Regular patterns in Coxeter groups
Theorem. Bédard's figure is generated by a finite automaton.

Proof :

  • The language of reduced words (as well as that of ShortLex words) for a Coxeter group is regular (Brink & Howlett). ShortLex tree for affine A2 and its minimal version
  • The set of words containing one side of a braid relation (a braid word) is regular.
  • The set of reduced words with a unique reduced expression is that of reduced words containing no braid words (Tits).
  • [ unique reduced words ] = [ reduced words ] AND NOT [ containing braid word ]