Toric matrix Schubert varieties and root polytopes ( extended abstract )

Start with a permutation matrix π and consider all matrices that can be obtained from π by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety Xπ . We characterize when the ideal definingXπ is toric (with respect to a 2n−1-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. We also show that these complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of β-Grothendieck polynomials. Subword complexes were introduced by Knutson and Miller in 2004, who showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations. Résumé. En partant d’une matrice de permutation π, considérons toutes les matrices qui peuvent être obtenues à partir de π en effectuant des opérations de ligne vers le bas et des opérations de colonne vers la droite ; l’adhérence de cet ensemble donne la variété Schubert de matrices Xπ . Nous caractérisons la situation où l’idéal définissant Xπ est torique et étudions le polytope associé de sa projectivisation. Nous construisons des triangulations régulières de ces polytopes et nous montrons qu’elles sont des réalisations géométriques d’une famille de complexes de sous-mots. Nous montrons également que ces complexes peuvent être réalisés géométriquement par des triangulations régulières de polytopes de racines. Cela implique qu’une famille de polynômes β-Grothendieck sont des cas particuliers de formes réduites dans l’algèbre de subdivision de polytopes des racines. On peut aussi écrire le volume et la série d’Ehrhart des polytopes des racines en termes de β-Grothendieck polynômes. Les complexes de sous-mots ont été introduits par Knutson et Miller en 2004, qui ont soulevé la question de leurs réalisations polytopales.


Introduction
This is an extended abstract based on Escobar and Mészáros (2015a) and Escobar and Mészáros (2015b).We study the geometry of matrix Schubert varieties and give geometric realizations of a family of subword complexes.Matrix Schubert varieties were introduced by Fulton (1992) to study the degeneraci loci of flagged vector bundles.Knutson and Miller (2005) showed that Schubert polynomials are multidegrees of matrix Schubert varieties.Knutson andMiller (2004, 2005) introduced subword complexes to illustrate the combinatorics of Schubert polynomials and determinantal ideals, building up on the work of Fomin and Kirillov (1994); Bergeron and Billey (1993).Knutson and Miller proved that any subword complex is homeomorphic to a ball or a sphere and asked about their geometric realizations.
Given a matrix Schubert variety X π , it can be written as X π = Y π × C q (where q is maximal possible).Our main results are as follows.We characterize when Y π is toric (with respect to a (C * ) 2n−1 -action) and study the polytope Φ(P(Y π )) corresponding to its projectivization.We construct a regular triangulation of Φ(P(Y π )), induced from a degeneration to a root polytope, which we show are geometric realizations of a family of subword complexes.The following papers have partially answered the question about the geometric realization of spherical subword complexes: Stump (2011); Ceballos (2012); Pilaud and Pocchiola (2012); Pilaud and Santos (2012); Serrano and Stump (2012); Ceballos et al. (2014); Bergeron et al. (2015).This submission is based on Escobar and Mészáros (2015a,b), where we give the first realizations of a family of subword complexes which are homeomorphic to balls.
The roadmap of this paper is as follows.In Section 2 we define matrix Schubert varieties X π and calculate the moment polytope Φ(P(Y π )) of the projectivization of Y π .In Section 3 we characterize when Y π is toric and construct a regular triangulation of Φ(P(Y π )).In Section 4 we define subword complexes, give geometric realizations of subword complexes homeomorphic to balls, and show how to express the volume and Ehrhart series of root polytopes in terms of Grothendieck polynomials.Finally, in Section 5 we give canonical triangulations of Φ(P(Y π )) and show they are geometric realizations of pipe dream complexes for all π such that Y π is toric.

