Toric Bruhat interval polytopes

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:179

Article

Lee, Eunjeong¹  Masuda, Mikiya²  Park, Seonjeong³ 

[1] Inst Basic Sci IBS, Ctr Geometry & Phys, Pohang 37673, South Korea

[2] Osaka City Univ, Grad Sch Sci, Dept Math, Sumiyoshi Ku, Sugimoto, Osaka 5588585, Japan

[3] Korea Adv Inst Sci & Technol, Dept Math Sci, Daejeon 34141, South Korea

关键词: Bruhat interval polytopes;    Richardson varieties;    Toric varieties;   

DOI  :  10.1016/j.jcta.2020.105387

来源: Elsevier

【 摘 要 】

For two elements v and w of the symmetric group S-n with v <= w in Bruhat order, the Bruhat interval polytope Q(v,w) is the convex hull of the points (z(1), ..., z(n)) is an element of R-n with v <= z <= w. It is known that the Bruhat interval polytope Q(v,w) is the moment map image of the Richardson variety X-w-1(v-1). We say that Q(v,w) is toricif the corresponding Richardson variety X-w-1(v-1) is a toric variety. We show that when Q(v,w) is toric, its combinatorial type is determined by the poset structure of the Bruhat interval [v, w] while this is not true unless Q(v,w) is toric. We are concerned with the problem of when Q(v,w) is (combinatorially equivalent to) a cube because Q(v,w) is a cube if and only if X-w-1(v-1) is a smooth toric variety. We show that a Bruhat interval polytope Q(v,w) is a cube if and only if Q(v,w) is toric and the Bruhat interval [v, w] is a Boolean algebra. We also give several sufficient conditions on vand wfor Q(v,w) to be a cube. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free