Brick manifolds and toric varieties of brick polytopes

Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We will see that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the"subword complexes"of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg and Lange. These stories connect in a nice way: the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.


Introduction
The Bott-Samelson varieties were first defined by Raoul Bott and Hans Samelson in [2]. Bott-Samelson varieties are a twisted product of CP 1 's with a map into G/B. These varieties have been studied mostly in the case in which the map into G/B is birational. In this paper we study some fibers of this map when it is not birational to the image. We show that for some Bott-Samelson varieties this fiber is a toric variety. In order to do so we translate this problem into a purely combinatorial one in terms of subword complexes. These simplicial complexes ∆(Q, w) depend on a word Q in the generators of the Weyl group W of G and an element w ∈ W . They were defined by Allen Knutson and Ezra Miller in [12] to describe the geometry of determinantal ideals and Schubert polynomials. In [16], Vincent Pilaud and Christian Stump defined the brick polytope and realized certain subword complexes as the boundary of a polytope dual to the brick polytope. In [5] Cesar Ceballos, Jean-Philippe Labbé and Stump showed that for a nice family of words, the brick polytope is the cluster polytope and for the Weyl group of type A it is an associahedron. In Theorem 3.5 we prove that for the words Pilaud and Stump define as "realizable", a fiber of the Bott-Samelson map is the toric variety of the brick polytope. We then get a description of the toric variety of the associahedron in terms of flags arranged in a poset.
Actually the toric case is just a shadow of a more general situation. We prove in Theorem 3.4 that for any word Q and element w ∈ W the brick polytope is the moment polytope of a fiber of the Bott-Samelson variety. This motivates us to define the brick manifold as the fiber studied here. In this paper we show a very nice connection between subword complexes, brick polytopes and brick manifolds. In Theorem 3.5 we classify the toric brick manifolds. We end the paper with two results about brick manifolds: we exhibit a stratification of the brick manifolds dual to the subword complex in Theorem 3.6 and following [4], show that brick manifolds provide resolutions for Richardson varieties in Theorem 3.8.
1. Some definitions 1.1. Subword complexes. Let W be the Weyl group of a complex Lie group G with respect to a torus T , i.e., W is a crystallographic Coxeter group, and let S = {s i : i ∈ I} denote its generators.
Some notation: Let Q = (q 1 , . . . , q m ) be a word in S, i.e. an ordered sequence of elements of S. A subword J = (r 1 , . . . , r m ) of Q is a word obtained from Q by replacing some of its letters by −. There are a total of 2 |Q| subwords of Q. Given a subword J, we denote by Q \ J the subword with k-th entry equal to − if r k = − and equal to q k otherwise for k = 1, . . . , m. For example, J = (s 1 , −, s 3 , −, s 2 ) is a subword of Q = (s 1 , s 2 , s 3 , s 1 , s 2 ) and Q \ J = (−, s 2 , −, s 1 , −). Given a subword J we denote by J (k) the product of the leftmost k letters in J with − behaving as the identity, if k ≥ 1, and J (0) = 1. Definition 1.1. Let Q = (q 1 , . . . , q m ) be a word in S and w ∈ W . The subword complex ∆(Q, w) is the simplicial complex on the vertex set Q whose facets are the subwords F of Q such that the product (Q \ F ) (m) is a reduced expression for w.
Example 1.2. Let Q = (s 1 , s 2 , s 1 , s 2 , s 1 ) and w = s 1 s 2 s 1 , then the simplicial complex ∆(Q, w) is In order to make the reduced expression more explicit, we are labeling the faces by their complements. Definition 1.3. We define the Demazure product of a word Q inductively as follows: • Dem(empty word) = id In [12] the authors prove that ∆(Q, w) is a sphere if and only if Dem(Q) = w. In this paper we only consider such pairs. If in addition we assume Q is reduced, then ∆(Q, w) = {∅}, the (−1)-sphere, so we will not consider reduced Q in this paper.
