Hereditary Matrix Models for Weyl Groups
This work was supported by NSF grant DMS-0245385.
University of Michigan
This web document provides access to a database of explicit matrices
for every irreducible representation of every irreducible Weyl
group of rank <= 8. The matrices are provided in a collection of highly
compressed text files suitable for reading into a Maple session.
We also provide the Maple programs that were used to create and verify
this database, and several files listing features of the models
in the database.
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:
In both cases, the models are hereditary in the following sense:
- Orthogonal: the matrices are real orthogonal, and have entries
that are rational on the diagonal, and are square roots of rationals off
- Rational seminormal: the matrices are rational, and the unique
invariant quadratic form is diagonal.
Let s,...,s[n] denote the simple reflections of the
Weyl group W, and let W[k] denote the
parabolic subgroup generated by s,...,s[k].
Then for each k=1,...,n,
the matrices A,...,A[k] representing
s,...,s[k] in a hereditary model have the property
that they decompose into a block-diagonal sum of irreducible matrix
representations of W[k], and furthermore, the blocks that
correspond to isomorphic representations are identical.
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 W[n-1],
W[n-1] to W[n-2], and so on, is
multiplicity-free. In such cases, hereditary matrix models are unique up
to diagonal transformations, and orthogonal hereditary models are
unique up to changes of sign.
II: How to use the database of models
First, you'll need Maple. Any version should work--the code has been
tested on versions ranging from Maple V Release 3 to Maple 9.
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
Third, follow the instructions in the accompanying
III: How the database was built
If you want to study the programs that were used to create the database,
you can browse the source directory. Or, if you
want to reproduce the computations on your hardware, everything you'll
need (except Maple itself) is included in this unix tar
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 ]:
The above programs depend on the following sub-programs:
- This is the top-level program that created the database of orthogonal
and rational seminormal models for all Weyl groups.
- This program built the statistical reports discussed in Part IV
below (except for the file of runtimes
that came from a particular run of
- If you have a fast machine with lots of RAM, read this file into a
Maple session to run the validity tests in
on every model in the database.
a copy of the vanilla edition of the Coxeter and Weyl packages.
For more information and documentation, see the
home page for Coxeter/Weyl.
build orthogonal and rational seminormal models for all
irreps of a given Weyl group.
build an orthogonal hereditary matrix model for a single irrep.
find all "clones" of a given irrep of a Weyl group W.
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 ].
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.
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 ]).
find an optimal orthogonal model for an irrep that is not totally free.
Reference: Section 4B of the paper
[ ps , dvi ].
tools for manipulating matrices expressed in the special sparse
format used by these programs, including fast_prod
(for matrix multiplication).
unpack one of the seminormal or orthogonal models from the database.
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
IV: Statistical reports on the models
For the irreps of the exceptional groups,
we have several files of information:
- 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 s[n], 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.
- 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 [N, t, e]
or [N, t1, t2, e], where N denotes the
dimension, t the trace of a reflection (or if there are
two conjugacy classes of reflections, the traces t1 and
t2 for both classes), and the sign e of
the trace of the longest element.
This suffices to distinguish all irreducible representations of the
exceptional groups, aside from two representations of F4 that both
have the label [6,0,0,+].
- This table reports the number of equations and variables that define
the variety of solutions for the matrix representing s[n]
in an orthogonal hereditary model for each irreducible representation of
the Weyl groups of E6, E7, and E8.
- 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 representation rho is another
representation rho' such that rho
and rho' are isomorphic when restricted to the parabolic
- 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