Emmanuel Tsukerman ; Lauren Williams - Bruhat interval polytopes

dmtcs:2507 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) - https://doi.org/10.46298/dmtcs.2507
Bruhat interval polytopesArticle

Authors: Emmanuel Tsukerman 1; Lauren Williams 1

  • 1 Department of Mathematics [Berkeley]

Let $u$ and $v$ be permutations on $n$ letters, with $u$ &le; $v$ in Bruhat order. A <i>Bruhat interval polytope</i> $Q_{u,v}$ is the convex hull of all permutation vectors $z=(z(1),z(2),...,z(n))$ with $u$ &le; $z$ &le; $v$. Note that when $u=e$ and $v=w_0$ are the shortest and longest elements of the symmetric group, $Q_{e,w_0}$ is the classical permutohedron. Bruhat interval polytopes were studied recently in the 2013 paper “The full Kostant-Toda hierarchy on the positive flag variety” by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety. In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality description and a dimension formula for Bruhat interval polytopes, and prove that every face of a Bruhat interval polytope is a Bruhat interval polytope. A key tool in the proof of the latter statement is a generalization of the well-known lifting property for Coxeter groups. Motivated by the relationship between the lifting property and $R$-polynomials, we also give a generalization of the standard recurrence for $R$-polynomials.


Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Imported on: November 21, 2016
Keywords: Bruhat order,symmetric group,$R$-polynomials,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Graduate Research Fellowship Program; Funder: National Science Foundation; Code: 1106400
  • CAREER: Cluster algebras, total positivity, and physical combinatorics; Funder: National Science Foundation; Code: 1049513

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