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On the Cyclically Fully Commutative Elements of Coxeter Groups

Tomas Boothby, Jeffrey Burkert, Morgan Eichwald, R. M. Green, Dana C. Ernst, Matthew Macauley

Abstract

Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in W. We say that w is "cyclically fully commutative" (CFC) if every cyclic shift of any reduced expression for w is fully commutative (i.e., avoids long braid relations). These generalize Coxeter elements in that their reduced expressions can be described combinatorially by acyclic directed graphs, and cyclically shifting corresponds to source-to-sink conversions. In this paper, we explore the combinatorics of the CFC elements and enumerate them in all Coxeter groups. Additionally, we characterize precisely which CFC elements have the property that powers of them remain fully commutative, via the presence of a simple combinatorial feature called a "band." This allows us to give necessary and sufficient conditions for a CFC element w to be "logarithmic," that is, l(w^k) = k l(w) for all k > 0, for a large class of Coxeter groups that includes all affine Weyl groups and simply-laced Coxeter groups. Finally, we give a simple non-CFC element that fails to be logarithmic under these conditions.

On the Cyclically Fully Commutative Elements of Coxeter Groups

Abstract

Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in W. We say that w is "cyclically fully commutative" (CFC) if every cyclic shift of any reduced expression for w is fully commutative (i.e., avoids long braid relations). These generalize Coxeter elements in that their reduced expressions can be described combinatorially by acyclic directed graphs, and cyclically shifting corresponds to source-to-sink conversions. In this paper, we explore the combinatorics of the CFC elements and enumerate them in all Coxeter groups. Additionally, we characterize precisely which CFC elements have the property that powers of them remain fully commutative, via the presence of a simple combinatorial feature called a "band." This allows us to give necessary and sufficient conditions for a CFC element w to be "logarithmic," that is, l(w^k) = k l(w) for all k > 0, for a large class of Coxeter groups that includes all affine Weyl groups and simply-laced Coxeter groups. Finally, we give a simple non-CFC element that fails to be logarithmic under these conditions.

Paper Structure

This paper contains 14 sections, 26 theorems, 27 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Coxeter groups
  3. Coxeter and cyclically fully commutative elements
  4. Properties of CFC elements
  5. Enumeration of CFC elements
  6. Type $A$
  7. Type $B$
  8. Type $D$
  9. Type $E$
  10. Type $F$
  11. Type $H$
  12. The Root Automaton
  13. Logarithmic CFC elements
  14. Conclusions and Future Work

Key Result

Theorem 2.1

In a Coxeter group $W$, any two reduced expressions for the same group element differ by braid relations. $\Box$

Figures (2)

  • Figure 3: Connected Coxeter graphs corresponding to CFC-finite groups.
  • Figure 4: The last remaining obstructions to Theorem \ref{['thm:largebands']}.

Theorems & Definitions (56)

  • Theorem 2.1: Matsumoto's theorem
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2: Eriksson--Eriksson Eriksson:09
  • Corollary 3.3
  • Definition 3.4
  • Example 3.5
  • ...and 46 more