1.2. Brick polytopes. Let ∆(W ) := {α s : s ∈ S} be the simple roots of W and let ∇(W ) := {ω i : s i ∈ S} be its fundamental weights. Pilaud and Stump define brick polytopes and study their properties in [16]. For them, the brick polytope is the convex hull of some conjugates of the fundamental weights of the Weyl group, one per each facet of the subword complex. Our definitions in this section are based on theirs, however we make the brick polytope be the convex hull of the brick vectors corresponding to all the faces in the subword complex such that the product of the complement is w. It turns out that the two definitions are equivalent as the proof of Theorem 3.4 exhibits.
Given a subword complex ∆(Q, w) with |Q| = m define the root function and the weight function Pilaud and Stump in [16] use the terminology realizing instead of root independent. They show that if Q is root independent, then the brick polytope is dual to the subword complex. One of the main theorems of this paper states that the brick manifold of a word Q is toric with respect to a maximal torus of the Lie group when Q is root independent.

Brick manifolds for SL n (C)
We start with the case G = SL n (C) both because it has beautiful combinatorial pictures and as a motivation to the general complex semi-simple Lie group case.
2.1. Brick polytopes in the SL n (C) case. The sorting network N Q of a word Q = (q 1 , . . . , q m ) consists of n horizontal lines (called the levels) and m vertical segments (called the commutators) drawn from left to right so that each commutator joins consecutive levels, no two commutators share a common endpoint, and if q k = s i then the k-th commutator connects levels i and i + 1. A brick of N Q is a connected component of its complement, bounded on the left by a commutator.
A pseudoline supported by N Q is a path on N Q traveling monotonically from left to right. A commutator of N Q is called a crossing between two pseudolines if it is crossed by the two pseudolines and it is called a contact otherwise. A pseudoline arrangement on N Q is a collection of n pseudolines such that each two have at most one crossing and no other intersection. Given a pseudoline arrangement supported by N Q , if we let J = (r 1 , . . . , r m ) be the subword of Q with r i = − precisely when there is a contact at the i-th commutator, then the product w = (Q \ J) (m) is an element of W and the pseudoline ending on the right at level i will start on the left at level w(i). We call such an arrangement a w-pseudoline arrangement. There is a one-to-one correspondence between faces J of ∆(Q, w) and (Q \ J) (m) -pseudoline arrangements supported by N Q . The pseudoline arrangement in the previous example corresponds to the subword J = (s 1 , −, −, −, s 1 ). In this setup, we have that w(J, j) is the characteristic vector of the pseudolines passing below the j-th brick of N Q . Moreover, the i-th coordinate of the brick vector B(J) is the number of bricks in N Q that lie above the i-th pseudoline with contacts J, and the brick polytope B(Q, w) is the following convex hull: For more pictures of brick polytopes of various Q and w, see [16].
A purpose of this paper is to assign geometry to these polytopes. To do so, we use the Bott-Samelson varieties which we define in the following section.

2.2.
Definition of Bott-Samelson varieties for SL n (C). Let G = SL n (C) and fix an ordered basis for C n . Let B be the subgroup SL n (C) consisting of upper triangular matrices with respect to this basis. We then get an ascending flag of B-invariant vector spaces e 1 ⊂ · · · ⊂ e 1 , . . . , e n , which we refer to as the base flag. Let T be the subgroup consisting of all diagonal matrices in G, so T is a maximal torus contained in B. Let P i be the minimal parabolic subgroup consisting of all matrices that are upper triangular except possibly at the position (i + 1, i). The quotient G/B is the flag variety, that is, the space We begin the definition of BS Q with an example.
In this example we have that More generally, if Q = (q 1 , . . . , q m ) then BS Q consists of a list of m + 1 flags where the zeroth one is the base flag and such that the k-th one agrees with the previous one except possibly on the k-th subspace V k . We can give a point in BS Q by giving the subspaces (V 1 , . . . , V m ) such that the incidence relations given by the flags hold. This carries a B-action, and the map BS Q mQ −→ G/B mapping the list to the last flag is B-equivariant.
