On the cyclically fully commutative elements of Coxeter groups

T. Boothby, J. Burkert, M. Eichwald, Dana C Ernst, R. M. Green, M. MacAuley

Research output: Contribution to journalArticle

4 Citations (Scopus)

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≥1, 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.

Original languageEnglish (US)
Pages (from-to)123-148
Number of pages26
JournalJournal of Algebraic Combinatorics
Volume36
Issue number1
DOIs
StatePublished - Aug 2012
Externally publishedYes

Fingerprint

Coxeter Group
Logarithmic
Affine Weyl Groups
Directed Acyclic Graph
Braid
Combinatorics
Necessary Conditions
Generalise
Sufficient Conditions
Arbitrary

Keywords

  • Coxeter groups
  • Cyclic words
  • Fully commutative elements
  • Root automaton

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Algebra and Number Theory

Cite this

On the cyclically fully commutative elements of Coxeter groups. / Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, Dana C; Green, R. M.; MacAuley, M.

In: Journal of Algebraic Combinatorics, Vol. 36, No. 1, 08.2012, p. 123-148.

Research output: Contribution to journalArticle

Boothby, T, Burkert, J, Eichwald, M, Ernst, DC, Green, RM & MacAuley, M 2012, 'On the cyclically fully commutative elements of Coxeter groups', Journal of Algebraic Combinatorics, vol. 36, no. 1, pp. 123-148. https://doi.org/10.1007/s10801-011-0327-z
Boothby, T. ; Burkert, J. ; Eichwald, M. ; Ernst, Dana C ; Green, R. M. ; MacAuley, M. / On the cyclically fully commutative elements of Coxeter groups. In: Journal of Algebraic Combinatorics. 2012 ; Vol. 36, No. 1. pp. 123-148.
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