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  1. Home > Articles & Issues >
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  3. Bruhat interval poly ...
Conference paper

Bruhat interval polytopes

Emmanuel Tsukerman (1), Lauren Williams (1)
(1) Department of Mathematics [Berkeley]
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Imported on
November 21, 2016
Published on
December 31, 2014
Last modified on
March 31, 2025
Volume
DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Proceedings
DOI
10.46298/dmtcs.2507
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Attribution 4.0 International (CC BY 4.0)
Graduate Research Fellowship Program
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Bruhat interval polytopes

Emmanuel Tsukerman (1), Lauren Williams (1)
(1) Department of Mathematics [Berkeley]
Abstract
English
Let $u$ and $v$ be permutations on $n$ letters, with $u$ ≤ $v$ 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 $u$ ≤ $z$ ≤ $v$. 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.
French
Soient $u$ et $v$ des permutations sur $n$ lettres, avec, $u$ ≤ $v$ dans l’ordre de Bruhat. Un <i>polytope d’intervalles de Bruhat</i> $Q_{u,v}$ est l’enveloppe convexe de tous les vecteurs de permutations $z=(z(1),z(2),...,z(n))$ avec $u$ ≤ $z$ ≤ $v$. Notons que lorsque $u=e$ et $v=w_0$ sont respectivement le plus court et le plus long élément du groupe symétrique, $Q_{e,w_0}$ est le permutoèdre classique. Les polytopes d’intervalles de Bruhat ont été étudiés récemment dans le papier de 2013 “The full Kostant-Toda hierarchy on the positive flag variety” par Kodama et le deuxième auteur, dans le contexte du treillis de Toda et la carte des moments sur la variété de drapeaux. Dans ce papier nous étudions des aspects combinatoires des polytopes d’intervalles de Bruhat. Par exemple, nous donnons une description par inégalités et une formule dimensionnelle pour les polytopes d’intervalles de Bruhat, et prouvons que chaque face d’un polytope d’intervalles de Bruhat est un polytope d’intervalles de Bruhat. Un outil essentiel dans la preuve de cette dernière affirmation est une généralisation de la célèbre propriété de lifting pour les groupes de Coxeter. Motivés par la relation entre la propriété de lifting et les $R$-polynômes, nous donnons aussi une généralisation de la récurrence standard pour les $R$-polynômes.
Keywords
  • [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
  • Bruhat order
  • symmetric group
  • $R$-polynomials
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