Mathematics: a branch of physics in
which the experiments are cheap.
--V. I. Arnold [not an exact quote]

# Archive of Research Data And Other Useful Stuff

## Coxeter Planes

There is a canonical two-dimensional subspace associated to any root system or reflection group; namely, one where a Coxeter element acts as rotation by 2Pi/h, where h denotes the Coxeter number.

Check out the interesting graphic images obtained by projecting the root system into this plane.

By popular demand, I've added some more graphic images obtained by projecting regular 4D-polytopes into their Coxeter planes.

For starters, here's the projection of the root system of D6 into its Coxeter plane:

## Explicit Matrices for Irreps of Weyl Groups

In connection with the
Atlas of Lie Groups Project, I've constructed a database of explicit matrices for every irreducible representation of every Weyl group of rank <= 8. The matrices are provided in a collection of highly compressed text files suitable for reading into a Maple session.

The database should work with all versions of Maple.

## The Kontsevich conjecture

In connection with my paper
Counting points on varieties over finite fields related to a conjecture of Kontsevich, I'm releasing a collection of Maple programs and data. The programs can be used to explicitly count the number of zeroes of a set of multivariate polynomials over the finite field GF(q), treating q as an indeterminate.

There must be a catch, you say.

Well, yes. The catch is that the program only works well when the polynomials have a very special form. Typically, you need many of the variables to occur linearly in the polynomials (e.g., sums of square-free monomials). And that's just for starters.

Kontsevich's conjecture says that if you take the generating function for all spanning trees of a graph (one variable for each edge), then the number of zeroes in GF(q) should be a polynomial in q. So along with the programs, there are several files of graphs that meet necessary conditions for a minimal counterexample to Kontsevich's conjecture and some related conjectures. If you want more details, you'll have to read the paper and the documentation.

If you grok unix, here's all the stuff you need, wrapped up in one gzip'd tar file: reduce.tar.gz (24K).
If you don't, here's a clickable list of what's available.

The Fine Print: All of the above should work with Maple V Releases 3, 4, 5.

Update [4-Sep-2009]: We've updated the software so that it should run in more recent versions of Maple, and to fix a bug found by Oliver Schnetz. See also Schnetz's paper in the Archive--he has discovered that the smallest counterexamples to the Kontsevich conjecture have 14 edges.

## Coxeter graph paper

If you've ever worked with affine reflection groups, you've probably wasted lots of time drawing the reflecting hyperplanes of the rank 2 groups on scraps of paper. You may also have wished you had pads of graph paper with these lines drawn in for you. If so, you've come to the right place. Behold! Coxeter graph paper!

Want more? Check out affhyp ( recently updated for Maple VR4), the Maple program that produced this paper. Requires the coxeter package. Affhyp can do lots more--here are some samples of snazzy 3D graphics it produced:

• The hyperplanes corresponding to the root systems A3 (27K gif) and B3 (23K gif).
• The "sandwich" arrangements for A3 (27K gif), B3 (27K gif), and C3 (26K gif).

## Poset lists

For the benefit of those of you who cannot use my posets package, here are the lists of posets on 8 or fewer vertices. The larger files are gzip images of plain text files, suitable for reading into a Maple session. If you are skilled in the use of a text editor, you can transform these files into any format you like.

A description of the format appears at the top of each file.

## Jack symmetric functions

These files contain tables of Jack's symmetric functions up to degree 16, in a highly compressed format. Each file is the gzip image of a plain text file, suitable for reading into a Maple session. Requires the SF package.

In order to use this data, you'll need jacktools, a Maple program for installing the data in these tables into an SF session. The tables were built using this code from the SF package.

It takes about 10 minutes to build these tables on a 2.4 GHz P4 running Maple 9 on Linux.

## q,t-Kostka polynomials

These files contain tables of q,t-Kostka polynomials up to degree 10, created with the SF package. Each file is the gzip image of a plain text file suitable for reading into a Maple session.