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Coxeter systems, left inversion sets, and higher dimensional cubes

Harrison Gimenez

TL;DR

The paper addresses when a left inversion set $\Φ_x$ can be mapped to another $\Φ_y$ by a Coxeter group element $w$, introducing Coxeter squares and higher-dimensional Coxeter $n$-cubes to encode these transfers. It develops a Brink-Howlett groupoid framework and shows, for finite Coxeter systems with $|S|=n$, that Coxeter $n$-cubes exist; in the $A_n$ case it gives an explicit bijection between such cubes (mod orientation) and binary trees with $n+1$ leaves, and identifies edges with bigrassmannian permutations. The results connect left inversion sets, the reflection cocycle, and weak right order to a combinatorial cube structure, revealing deep links between root-poset geometry and tree-like combinatorics. These findings illuminate morphisms in generalized Brink-Howlett groupoids and provide canonical decompositions of morphisms into elementary generators, with potential implications for representation theory and combinatorial models of Coxeter group actions.

Abstract

Let $ (W,S)$ be a Coxeter system. We investigate the equation $ w(Φ_{x}) = Φ_{y}$ where $ w,x,y\in W$ and $ Φ_{x}$, $Φ_{y}$ denote the left inversion sets of $ x$ and $ y$. We then define a commutative square diagram called a Coxeter square which describes the relationship between 4 non-identity elements of the Coxeter group $ W$ and the equation $ w(Φ_{x}) = Φ_{y}$. Coxeter squares were first introduced by Dyer, Wang in \cite{dyer2011groupoids2} and \cite{dyer2019characterization}. Coxeter squares can be \textquotedblleft glued" together by compatible edges to form commutative diagrams in the shape of higher dimensional cubes called Coxeter $n$-cubes, which were first defined by Dyer in Example 12.5 of \cite{dyer2011groupoids2}. When $ |W| < \infty$ and $ |S| = n$, we show that Coxeter $n$-cubes must exist within $ (W,S)$. We then prove results about Coxeter $n$-cubes in the $A_{n}$ Coxeter system. We establish an explicit bijection between Coxeter $n$-cubes (modulo orientation) in $ A_{n}$ and binary trees with $n+1$ leaves. We also show that an element $x$ of $ A_{n}$ appears as the edge of some Coxeter $n$-cube if and only if $ x$ is a bigrassmannian permutation.

Coxeter systems, left inversion sets, and higher dimensional cubes

TL;DR

The paper addresses when a left inversion set can be mapped to another by a Coxeter group element , introducing Coxeter squares and higher-dimensional Coxeter -cubes to encode these transfers. It develops a Brink-Howlett groupoid framework and shows, for finite Coxeter systems with , that Coxeter -cubes exist; in the case it gives an explicit bijection between such cubes (mod orientation) and binary trees with leaves, and identifies edges with bigrassmannian permutations. The results connect left inversion sets, the reflection cocycle, and weak right order to a combinatorial cube structure, revealing deep links between root-poset geometry and tree-like combinatorics. These findings illuminate morphisms in generalized Brink-Howlett groupoids and provide canonical decompositions of morphisms into elementary generators, with potential implications for representation theory and combinatorial models of Coxeter group actions.

Abstract

Let be a Coxeter system. We investigate the equation where and , denote the left inversion sets of and . We then define a commutative square diagram called a Coxeter square which describes the relationship between 4 non-identity elements of the Coxeter group and the equation . Coxeter squares were first introduced by Dyer, Wang in \cite{dyer2011groupoids2} and \cite{dyer2019characterization}. Coxeter squares can be \textquotedblleft glued" together by compatible edges to form commutative diagrams in the shape of higher dimensional cubes called Coxeter -cubes, which were first defined by Dyer in Example 12.5 of \cite{dyer2011groupoids2}. When and , we show that Coxeter -cubes must exist within . We then prove results about Coxeter -cubes in the Coxeter system. We establish an explicit bijection between Coxeter -cubes (modulo orientation) in and binary trees with leaves. We also show that an element of appears as the edge of some Coxeter -cube if and only if is a bigrassmannian permutation.

Paper Structure

This paper contains 15 sections, 36 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Background Material on Coxeter Systems
  3. Coxeter systems
  4. Standard parabolic subgroups
  5. The length function
  6. The canonical root system
  7. Left Inversion Sets
  8. The weak right order
  9. The reflection cocycle
  10. Background Material on Brink-Howlett groupoids
  11. Definition of Brink-Howlett groupoids
  12. Generators of the groupoid
  13. Decomposing morphisms
  14. The equation $w(\Phi_{x}) = \Phi_{y}$
  15. Coxeter Squares and Coxeter $n$-Cubes

Key Result

Proposition 2.1

Let $w = s_{1}s_{2} \dots s_{r}$. Let $t\in T$ such that $\ell(wt) < \ell(w)$. Then the following are true:

Theorems & Definitions (83)

  • Proposition 2.1: Generalized Exchange Condition
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 73 more