Hereditary Matrix Models for Weyl Groups

John Stembridge
University of Michigan
March 2004

This work was supported by NSF grant DMS-0245385.

I: Overview

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:

• 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.
In both cases, the models are hereditary in the following sense:

Let s[1],...,s[n] denote the simple reflections of the Weyl group W, and let W[k] denote the parabolic subgroup generated by s[1],...,s[k]. Then for each k=1,...,n, the matrices A[1],...,A[k] representing s[1],...,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 these models.

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 file: models.tar.gz.

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.

The above programs depend on the following sub-programs:

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 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 ].

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 fast_prod (for matrix multiplication).

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.

IV: Statistical reports on the models

For the irreps of the exceptional groups, we have several files of information:

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 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.

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 [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,+].

complexity
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.

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 representation rho is another representation rho' such that rho and rho' are isomorphic when restricted to the parabolic subgroup 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.