W-cells
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.
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