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 with in Bruhat order, the Bruhat interval polytope is the convex hull of the points with . It is known that the Bruhat interval polytope is the moment map image of the Richardson variety . We say that is toric if the corresponding Richardson variety is a toric variety. We show that when is toric, its combinatorial type is determined by the poset structure of the Bruhat interval while this is not true unless is toric. We are concerned with the problem of when is (combinatorially equivalent to) a cube because is a cube if and only if is a smooth toric variety. We show that a Bruhat interval polytope is a cube if and only if is toric and the Bruhat interval is a Boolean algebra. We also give several sufficient conditions on v and w for 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v和w 与 按照Bruhat顺序,Bruhat区间多义词 是点的凸包 与 。已知Bruhat间隔多态性 是理查森系列的瞬间地图图像 。我们说如果对应的Richardson品种是复曲面是复曲面的品种。我们表明 是复曲面,其组合类型由Bruhat区间的波塞结构决定 虽然这不是真的,除非 是复曲面。我们担心什么时候 是(在组合上等效于)多维数据集,因为 是一个多维数据集,当且仅当 是光滑的复曲面品种。我们证明了Bruhat区间多态性 是一个多维数据集,当且仅当 是复曲面和Bruhat间隔 是布尔代数。我们也给几个充分条件v和w ^为 成为一个立方体。