John Stembridge

University of Michigan

March 2004

**I: Overview****II: How to use the database of models****III: How the database was built****IV: Statistical reports on the models**

For a more detailed account of the models and the algorithms used to produce them, see the paper "Explicit matrices for irreducible representations of Weyl groups" [ ps , dvi ].

For each irreducible representation, we provide two models:

**Orthogonal**: the matrices are real orthogonal, and have entries that are rational on the diagonal, and are square roots of rationals off the diagonal.

**Rational seminormal**: the matrices are rational, and the unique invariant quadratic form is diagonal.

Let ** s[1],...,s[n]** denote the simple reflections of the
Weyl group

Note that this property depends on the ordering of the simple reflections.

In all cases except for **E8**, it turns out that the simple reflections
may be ordered so that the restriction of irreducible representations
from ** W[n]** to

Second, you'll need the data files: either download everything in the
data directory one file at a time (including the
unpack program), or grab this unix
` tar` file: models.tar.gz.

You'll probably also want to download a copy of sparseops, a collection of procedures for manipulating matrices in the special sparse format used by these models.

Third, follow the instructions in the accompanying READ_ME file.

Here is a brief summary of the purpose of each program. Further documentation is provided within the files. Theoretical justifications are provided in the accompanying paper [ ps , dvi ]:

- makemodels
- This is the top-level program that created the database of orthogonal
and rational seminormal models for all Weyl groups.
- makestats
- This program built the statistical reports discussed in Part IV
below (except for the file of runtimes
that came from a particular run of
makemodels).
- runchecks
- If you have a fast machine with lots of RAM, read this file into a
Maple session to run the validity tests in
checkreps
on every model in the database.

- coxeter2.4v.txt:
a copy of the vanilla edition of the Coxeter and Weyl packages.
For more information and documentation, see the
home page for Coxeter/Weyl.
- buildall:
build orthogonal and rational seminormal models for all
irreps of a given Weyl group.
- orthogonal:
build an orthogonal hereditary matrix model for a single irrep.
- clone_test:
find all "clones" of a given irrep of a Weyl group
. A list of elements of`W`of "para-Coxeter type" is returned whose traces distinguish the desired representation from its clones. Reference: Section 2E of the paper [ ps , dvi ].`W` - quadsolve:
an implementation of the quadratic reduction algorithm described in
Section 3D of the paper
[ ps , dvi ].
It uses Grobner-like reductions to find a solution of a (typically)
quadratic system of equations defining a 0-dimensional variety over a
ground field that includes the square roots of zero or more rationals.
- seminormal:
find a diagonal change of basis that converts an orthogonal hereditary
model for an irrep into rational seminormal form, and then apply linear
programming methods to optimize the change of basis. Reference: Sections 3E-F
of the paper
[ ps , dvi ]).
- orthopt:
find an optimal orthogonal model for an irrep that is not totally free.
Reference: Section 4B of the paper
[ ps , dvi ].
- sparseops:
tools for manipulating matrices expressed in the special sparse
format used by these programs, including
(for matrix multiplication).`fast_prod` - unpack:
unpack one of the seminormal or orthogonal models from the database.
- checkreps:
test the validity of the database of matrix models. In particular,
verify that each orthogonal and seminormal model satisfies the Coxeter
relations, and check enough traces to be certain that a model has the
desired character.

- quality
- A report on the quality of each of the orthogonal and rational
seminormal models for
**F4**,**E6**,**E7**, and**E8**, including the average number of nonzero entries in each row of the matrix representing, as well as the largest numerator, largest denominator, and least common denominator for the entries in the matrix representing`s[n]`. Note that the data in the (totally free) orthogonal cases is canonical, and provide lower bounds for what can be achieved by the rational seminormal models.`s[n]` - canonical_names
- This table provides a translation between the numbering of the
irreducible representations of the exceptional Weyl groups
[specifically,
**G2**,**F4**,**E5**,**E6**,**E7**, and**E8**] used by the programs, and the coordinate-free, canonical names used in the paper [ ps , dvi ].For each representation, we report the tuple

or`[N, t, e]`, where`[N, t1, t2, e]`denotes the dimension,`N`the trace of a reflection (or if there are two conjugacy classes of reflections, the traces`t`and`t1`for both classes), and the sign`t2`of the trace of the longest element. This suffices to distinguish all irreducible representations of the exceptional groups, aside from two representations of`e`**F4**that both have the label.`[6,0,0,+]` - complexity
- This table reports the number of equations and variables that define
the variety of solutions for the matrix representing
in an orthogonal hereditary model for each irreducible representation of the Weyl groups of`s[n]`**E6**,**E7**, and**E8**. - clonelist
- For the Weyl groups of
**F4**and**E4**through**E8**, we list each nontrivial clone of each irreducible representation. (Recall that a clone of a representationis another representation`rho`such that`rho'`and`rho`are isomorphic when restricted to the parabolic subgroup`rho'`.)`W[n-1]` - runtimes
- Here, we list the running times, space used, and other diagnostic
information reported during the building of the orthogonal and
seminormal models of the irreps of the Weyl groups of
**E5**,**E6**,**E7**, and**E8**. The host CPU was a 2.8GHz Pentium IV running Red Hat Linux 9 and Maple 9.

This page last modified Tue Mar 9 16:29:26 EST 2004