Toric Bruhat interval polytopes

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Abstract

For two elements v and w of the symmetric group Sn with vw in Bruhat order, the Bruhat interval polytope Qv,w is the convex hull of the points (z(1),,z(n))Rn with vzw. It is known that the Bruhat interval polytope Qv,w is the moment map image of the Richardson variety Xw1v1. We say that Qv,w is toric if the corresponding Richardson variety Xw1v1 is a toric variety. We show that when Qv,w is toric, its combinatorial type is determined by the poset structure of the Bruhat interval [v,w] while this is not true unless Qv,w is toric. We are concerned with the problem of when Qv,w is (combinatorially equivalent to) a cube because Qv,w is a cube if and only if Xw1v1 is a smooth toric variety. We show that a Bruhat interval polytope Qv,w is a cube if and only if Qv,w is toric and the Bruhat interval [v,w] is a Boolean algebra. We also give several sufficient conditions on v and w for Qv,w to be a cube.

Introduction

The permutohedron Permn1 is an (n1)-dimensional simple polytope in Rn defined by the convex hull of all points (u(1),u(2),,u(n))Rn for u in the symmetric group Sn on the set {1,2,,n}. It was first investigated by Schoute in 1911 (see [25] and references therein), and later Guilbaud and Rosenstiehl gave the name “permutohedron” in [11]. There are many works on generalizations of the notion of permutohedra such as generalized permutohedra in [18], graphicahedra in [2], Bruhat interval polytopes in [23], and so on.

On the other hand, the permutohedra have appeared in not only combinatorics but also the geometries of flag varieties. The flag variety Fn is a smooth projective variety which consists of chains {0}V1V2Vn=Cn of subspaces of Cn with dimCVi=i. It is known that the algebraic torus T=(C)n acts on Fn and there is a one-to-one correspondence between the set of fixed points of T on Fn and the elements of Sn. Moreover, with a symplectic form coming from the Plücker embedding of G/B, a moment map image of Fn is the permutohedron Permn1 (cf. Section 3), and the closure of the T-orbit of a generic point of Fn is known to be the permutohedral variety, which is the toric variety whose fan is the normal fan of Permn1 (see [14], [19]).

In this manuscript, we are studying Bruhat interval polytopes that were introduced by Tsukerman and Williams [23] in 2015. For two elements v and w of the symmetric group Sn with vw in Bruhat order, the Bruhat interval polytope Qv,w is defined to be the convex hull of all points (z(1),,z(n))Rn with vzw. Bruhat interval polytopes are one of the generalizations of permutohedra. Indeed, the Bruhat interval polytope Qe,w0 is the permutohedron Permn1 where e is the identity element and w0 is the longest element in Sn.

As in the case of permutohedra and flag varieties, Bruhat interval polytopes are related to Richardson varieties. For vw, the Richardson variety Xwv is defined to be the intersection of the Schubert variety Xw and the opposite Schubert variety w0Xw0v. It is an irreducible T-invariant subvariety of the flag variety Fn. It is known that there is a one-to-one correspondence between the set of fixed points of T on the Richardson variety Xwv and the set {z|vzw}, and it leads naturally to consider the convex hull of the points (z(1),,z(n))Rn with vzw. Note that the moment map image of the Richardson variety Xwv is the Bruhat interval polytope Qv1,w1 not Qv,w (see Lemma 3.1).

It should be noted that Bruhat interval polytopes Qv,w and Qv1,w1 are not combinatorially equivalent in general even though the Bruhat intervals [v,w] and [v1,w1] are isomorphic as posets. Moreover, even if two Bruhat interval polytopes Qv,w and Qv1,w1 are combinatorially equivalent, the fact that a subinterval [x,y][v,w] is realized as a face of Qv,w does not imply that the subinterval [x1,y1] is realized as a face of Qv1,w1 (see Remark 4.5).

