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Toric Bruhat interval polytopes
Journal of Combinatorial Theory Series A ( IF 1.1 ) Pub Date : 2020-12-18 , DOI: 10.1016/j.jcta.2020.105387
Eunjeong Lee , Mikiya Masuda , Seonjeong Park

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.



中文翻译:

Toric Bruhat区间多态性

对于对称群的两个元素vw小号ñvw 按照Bruhat顺序,Bruhat区间多义词 vw 是点的凸包 ž1个žñ[Rñvžw。已知Bruhat间隔多态性vw 是理查森系列的瞬间地图图像 Xw-1个v-1个。我们说vw如果对应的Richardson品种是复曲面Xw-1个v-1个是复曲面的品种。我们表明vw 是复曲面,其组合类型由Bruhat区间的波塞结构决定 [vw] 虽然这不是真的,除非 vw是复曲面。我们担心什么时候vw 是(在组合上等效于)多维数据集,因为 vw 是一个多维数据集,当且仅当 Xw-1个v-1个是光滑的复曲面品种。我们证明了Bruhat区间多态性vw 是一个多维数据集,当且仅当 vw 是复曲面和Bruhat间隔 [vw]是布尔代数。我们也给几个充分条件vw ^vw 成为一个立方体。

更新日期:2020-12-20
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