Regular patterns in Coxeter groups


The right cells of W are the equivalence classes in the right W-graph.

The identity element has links from 1 to s, but there are never any links from any element to 1. The identity element is therefore a singleton cell.

There is always a trivial link from x to xs > x. There will be a reverse link going from xs to x if Rx is not contained in Rxs - in particular if x != 1 and xs has a unique reduced word.

  • The elements of W with a unique reduced word starting with s make up a single right cell (Lusztig). Bédard's diagram
Even in complicated groups, the cells tend to be reasonable regions. Many interesting regions of a Coxeter group are described by regular expressions, and as far as I know there is no evidence against the idea that the cells are among these. Of course this is a rather vague statement.

The W-graph will not usually be a regular structure, but there is eveidence that its restriction to a right cell is regular.