We are particularly interested in Bruhat interval polytopes whose corresponding Richardson varieties are toric varieties with respect to the T-action. In fact, Xwv is a toric variety with respect to the T-action if and only if so is Xw1v1 (see Proposition 3.4). We call such a Bruhat interval polytope toric. It is known that dim Qv,wdimXw1v1=(w)(v) in general, where () denotes the length of a permutation, and we have that dimQv,w=(w)(v) if and only if Qv,w is toric (see Section 3). Toric Bruhat interval polytopes have a bunch of nice properties that an arbitrary Bruhat interval polytope does not have. Furthermore, those nice properties give us topological and geometric information of toric Richardson varieties.

Theorem 1.1 Theorem 5.1

A Bruhat interval polytope Qv,w is toric with respect to the T-action if and only if every subinterval [x,y] of [v,w] is realized as a face of Qv,w.

The above theorem implies that if Qv,w is toric, then its combinatorial type is determined by the poset structure of [v,w], and hence Qv,w and Qv1,w1 are combinatorially equivalent.

Combinatorial properties of a toric Bruhat interval polytope Qv,w give us some geometric information about the toric Richardson variety Xwv. The toric Richardson variety Xwv is smooth at a T-fixed point uB for vuw if and only if the vertex u of the Bruhat interval polytope Qv,w is simple, that is, the number of edges meeting at the vertex u is same as the dimension of the polytope Qv,w (see Proposition 4.8). Hence a Richardson variety is a smooth toric variety if and only if the corresponding Bruhat interval polytope is toric and a simple polytope.

Note that every toric Schubert variety is smooth and its corresponding Bruhat interval polytope is combinatorially equivalent to a cube (see [8], [13], [16]). But not every toric Bruhat interval polytope is a simple polytope and hence not every toric Richardson variety is smooth. See Fig. 1, Fig. 3. By restricting our attention to toric Bruhat interval polytopes, we get the following.

Proposition 1.2 Proposition 5.6

A toric Bruhat interval polytope is a simple polytope if and only if it is combinatorially equivalent to a cube.

It is well-known in toric topology that every smooth toric variety whose fan is the normal fan of a combinatorial cube has a sequence of CP1-fiber bundles, so called a Bott tower.1 Hence the above proposition implies that every smooth toric Richardson variety is a Bott manifold, that is a manifold in a Bott tower. We can further show the following whose geometric meaning is that a Richardson variety Xwv is a Bott manifold if and only if it is toric and the Bruhat interval [v,w] is a Boolean algebra.

Theorem 1.3 Theorem 5.7

A Bruhat interval polytope Qv,w is combinatorially equivalent to a cube if and only if it is toric and the Bruhat interval [v,w] is a Boolean algebra.

In the above theorem, we cannot drop the toric condition. There exist permutations v and w in Sn (n4) such that the Bruhat interval [v,w] is a Boolean algebra but the combinatorial type of the Bruhat interval polytope Qv,w is not a cube. See Fig. 8 and Section 6.

We also study necessary and sufficient conditions on v and w such that the Bruhat interval polytope Qv,w is toric or combinatorially equivalent to a cube. It was shown in [8] that a Bruhat interval polytope Qe,w is combinatorially equivalent to a cube if and only if w is a product of distinct simple transpositions. But the similar extension does not hold for general v. That is, even if there exist reduced expressions r(v) and r(w) for v and w such that the subword r(w)r(v) of r(w) consists of distinct simple transpositions, we cannot conclude that Qv,w is combinatorially equivalent to a cube (see Example 5.9) nor toric (see Example 7.10). So, it seems difficult to characterize v and w for which Qv,w is toric or combinatorially equivalent to a cube. We find some sufficient conditions on v and w for Qv,w to be toric, and give a necessary and sufficient condition on v and w for Qv,w to be a cube when v and w satisfy some special condition.

This manuscript is organized as follows. In Section 2, we compile some basic facts on posets, polytopes and toric varieties, and introduce Bruhat interval polytopes. In Section 3, we show that the Bruhat interval polytope Qv,w is the moment map image of the Richardson variety Xw1v1. In Section 4, we interpret combinatorial properties of Bruhat interval polytopes in terms of graphs defined by Bruhat intervals. Section 5 deals with properties of toric Bruhat interval polytopes and contains the proof of Theorem 1.3. In Section 6, we show that there are infinitely many non-simple toric Bruhat interval polytopes. In Section 7, we find some sufficient conditions on v and w for Qv,w to be toric, and then for such toric Bruhat interval polytopes Qv,w we find a sufficient condition to be a cube. In Section 8, we will find all coatoms of the Bruhat interval [v,w] when v and w satisfy some special condition, and then describe when Qv,w is a cube for such special cases.