Example 2.4. Continuing with the previous example, we have that We now define the main object of study in this paper.
Definition 2.5. Let Q = (q 1 , . . . , q m ) be a word in the generators of W and w = Dem(Q), then the brick manifold is the fiber m −1 Q (wB/B).
Note that the B-action restricted to T is just the extension to BS Q of T acting on C n by multiplication. There is a 1-1 correspondence between T -fixed points on corresponding to the subword J = (r 1 , . . . , r m ) is determined by deciding between = and = in each diamond using the rule: for Q = (q 1 , . . . , q m ), we pick "=" if r j = q j and " =" if r j = −. We illustrate this correspondence by an example. This correspondence motivates the relation between fibers of the map m Q and subword complexes. The main tool connecting brick polytopes with fibers of Bott-Samelson varieties will be moment maps of symplectic manifolds. We will discuss the symplectic manifold structure on general BS Q in Section 3.1. Namely, we will show that Bott-Samelson varieties are Hamiltonian symplectic manifolds with respect to the torus action described above. Therefore, a Bott-Samelson variety comes equipped with a moment map associated to the torus action. The image of this map is the moment polytope and it equals the convex hull of the images of the T -fixed points. Every toric variety is a Hamiltonian symplectic manifold with respect to the torus action. Moreover, if X is the toric variety associated to a Delzant polytope P then the image of the moment map is the polytope P .
In order to motivate latter sections and, more importantly, to be able to state the theorem connecting Bott-Samelson varieties and brick polytopes, we now describe the moment map of BS Q for the current case of interest, G = SL n (C). The moment map is a map where R ∇(W ) is the real span of the fundamental weights of W . Let π V : C n → V denote the orthogonal projection onto V and let P V be the corresponding matrix with respect to the basis e 1 , . . . , e n . Given p = (V 1 , . . . , V m ) ∈ BS Q the moment map is In the following section we give a precise statement about the relation between brick polytopes and Bott-Samelson varieties.   BS (s1,s2,s1,s2,s1) . The diagram below exhibits this correspondence. Each brick of the sorting network corresponds to a coordinate subspace of a point in the Bott-Samelson variety. Given a pseudoline arrangement supported in the sorting network of Q, the j-th subspace corresponding to the j-th brick is the coordinate subspace spanned by the e i where i ranges over those pseudolines passing below the j-th brick. Note then that two bricks share a crossing if and only if the corresponding coordinate spaces are equal. This will be proven in the theorem that follows. We have proved the if part of this theorem; however the only if part will follow from Theorem 3.5. The following corollary follows from the work of Pilaud and Santos in [15]. We define a Coxeter element c to be the product of all simple reflections in some order using each reflection only once. Define the c-sorting word of w to be the lexicographically first subword of c ∞ that is a reduced expression for w. Corollary 2.10. If Q is the concatenation of a word c representing a Coxeter element c and the c-sorting word for w 0 , then m −1 Q (w 0 B/B) is the toric variety of the associahedron as realized in [9] and in [15].
Example 2.11. The toric variety of the pentagon from example 2.2, i.e. the associahedron corresponding to the Coxeter element c = (s 1 , s 2 ), is

Brick manifolds in the general case
Let G be a complex semisimple Lie group, let B be a Borel subgroup of G, i.e., a maximal solvable subgroup, and T be the maximal torus contained in B. Let W be the Weyl group of G with generators S = {s 1 , . . . , s n }, which correspond to the simple roots ∆(W ) = {α 1 , . . . , α n }. Let P be a parabolic subgroup of G, i.e., a subgroup of G for which the quotient B/P is a projective algebraic variety; this condition is equivalent to P contains B. We denote by P i the minimal parabolic subgroup corresponding to s i , we then have that P i /B ∼ = CP 1 . The torus T acts on this quotient and this action has exactly two T -fixed points: one corresponding to the identity element and one corresponding to the generator s i . Definition 3.1. Let Q = (s i1 , . . . , s im ) be a word in the generators of W . Then the product P i1 × · · · × P im has an action of B m given by: The Bott-Samelson variety of Q is the quotient of the product of the P i 's by this action BS Q := (P i1 × · · · × P im )/B m .
