f_2 = 0.80[ 0 0 0; 0 0 0; 0 1 0 ], 
    x_2(t) = 0.80[ 1 0 0; 0 1 t; 0 0 1 ], 
    y_2(t) = 0.80[ 1 0 0; 0 1 0; 0 t 1 ], α_2^∨(t) = 0.80[    1    0    0;    0    t    0;    0    0 t^-1 ].

Then B^+, B^-, and T are the subgroups of upper-triangular, lower-triangular, and diagonal matrices, respectively. Taking J:=∅, we can identify ^= with the space of complete flags in ^3. Explicitly, if g = [g_1  |  g_2 |  g_3]∈ G, then gB^+g^-1∈^ corresponds to the flag {0}⊂⟨ g_1⟩⊂⟨ g_1, g_2⟩⊂^3. One can check that the totally nonnegative part ^ consists of flags {0}⊂ V_1 ⊂ V_2 ⊂^3 where V_1 is spanned by a vector (v_1, v_2, v_3)∈^3 with v_1, v_2, v_3≥ 0, and V_2 is orthogonal to a vector (w_1, -w_2, w_3)∈^3 with w_1, w_2, w_3 ≥ 0. Hence we may identify ^ with the subset of ^6 of points (v_1, v_2, v_3, w_1, w_2, w_3) satisfying the (in)equalities

    v_1 + v_2 + v_3 = 1,    w_1 + w_2 + w_3 = 1,    v_1w_1 - v_2w_2 + v_3w_3 = 0,    v_1, v_2, v_3, w_1, w_2, w_3 ≥ 0.

<ref> implies that this region is homeomorphic to a 3-dimensional closed ball. Its cell decomposition (determined by which of the six coordinates are zero) is shown in <ref>.


  
  
  
  

  2pt
  0.6
  1
  
  0.5
  0.2

  



 §.§ Canonical basis


We now assume that G is simply laced. Let Y (respectively, X) be the free abelian group of homomorphisms of algebraic groups ^∗→ T (respectively, T→^∗), with standard pairing ·,·: Y× X→. Let  be the set of ł∈ X such that α_i^∨,ł≥ 0 for all i∈, and for such ł let (ł):={i∈ : α_i^∨,ł>0} be its support. 

For each ł∈, there is a unique irreducible G-module _ł with highest weight ł. Lusztig 3.1 defined a canonical basis  of _ł; see <cit.> for more details. For  in the projective space , we write >0 (respectively, ≥ 0) if the line  is spanned by a vector x∈_ł whose coordinates in the canonical basis  are all positive (respectively, nonnegative). We denote

    :={∈ : >0},  :={∈ : ≥0}.


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 §.§ Contractive flows

We recall a topological lemma from <cit.> that we used to show that various spaces appearing in total positivity (including the totally nonnegative part of a type A Grassmannian) are homeomorphic to closed balls.

A map :×^N→^N is called a contractive flow if the following conditions are satisfied:


  *  the map  is continuous;

  *  for all p∈^N and t_1,t_2∈, we have 0 p=p and t_1+t_2p=t_1t_2p; and

  *  for all p≠ 0 and t > 0, we have tp < p.


Here p denotes the Euclidean norm[In <cit.>, <ref> was proved more generally for an arbitrary norm · on ^N.] of p∈^N. For a subset K⊂^N and t∈, let tK denote { t p : p∈ K}.


Let ⊂^N be a smooth embedded submanifold of dimension d ≤ N, and :×^N→^N a contractive flow. Suppose that  is bounded and satisfies the condition

    t⊂   for t > 0.

Then the closure  is homeomorphic to a closed ball of dimension d, and ∖ is homeomorphic to a sphere of dimension d-1.




§ PROOF OF THEOREM 1


μ
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We first prove <ref> in the case that G is simply laced, in <ref>. We will then deduce the result for all G using the folding technique, in <ref>. Our arguments employ the vector field τ := ∑_i∈ I(e_i+f_i)∈, and

    :=exp(tτ) defined for t∈.

