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Some open problems on Coxeter groups and unimodality

Francesco Brenti

Abstract

In this paper I present some open problems on Coxeter groups and unimodality, together with the main partial results, and computational evidence, that are known about them.

Some open problems on Coxeter groups and unimodality

Abstract

In this paper I present some open problems on Coxeter groups and unimodality, together with the main partial results, and computational evidence, that are known about them.

Paper Structure

This paper contains 2 sections, 10 theorems, 30 equations.

Table of Contents

  1. Coxeter groups
  2. Unimodality

Key Result

Theorem 1.2

Let $(W,S)$ be a Coxeter system, $J \subseteq S$, and $x \in \{ -1,q \}$. Then there is a unique family of polynomials $\{ R_{u,v}^{J,x} \} _{u,v \in W^{J}} \subseteq {\mathbb Z}[q]$ such that, for all $u,v \in W^{J}$:

Theorems & Definitions (19)

  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Conjecture 1.8
  • Theorem 1.11
  • Theorem 1.12
  • Theorem 2.1
  • ...and 9 more