Bott-Samelson varieties are smooth, irreducible and |Q|-dimensional algebraic varieties. They have a B action given by b · (p 1 , p 2 , . . . , p m ) = (b · p 1 , p 2 , . . . , p m ). and they come equipped with a natural B-equivariant map The image of this map is the opposite Schubert variety X w := BwB/B, where w = Dem(Q). In the case in which Q is reduced, this map is a resolution of singularities for X w , however in this paper we will study cases in which Q is not reduced.  Proof. We can write the fiber as the fibered product (wB/B) × X w BS Q , so by Kleiman's transversality theorem, see [10], we have that this fiber is a smooth variety of the desired dimension. Let N be the unipotent subgroup corresponding to B and N − the opposite unipotent subgroup. A consequence of the Bruhat decomposition of G/B is that if N w := N ∩ wN − w −1 , then N w · wB/B is a free dense orbit in X w . Since BS Q maps B-equivariantly to X w , the preimage of N w · wB/B is isomorphic to m −1 Q (wB/B) × N w . Since BS Q is irreducible, it follows that the brick manifold is irreducible.
3.1. Symplectic structure on Bott-Samelson varieties and brick manifolds. A reference for toric moment maps of coadjoint orbits is Chapter 5 of [8]. Let Pî be the maximal parabolic subgroup of G corresponding to the generators Sî := {s 1 , . . . ,ŝ i , . . . , s n }. Note that for G = SL n (C) each quotient G/Pî is a Grassmannian. Let K be the maximal compact subgroup of G. Then we can view G/Pî as a coadjoint orbit, i.e., a K-orbit through the fundamental weight ω i ∈ k * , where k is the Lie algebra of K. This interpretation gives us a symplectic structure on G/Pî with respect to the action of K such that the inclusion G/Pî ֒−→ k * is a moment map for the K-action. Then the composition is the moment map of G/Pî with respect to the torus action, where t is the Lie algebra of the torus. Moreover, the moment map for the diagonal T -action on a product G/Pî is the sum of the moment maps G/Pî −→ t * .
Given Q = (q 1 , . . . , q m ) we have a T -equivariant inclusion where ϕ = (ϕ 1 , . . . , ϕ m ) and the k-th component is This map makes BS Q a symplectic submanifold. The composition where µ k (Pk) = ω k , the fundamental weight corresponding to s k , and it maps a general element to a Weyl conjugate of this fundamental weight. Before we finish describing the maps µ k , we note that the moment map of BS Q is then Consider the fixed point (p 1 , . . . , p m ) in BS Q corresponding to the subword J of Q then under the moment map µ k each p j corresponds to either the reflection s ij if q j ∈ J or to the identity in W . In other words, p j corresponds to s ij if p j / ∈ B and to the identity in W otherwise. In conclusion we have that for J subword of Q and p J = the fixed point corresponding to J It then follows that 3.2. Moment polytopes of brick manifolds. We now state and prove the main results of the paper. Note that this theorem does not assume that the fiber is a toric variety so the relation between brick polytopes and brick manifolds is quite strong. The following theorem classifies toric brick manifolds. Proof. Note that dim(m −1 Q (wB/B)) ≤ dim(T ). However, if we have < then we can make the torus smaller and so without loss of generality we can assume the dimensions are equal. It suffices to show that T doesn't have generic stabilizer of positive dimension. This is true if and only if µ(m −1 Q (wB/B)) spans R n and this happens precisely when Q is root independent.
3.3. Stratification of the brick manifold. We give a stratification whose dual, in some sense, is the subword complex. We now introduce and recall some notation.