By <cit.>, we have ∈ for t>0.


In our earlier proof <cit.> that the totally nonnegative part of a type A Grassmannian is homeomorphic to a closed ball, we worked with a closely related vector field, which is a cyclically symmetric (or loop group) analogue of τ. We did so because exploiting cyclic symmetry appears to be necessary to show that the compactification of the space of electrical networks is homeomorphic to a ball, as we proved in <cit.>.




 §.§ Simply-laced case


ψ
Choose ł∈ such that (ł)=I∖ J. Then by 1.6, for each P∈^J, there is a unique line _P^ł in _ł that is stable under the action of P on _ł.  Moreover, the map P↦_P^ł is a smooth embedding of ^J into (_ł), which we denote by _J:^J→(_ł). Lusztig <cit.> showed that we may select λ so that the following hold:


  * we have P∈^J if and only if _P^ł>0; and

  * we have P∈^J if and only if _P^ł≥0.


Since ∈ for t>0, by <cit.> we have that τ acts on _ł as a regular and semisimple endomorphism over ℝ. Let v_0,v_1,…,v_N∈_ł be an eigenbasis of τ with eigenvalues _0 ≥_1 ≥⋯≥_N in . We have a smooth open chart

    ϕ:↪,

which sends =(_1,…,_N)∈ to the line spanned by v_0+_1v_1+…+_Nv_N. By <cit.> again, we have _k < _0 for 1≤ k≤ N, and the image of ϕ contains . Consider the sets ⊂⊂̋ defined by

    :=ϕ^-1(_J(^J))≃^J,  :̋=ϕ^-1(_J(^J))≃^J,

so that $̋ is compact and is the closure of.

Consider a map:×^N→^Ndefined fort∈andp=(p_1,…,p_N)∈^Nby

    t p:= (e^t(_1-_0)_1,…,e^t(_N-_0)_N).

We haveϕ(t p) = ·ϕ(p). We claim thatis a contractive flow. Indeed, parts (<ref>) and (<ref>) of <ref> hold for. By (<ref>), we see thatsatisfies the property

    t p ≤ C^-t p    for all t > 0 and p∈^N,

whereC>1is the minimum ofe^_0-_jfor1 ≤j ≤N. Thereforefsatisfies part (<ref>) of <ref>, as claimed.

Lusztig <cit.> showed that^_>0is an open submanifold of^. Anyg∈sendsinside<cit.>, which implies that (<ref>) holds. Therefore <ref> in the simply-laced case follows by applying <ref>.


Adopting the setup of <ref>, we have τ = 0.72[ 0 1 0; 1 0 1; 0 1 0 ]. The flow f contracts ^_≥ 0 onto the flag generated by the eigenvectors of τ, ordered by decreasing eigenvalue. Its coordinates (see <ref>) are v_1 = v_3 = w_1 = w_3 = 1/2+√(2), v_2 = w_2 = √(2)/2+√(2).


 §.§ General case, via folding

For generalG, by 1.6 there exists an algebraic groupof simply-laced type with pinning(,^+,^-,_i,_i;i∈)and an automorphismσ:→that extends to an automorphism of(also denotedσ), such thatG≅^σ. (HereS^σdenotes the set of fixed points ofσinS.) We have

    =()^σ,  =()^σ,  =()^σ,  =()^σ;

see 8.8. It follows from (<ref>) that

    ^=(^)^σ,  ^=(^)^σ,

where we identifywith the set of equivalence classes ofmodulo the action ofσ, andis defined to be the preimage ofunder the projection map→.

Note that∑_i∈I(e_i+f_i)∈, so∈Gfor allt∈. Thusacts on^=(^)^σ. The smooth embedding^↪restricts to a smooth embedding^↪. Therefore we may apply <ref> again to deduce that^is homeomorphic to a closed ball, completing the proof of <ref> in the general case.

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