Matrix Schubert varieties
Given a matrix Schubert variety X π we define a variety Y π → X π and characterize for which π, the variety Y π is toric using the diagram of π.For such π, we construct a regular triangulation of its corresponding polytope, which we show is a geometric realization of a family of subword complexes, see Proposition 3.1 and Theorem 5.3.
Let M n denote n × n matrices over C, B + denote upper triangular invertible n × n matrices and B − denote lower triangular invertible n × n matrices.We let π ∈ S n denote both a permutation and its corresponding permutation matrix, where its (i, j)-th entry is The multiplication on the left by matrices in B − corresponds to downward row operations and multiplication on the right by matrices in B + corresponds to rightward column operations.This multiplication gives a left action of Definition 2.1 The matrix Schubert variety of π is X π := B − πB + , i.e. the Zariksi closure of its Fulton studied this affine variety in Fulton (1992).We summarize some of his results here.
Theorem 2.2 (Fulton, 1992, Proposition 3.3) The matrix Schubert variety X π is an irreducible variety of dimension n 2 − (π) defined as a scheme by the equations r Some of these inequalities are implied by others, and Fulton described the minimal set of rank conditions.
Definition 2.3 The (Rothe) diagram of a permutation π is the collection of boxes D(π) = {(π j , i) : i < j, π i > π j }.It can be visualized by considering the boxes left in the n × n grid after we cross out the boxes appearing south and east of each 1 in the permutation matrix for π.Definition 2.4 Fulton's essential set Ess(π) is the set consisting of the south-east corners of D(π).
Theorem 2.5 (Fulton, 1992, Lemma 3.10) The ideal defining the variety X π is generated by the equations r M (a, b) ≤ r π (a, b) for all (a, b) ∈ Ess(π).
We now define some regions inside the (n × n)-grid and some varieties corresponding to these regions, including Y π .
Definition 2.6 The dominant piece, denoted dom(π), of a permutation π is the connected component of the diagram of π containing the box (1, 1), or empty if π(1) = 1.
See Figure 2 for an example.Definition 2.8 Given a permutation π, let Y π be the projection of X π onto the entries inside L(π) and let V π be the projection onto the entries not north-west of any box of D(π).
Theorem 2.5 implies that the entries in V π are free in X π and thus V π ∼ = C q , where q is the number of boxes in the region defining V π , and that X π = Y π × V π .This, together with Theorem 2.2, imply that dim(Y π ) = |L (π)| and that Y π is irreducible.
Let T n consist of n × n diagonal invertible matrices.The action defined in Equation (1) restricts to a (T n ×T n )-action on M n .This yields a (T n ×T n )-action on X π with Stab(T 2n ) = {(a•I, a•I) : a ∈ C * }, as well as an action on Y π and V π .In Theorem 3.3 we characterize the π for which Y π is a toric variety with respect to T 2n /Stab(T 2n ) in terms of the shape of L (π).In other words, we characterize the π such that Y π has a dense T 2n -orbit.We denote the quotient T 2n /Stab(T 2n ) by T 2n−1 .Note that X π and Y π are normal varieties by Fulton's realizations in Fulton (1992) as subvarieties of Schubert varieties, which are normal by De Concini and Lakshmibai (1981); Ramanan and Ramanathan (1985).
Since Y π is an irreducible variety and toric varieties are also irreducible, in order to show that Y π is a toric variety with respect to T 2n−1 , it suffices to show that it has the same dimension as some To compute the dimension of the cone Φ(Y π ), we start by describing the cone Φ(X π ) corresponding to a T 2n -orbit of a general point q in X π ; without loss of generality q = (1, . . ., 1).The orbit T 2n • q is the Zariski closure of the image of a map ϕ : where ϕ(t) = (t a (1,1) q (1,1) , . . ., t a (n,n) q (n,n) ) and Φ(X π ) is the cone spanned by the exponents a (i,j) of the monomials.Notice that the exponents are x i −y j , where the x i are the standard basis for R n × 0, and the y j are the standard basis for 0 × R n , because if A and B are the diagonal matrices with diagonal entries (a 1 , . . ., a n ) and (b 1 , . . ., b n ), respectively, then for any matrix M the (i, j)-th The variety X π is a cone, meaning that for any z ∈ X π and c ∈ C, we have that cz ∈ X π .We can therefore projectivize it, that is, we can take the projective variety and the same is true for Y π .In this paper we study the moment (ii) polytope Φ(P(Y π )) of the projectivization of Y π .This polytope is the convex hull of (x i − y j ) for (i, j) inside L(π).The next section studies the properties of the moment polytopes Φ(P(Y π )).
3 Understanding the polytope Φ(P(Y π )) In this section we describe the polytope the moment polytope of the projectivization of Y π .This polytope is a root polytope, since its vertices are positive roots of type A n−1 .We will encounter slightly different root polytopes (acyclic root polytopes) in Section 4.2 when describing the realizations of a family of pipe dream complexes.In Section 5 we will give a map that transforms the root polytope Φ(P(Y π )) into an acyclic root polytope for π = 1π with π dominant.We set our notation for the first root polytopes now. (i) The reason we use the word moment for these convex objects is because they arise in the context of symplectic and pre-symplectic geometry.For readers interested in the connection, we refer them to Cannas da Silva (2001Silva ( , 2003)); Berline and Vergne (2011). (ii) See footnote (i).

