Toric Bruhat interval polytopes
Introduction
The permutohedron is an -dimensional simple polytope in defined by the convex hull of all points for u in the symmetric group on the set . 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 is a smooth projective variety which consists of chains of subspaces of with . It is known that the algebraic torus acts on and there is a one-to-one correspondence between the set of fixed points of on and the elements of . Moreover, with a symplectic form coming from the Plücker embedding of , a moment map image of is the permutohedron (cf. Section 3), and the closure of the -orbit of a generic point of is known to be the permutohedral variety, which is the toric variety whose fan is the normal fan of (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 with in Bruhat order, the Bruhat interval polytope is defined to be the convex hull of all points with . Bruhat interval polytopes are one of the generalizations of permutohedra. Indeed, the Bruhat interval polytope is the permutohedron where e is the identity element and is the longest element in .
As in the case of permutohedra and flag varieties, Bruhat interval polytopes are related to Richardson varieties. For , the Richardson variety is defined to be the intersection of the Schubert variety and the opposite Schubert variety . It is an irreducible -invariant subvariety of the flag variety . It is known that there is a one-to-one correspondence between the set of fixed points of on the Richardson variety and the set , and it leads naturally to consider the convex hull of the points with . Note that the moment map image of the Richardson variety is the Bruhat interval polytope not (see Lemma 3.1).
It should be noted that Bruhat interval polytopes and are not combinatorially equivalent in general even though the Bruhat intervals and are isomorphic as posets. Moreover, even if two Bruhat interval polytopes and are combinatorially equivalent, the fact that a subinterval is realized as a face of does not imply that the subinterval is realized as a face of (see Remark 4.5).
We are particularly interested in Bruhat interval polytopes whose corresponding Richardson varieties are toric varieties with respect to the -action. In fact, is a toric variety with respect to the -action if and only if so is (see Proposition 3.4). We call such a Bruhat interval polytope toric. It is known that dim in general, where denotes the length of a permutation, and we have that if and only if 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 is toric with respect to the -action if and only if every subinterval of is realized as a face of .
The above theorem implies that if is toric, then its combinatorial type is determined by the poset structure of , and hence and are combinatorially equivalent.
Combinatorial properties of a toric Bruhat interval polytope give us some geometric information about the toric Richardson variety . The toric Richardson variety is smooth at a -fixed point uB for if and only if the vertex u of the Bruhat interval polytope is simple, that is, the number of edges meeting at the vertex u is same as the dimension of the polytope (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 -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 is a Bott manifold if and only if it is toric and the Bruhat interval is a Boolean algebra.
Theorem 1.3 Theorem 5.7 A Bruhat interval polytope is combinatorially equivalent to a cube if and only if it is toric and the Bruhat interval is a Boolean algebra.
In the above theorem, we cannot drop the toric condition. There exist permutations v and w in () such that the Bruhat interval is a Boolean algebra but the combinatorial type of the Bruhat interval polytope 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 is toric or combinatorially equivalent to a cube. It was shown in [8] that a Bruhat interval polytope 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 and for v and w such that the subword of consists of distinct simple transpositions, we cannot conclude that 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 is toric or combinatorially equivalent to a cube. We find some sufficient conditions on v and w for to be toric, and give a necessary and sufficient condition on v and w for 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 is the moment map image of the Richardson variety . 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 to be toric, and then for such toric Bruhat interval polytopes we find a sufficient condition to be a cube. In Section 8, we will find all coatoms of the Bruhat interval when v and w satisfy some special condition, and then describe when is a cube for such special cases.
Acknowledgments. The authors thank Akiyoshi Tsuchiya for his computer program to check Conjecture 5.11 for and . 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 , the set of upper triangular matrices, and the set of diagonal matrices. Let be the set of lower triangular matrices. Then and . The manifold can be identified with the flag variety which is defined to be
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 and have the same dimension.
We first set up notations and terminologies related to digraphs (or directed graphs). A digraph is an ordered pair
Toric Bruhat interval polytopes
Recall that a Bruhat interval polytope is toric if . In this section, we show that the combinatorial type of a toric Bruhat interval polytope is determined by the poset structure of the interval (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 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 , we associate an element in , denoted by , as follows: express , , and define Since are less than or equal to while are greater than or equal to p, is well-defined, that is, independent of the
Conditions on v and w for to be toric
In this section, we first find some sufficient conditions on v and w for to be toric, and then find a sufficient condition for such a toric Bruhat interval polytope to be a cube.
It was shown in [12, §5 and §6] that is toric (in fact, a cube) if and or and . In these cases, These examples motivate us to study the following case:
Finding all coatoms in some special cases
We already have seen that studying the combinatorics of a toric Bruhat interval polytope is closely related to observing atoms and coatoms of the interval . Indeed, we proved in Proposition 5.10 that if is toric and either v or w is a simple vertex, then the whole polytope is simple. In this section, we will find all coatoms in some special cases. We first find a necessary and sufficient condition for to be a coatom of when , and then conclude that
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