`ShortLex`

rewite rules with
Mathematica.One may generate the

`ShortLex`

rewrite rules for finite Coxeter groups a little
more intelligently than described in our note
Computing some KL-polynomials for the poset
of B×B-orbits in
group compactifications
. See alsoFokko du Cloux, A transducer approach to Coxeter groups, J. of Symbolic Computation 27 (1999), no. 3, 311-324.

For type E

As our programs are quite slow, we have collected some more rules in this directory.

Here is one possibility for the main part that is called in the programs below that generate rules for

- type A
_{n} - type B
_{n} - type D
_{n} - type E
_{n} - type F
_{4} - type H
_{4}

- type affine A
_{n} - type affine B
_{n} - type affine C
_{n} - type affine D
_{n} - type affine E
_{n} - type affine F
_{4} - type affine G
_{2}.

`ShortLex`

or not.
For affine G`ShortLex`

normal forms.
One may prove completeness of the rules with a program for checking the Church Rosser conditions. This program tells if the Knuth Bendix algorithm does not change the rewriting system. In fact, it seems to be advantageous to replace the "main part" above with a main part that is based entirely on searching, by length, for failures of the Church Rosser condition. (Some kind of Knuth Bendix by length, relying on the theorem of Tits.) This is better than just Knuth Bendix, but of course it gets quite slow when the number of rules is large. One should at least turn to a compiled language. Then again, if the number of rules gets really large, why would you want them?

Wilberd van der Kallen