Acknowledgments. The authors thank Akiyoshi Tsuchiya for his computer program to check Conjecture 5.11 for S5 and S6. Lee was supported by Institute for Basic Science IBS-R003-D1. Masuda was supported in part by JSPS Grant-in-Aid for Scientific Research 16K05152, 19K03472, and the HSE University Basic Research Program. Park was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Government of Korea (NRF-2018R1A6A3A11047606 and NRF-2016R1D1A1A09917654). This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).

Section snippets

Preliminaries

In this section, we prepare some notions and basic facts about posets and polytopes, and then introduce the notion of Bruhat interval polytopes.

Relation with Richardson varieties

In this section, we review the relation between Bruhat interval polytopes and Richardson varieties, and introduce the connection between combinatorial properties of Bruhat interval polytopes and geometric properties of Richardson varieties.

Let G=GLn(C), BG the set of upper triangular matrices, and TG the set of diagonal matrices. Let BG be the set of lower triangular matrices. Then T:=BB and B=w0Bw0. The manifold G/B can be identified with the flag variety Fn which is defined to beFn:={

Properties of Bruhat interval polytopes

In this section, we review some notations and facts about Bruhat interval polytopes and related graphs introduced in [16] and [23]. Then we interpret combinatorial properties of Bruhat interval polytopes using these graphs. Using this interpretation, we provide a proof of Proposition 3.4 which shows that Bruhat interval polytopes Qv,w and Qv1,w1 have the same dimension.

We first set up notations and terminologies related to digraphs (or directed graphs). A digraph is an ordered pair G=(V(G),E(G

Toric Bruhat interval polytopes

Recall that a Bruhat interval polytope Qv,w is toric if dimQv,w=(w)(v). In this section, we show that the combinatorial type of a toric Bruhat interval polytope Qv,w is determined by the poset structure of the interval [v,w] (see Theorem 5.1). Furthermore, a toric Bruhat interval polytope is simple if and only if it is combinatorially equivalent to a cube (see Corollary 5.13).

We already have seen in Theorem 4.2 that every face of a Bruhat interval polytope Qv,w can be realized by a

Product of Bruhat intervals

In this section, we will show that there are infinitely many non-simple toric Bruhat interval polytopes (see Proposition 6.4).

Let r be a non-negative integer. To a pair (x,y)Sp×Sq, we associate an element in Sp+q+r1, denoted by xry, as follows: express x=si1sikSp, y=sj1sjSq, and definexry:=(si1sik)(sj1+p+r1sj+p+r1)Sp+q+r1. Since i1,,ik are less than or equal to p1 while j1+p+r1,,j+p+r1 are greater than or equal to p, xry is well-defined, that is, independent of the

Conditions on v and w for Qv,w to be toric

In this section, we first find some sufficient conditions on v and w for Qv,w to be toric, and then find a sufficient condition for such a toric Bruhat interval polytope Qv,w to be a cube.

It was shown in [12, §5 and §6] that Qv,w is toric (in fact, a cube) if v=[a1,,an1,n] and w=[n,a1,,an1] or v=[1,b2,,bn] and w=[b2,,bn,1]. In these cases,w=vsn1sn2s1and(w)(v)=n1,w=vs1s2sn1and(w)(v)=n1. These examples motivate us to study the following case:w=vsj1sj2sjm where (w)(v)=m and j

Finding all coatoms in some special cases

We already have seen that studying the combinatorics of a toric Bruhat interval polytope Qv,w is closely related to observing atoms and coatoms of the interval [v,w]. Indeed, we proved in Proposition 5.10 that if Qv,w is toric and either v or w is a simple vertex, then the whole polytope Qv,w is simple. In this section, we will find all coatoms in some special cases. We first find a necessary and sufficient condition for w(i,j) to be a coatom of [v,w] when w=vs(1,n1), and then conclude that

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