Root polytopes and their triangulations
A root polytope (of type A n−1 ) is the convex hull of some of the points e i − e j for 1 ≤ i < j ≤ n.Given a graph G on the vertex set [n] we associate to it the root polytope (2) Note that for every π ∈ S n we have that L(π) is a skew Ferrers diagram.Given a skew Ferrers diagram D with r rows and c columns, label its rows by 1, 2, . . ., r from top to bottom and its columns by 1, 2, . . ., c from left to right.Define (3) Note that an edge (x i , y j ) ∈ G D yields the vertex e i − e r+j of the root polytope Given a drawing of a graph G so that its vertices v 1 , . . ., v n are arranged in this order on a horizontal line, and its edges are drawn above this line, we say that G is noncrossing if it has no edges (v i , v k ) and (v j , v l ) with i < j < k < l.A vertex v i of G is said to be nonalternating if it has both an incoming and an outgoing edge; it is called alternating otherwise.The graph G is alternating if all its vertices are alternating.
Since being noncrossing depends on the drawing of the graph it is essential that we set a way to draw G D .For the purposes of this paper the vertices of G D are drawn from left to right in the following order: x r , . . ., x 1 , y c , . . ., y 1 .We call the triangulation of Q G D given in Lemma 3.1 the noncrossing alternating triangulation, or NAT for short.These triangulations are closely related to the triangulations appearing in Gelfand et al. (1997) and Cellini and Marietti (2014).Recall that a triangulation of a polytope P is regular if there exists a concave piecewise linear function f : P → R such that the regions of linearity of f are the maximal simplices in the triangulation.
Proposition 3.1 For a skew diagram D, the NAT triangulation of Q G D described in Lemma 3.1 is a regular triangulation.

Characterizing when Y π is a toric variety
Now we are ready to use the above lemmas in order to characterize when Y π is a toric variety.
Lemma 3.2 Given a skew diagram D with r rows and c columns, for which G D has k components, the dimension of Theorem 3.3 Y π is a toric variety with respect to the T 2n−1 -action if and only if L (π) consists of disjoint hooks that do not share a row or a column with each other.
Proof: We have that dim(Y π ) = |L (π)|.Lemma 3.2 yields that the dimension of Φ(P(Y π )) equals |L (π)| − 1 if and only if L (π) consists of disjoint hooks that do not share a row or a column with each other.This suffices to prove the theorem. 2 A dominant permutation is one for which its diagram has empty dominant piece and is in the shape of a partition.An immediate corollary of Theorem 3.3 is the following.

On geometric realizations of subword complexes
In this section we give some geometric realizations of subword complexes homeomorphic to balls.In Section 4.1 we show that the NAT triangulations studied in the previous section geometrically realize certain subword complexes.In Section 4.2 we use acyclic root polytopes to give geometric realizations for pipe dream complexes of permutations π = 1π with π dominant.
The symmetric group S n is generated by the adjacent transpositions s 1 , . . ., s n−1 , where s i transposes i ↔ i + 1.Let Q = (q 1 , . . ., q m ) be a word in {s 1 , . . ., s n−1 }.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 the product of the letters in J, from left to right, with − behaving as the identity.

