Mathematics: a branch of physics in

which the
experiments are cheap.

--V. I. Arnold [not an exact quote]

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:

- The main entry point into the Models Database.
- A related paper: Explicit matrices for irreducible representations of Weyl groups (gzipped PostScript).
- A
**tar**file that includes all of the above: models.tar.gz (if you grok unix). - A clickable directory of data (if you don't).

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.

Want more? Check out affhyp ( recently updated for Maple VR4), the Maple program that produced this paper. Requires the

- 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).

- The 24 posets on 1-4 vertices (544 bytes, plain text).
- The 63 posets on 5 vertices (404 bytes, gzip'd text).
- The 318 posets on 6 vertices (1.4K, gzip'd text).
- The 2045 posets on 7 vertices (9K, gzip'd text).
- The 16999 posets on 8 vertices (77K, gzip'd text).

- Degrees 1-9 (7K, gzip'd text).
- Degree 10 (8K, gzip'd text).
- Degree 11 (17K, gzip'd text).
- Degree 12 (34K, gzip'd text).
- Degree 13 (70K, gzip'd text).
- Degree 14 (140K, gzip'd text).
- Degree 15 (272K, gzip'd text).
- Degree 16 (530K, gzip'd text).

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

- Degree 4 (0.2K, gzip'd text).
- Degree 5 (0.5K, gzip'd text).
- Degree 6 (1.7K, gzip'd text).
- Degree 7 (5K, gzip'd text).
- Degree 8 (19K, gzip'd text).
- Degree 9 (58K, gzip'd text).
- Degree 10 (185K, gzip'd text).

Back to John Stembridge's Home Page This page last modified Mon Sep 7 15:19:29 EDT 2009