Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Length enumeration of fully commutative elements in finite and affine Coxeter groups

Riccardo Biagioli, Mireille Bousquet-Mélou, Frédéric Jouhet, Philippe Nadeau

TL;DR

The paper advances length enumeration for fully commutative elements in finite and affine classical Coxeter groups by developing two recursive peeling approaches on heap representations. It derives explicit closed-form generating functions in terms of the q-series $J(x)$, $K(x)$ and derivatives of $J$, $H$, and their even/odd parts, for types $A,B,D$ and their affine analogues, including fc involutions; these formulas yield rational or Laurent-polynomial coefficients and reveal coefficient periodicity. Beyond reproducing known nonlinear $q$-equation frameworks, the authors connect fc elements to staircase polyominoes and provide new path/column-based generating-function expressions that streamline computations. The work extends to affine types $ ilde{A}, ilde{B}, ilde{C}, ilde{D}$ with structured decompositions into linear combinations of canonical path-heap series, offering compact, checkable forms and broad consistency checks via GAP computations. Collectively, the results deliver closed-form, verifiable generating functions and periodicity phenomena across finite and infinite Coxeter families, enriching the combinatorial and algebraic understanding of fc elements.

Abstract

An element w of a Coxeter group W is said to be fully commutative, if any reduced expression of w can be obtained from any other by transposing adjacent pairs of generators. These elements were described in 1996 by Stembridge in the case of finite irreducible groups, and more recently by Biagioli, Jouhet and Nadeau (BJN) in the affine cases. We focus here on the length enumeration of these elements. Using a recursive description, BJN established for the associated generating functions systems of non-linear q-equations. Here, we show that an alternative recursive description leads to explicit expressions for these generating functions.

Length enumeration of fully commutative elements in finite and affine Coxeter groups

TL;DR

The paper advances length enumeration for fully commutative elements in finite and affine classical Coxeter groups by developing two recursive peeling approaches on heap representations. It derives explicit closed-form generating functions in terms of the q-series , and derivatives of , , and their even/odd parts, for types and their affine analogues, including fc involutions; these formulas yield rational or Laurent-polynomial coefficients and reveal coefficient periodicity. Beyond reproducing known nonlinear -equation frameworks, the authors connect fc elements to staircase polyominoes and provide new path/column-based generating-function expressions that streamline computations. The work extends to affine types with structured decompositions into linear combinations of canonical path-heap series, offering compact, checkable forms and broad consistency checks via GAP computations. Collectively, the results deliver closed-form, verifiable generating functions and periodicity phenomena across finite and infinite Coxeter families, enriching the combinatorial and algebraic understanding of fc elements.

Abstract

An element w of a Coxeter group W is said to be fully commutative, if any reduced expression of w can be obtained from any other by transposing adjacent pairs of generators. These elements were described in 1996 by Stembridge in the case of finite irreducible groups, and more recently by Biagioli, Jouhet and Nadeau (BJN) in the affine cases. We focus here on the length enumeration of these elements. Using a recursive description, BJN established for the associated generating functions systems of non-linear q-equations. Here, we show that an alternative recursive description leads to explicit expressions for these generating functions.

Paper Structure

This paper contains 19 sections, 22 theorems, 178 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Fully commutative elements and heaps of generators
  3. Generating functions for finite types
  4. All fully commutative elements
  5. Fully commutative involutions
  6. First recursive approach: Peeling a diagonal
  7. Type $\boldsymbol A$
  8. Type $\boldsymbol B$
  9. Type $\boldsymbol D$
  10. New expressions for some path generating functions
  11. Affine type $\boldsymbol{\tilde{A}}$
  12. All fully commutative elements
  13. Fully commutative involutions
  14. Another recursive approach: Peeling a column
  15. Alternating heaps over a path
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $A(x,q)\equiv A$ and $\tilde{A}(x,q)\equiv \tilde{A}$ be the generating functions of fully commutative elements of type $A$ and $\tilde{A}$, respectively defined by Then where $J(x)$ is the following series:

Figures (7)

  • Figure 1: Coxeter graphs for classical types.
  • Figure 2: Left: A heap over the 6-point path. Center: An alternating heap over the 12-point path. Right: an alternating heap over the 11-point cycle.
  • Figure 3: Two types of alternating heaps with one point in the last column.
  • Figure 4: Peeling the rightmost diagonals in a self-dual alternating heap.
  • Figure 5: Left: A connected fc element of $A_9$ and the corresponding staircase polyomino, formed of square cells. Right: A connected fc involution of $A_9$ and the corresponding Dyck path, with the heights of the vertices shown.
  • ...and 2 more figures

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • Proposition 3.4
  • ...and 31 more