Realization by NAT triangulations
Given a permutation π ∈ S n , let L(π) be the mirror image of the skew shape L(π).Fill in the boxes of L(π) with transpositions starting with s 1 , s 2 , . . . on the first column, s 2 , s 3 , . . . on the second column, and so on.Let Q(L(π)) be the word given by reading the transpositions in the boxes of L(π) from left to right and from bottom to top.Let P (π) = L(π) − B(π) where B(π) is as follows.In each connected part of L(π) draw the lowestmost path from its top left box to its bottommost rightmost box.These boxes constitute B(π).Let p(π) be the permutation obtained from reading the transpositions in the boxes of P (π) from left to right and from bottom to top.See Figure 4. Theorem 4.2 For π ∈ S n , the noncrossing alternating triangulation of the root polytope Q G L(π) is a geometric realization of the subword complex ∆(Q(L(π)), p(π)).

Realizations for pipe dream complexes of dominant permutations by acyclic root polytopes
In this section we give a geometric realization for a different family of subword complexes using acyclic root polytopes.We show that the pipe dream complex P D(π) of a permutation π = 1π , with π dominant, can be geometrically realized as the canonical triangulation of an acyclic root polytope P(T (π)).
These polytopes are closely related to the root polytopes of Section 3.1.We begin by defining acyclic root polytopes.Let G be an acyclic graph on the vertex set , the cone associated to G; and where Φ + = {e i − e j | 1 ≤ i < j ≤ n + 1} is the set of positive roots of type A n .The acyclic root polytope P(G) associated to the acyclic graph G is Theorem 4.3 Mészáros (2011) Let T 1 , . . ., T k be the noncrossing alternating spanning trees of the directed transitive closure of the acyclic graph G. Then P(T 1 ), . . ., P(T k ) are top dimensional simplices in a regular triangulation of P(G) called the canonical triangulation.
When P D(π) is not a ball, it is usually a cone over a list of its vertices, namely those that are in all its facets.Let cone(π) denote the set of vertices of P D(π) that are in all its facets.We define the core of π to be the restriction of P D(π) to the set of vertices not in cone(π).Then P D(π) is obtained from its core by iteratively coning core(π) over the vertices in cone(π).Translating to pipe dream complexes, the core is the restriction to the entries in the n × n matrix that are a cross in some reduced pipe dream for π.We refer to the region itself as the core region, and denote it by cr(π).Let π = 1π , where π is dominant.Denote by S(π) the subword complex which is the core(π) coned over the vertex of P D(π) corresponding to the entry (1, 1).Denote the region which is the union of (1, 1) and cr(π) by R(π).
The SE boundary of the core region starting from the southwest (SW) corner of it to the northeast (NE) corner can be described as a series of east (E) and north (N) steps.We construct the graph T (π) by looking at the E and N steps bounding the SE boundary of N W (π) − Ess(π).Let A be the set consisting of all the N steps together with the E steps that do not bound a box in Ess(π).Suppose |A| = m, as we travel the SE boundary from the SW corner, we label, in order, the E steps and N steps in A with the elements of the sequence (α 1 , . . ., α m ).For the E steps that we did not assign an α i , we consider their label to be the α i assigned to the N step directly preceding them.See Figure 6 for an example of the labelling.
Given a diagram of a permutation there are two reduced pipe dreams for π with special names: the bottom reduced pipe dream of π obtained by aligning the diagram to the left and replacing the boxes with crosses, and the top reduced pipe dream of π obtained in a similar fashion, but by aligning the diagram up.See Figure 5 for an example of the bottom reduced pipe dream.Consider the bottom reduced pipe dream drawn inside R(π) and with elbows replaced by dots.Drop these dots south.Define T (π) to be the tree with vertices V = {α 1 , . . ., α m } such that there is an edge between vertices α i and α j with i < j if there is a dot in the entry in the column of the E step labeled α i i and in the row of the N step labeled α i j.See Figure 6 for an example.We now mention some of the corollaries to this Theorem.
Corollary 4.5 The volume of the root polytope P(T (1π )), for π dominant is equal to the number of reduced pipe dreams of 1π .
Recall that for a polytope P ⊂ R N , the t th dilate of P is tP = {(tx 1 , . . ., tx N ) | (x 1 , . . ., x N ) ∈ P}.The number of lattice points of tP, where t is a nonnegative integer and P is a convex polytope, is given by the Ehrhart function i(P, t).If P has integral vertices then i(P, t) is a polynomial.
In the spirit of Knutson and Miller (2004); Fomin and Kirillov (1994), the double β-Grothendieck polynomial G β w (x, y) for w ∈ S n , where x = (x 1 , . . ., x n−1 ) and y = (y 1 , . . ., y n−1 ) is β codim P D(w) F (P ) wt x,y (P ), where Pipes(w) is the set of all pipe dreams of w (both reduced and nonreduced), F (P ) is the interior face in P D(w) labeled by the pipe dream P , codim P D(w) F (P ) denotes the codimension of F (P ) in P D(w) and wt x,y (P ) = (i,j)∈cross(P ) (x i − y j ), with cross(P ) being the set of positions where P has a cross.Note that in the product (i,j)∈cross(P ) (x i − y j ) we are assuming a certain labeling of rows and columns.Conventionally, rows are labeled increasingly from top to bottom and columns are labeled increasingly from left to right.
Corollary 4.6 Let π = 1π , where π is a dominant permutation.Then 5 Degeneration of moment polytopes into acyclic root polytopes In this section we explain how to map the root polytope Φ(P(Y π )) to the acyclic root polytope P(T (π)).
We then use this map to triangulate Φ(P(Y π )) based on the triangulation of P(T (π)).
Proof: Consider the linear map from Φ(P(Y π )) → P(T (π)) that is the composition of the maps K and L, where L is the map L(x i ) = −e j , where α j is the label of step N on row i, and L(y i ) = 0 if (a, i) ∈ Ess(π) for some a, −e j where α j is the label of step E on column i, and K is the map given by K(y j ) = x i if (i, j) ∈ Ess(π), y j if there is no a such that (a, j) ∈ Ess(π).
K(x i ) = x i .
See Figure 8 for an example of these maps.Then this maps Φ(P(Y π )) to P(T (π)). 2 The degeneration L • K : Φ(P(Y π )) → P(T (π)) consists of contracting the face of Φ(P(Y π )) corresponding to Ess(π) to a point and moving this point to the origin while tweaking the vertices of Φ(P(Y π )) that are of the form 1 2 (x i − y j ) where (i, j) is north of an entry of Ess(π) and not in dom(π).
Denote by C(π) the core of the pipe dream complex P D(π) and by C i (π) the core C(π) coned over i times.
Theorem 5.3 The canonical triangulation of Φ(P(Y π )), for π = 1π , with π dominant, is a geometric realization of C |Ess(π)|+1 (π).Using the characterization of toric Y π of Theorem 3.3, one can extend this geometric realization to realizations of pipe dream complexes for all π such that Y π is toric.

Lemma 3. 1
Given a skew diagram D for which G D has k components, the root polytope Q G D = F Q F , where the union runs over all noncrossing alternating spanning forests of G D with |V (G D )| − k edges and the simplices Q F are interior disjoint and of the same dimension as Q G D .

Fig. 3 :
Fig.3: For D the unshaded region, GD is disconnected.In this case, D equals D together with the shaded square.

Fig. 5 :
Fig. 5: The bottom reduced pipe dream for [14523] obtained by aligning the diagram to the left.

Fig. 6 :
Fig.6: On the left we have N W (π) − Ess(π) for π = [14523] with its SW boundary labelled by (α1, . . ., αm) and the bottom reduced pipe dream drawn inside N W (π) − Ess(π) with dots instead of elbows.We then drop the dots to the south to get the edges of T (π), which is depicted on the right.