[Introduction] [Software] [Tables] [Diagrams]
This web-page is an introduction to a package of C-programs called Dynkin for computing in the modular representation theory of algebraic groups. We are making this available, because we believe that some of the programs are quite effective and useful (especially when it comes to computing Kazhdan-Lusztig polynomials for the affine Weyl group and characters of small rank simple modules).
The project ``Dynkin'' was begun in the spring of 92 (in Urbana) in an attempt to write a Mathematica package to carry out computations associated with non-reduced parabolic subgroup schemes in algebraic groups. This effort was prompted by a remark by A. Ramanathan, that one should look for a counterexample to Kodaira vanishing in prime characteristic among proper homogeneous spaces with non-reduced stabilizers. Using the initial Mathematica package ``dynkin.m'' a counterexample was found in June 92 (Niels Lauritzen, ``The Euler characteristic of a homogeneous line bundle'', C. R. Acad. Sci. Paris 315 (1992), 715-718).
It soon became apparent that more serious computations in modular representations of algebraic groups required the simplicity and speed of a lower level language like C (especially when computing Kazhdan-Lusztig polynomials).
In an effort to verify (or disprove) the modular Lusztig conjecture for the prime 5 for SL5, Anders Buch started rewriting ``Dynkin'' in C in the fall of 94, beginning his graduate studies in Aarhus. He significantly extended and rewrote all parts of ``dynkin'' to form a coherent archive of C-programs to find dimensions of weight spaces in the simple modules, compute with Jantzens sum formula, compute Kazhdan-Lusztig polynomials for affine Weyl groups and finally compare the left and right hand side of the modular Lusztig conjecture
where 
is dominant.
For further explanation of notation we refer to Jantzens book ``Representations of Algebraic Groups'', p. 294 (there is an error there, which seems somehow to have propagated out through the mathematical literature: the long word w0 succeeds y and w. We have corrected this above. We discovered this testing our programs for SL4). The algorithm for computing the dimensions of the weight spaces of the irreducible modules is documented in Gilkey and Seitz, ``Some representations of exceptional Lie algebras'', Geometriae Dedicata 25 (1988), 407-416.
In April 95 we were finally able to verify the Lusztig conjecture in the SL5, p =5 case using the super computer (SGI Power Challenge) at Aarhus Universitet. This came as a result of struggling with the top restricted alcove for more than two months.
The SL5, p=5 case was verified independently by L. Scott et al. in the summer of 98 using more effective algorithms for the left hand side (computing the maximal submodule in a baby Verma module). We have had occasion to compare our results with Scott et al. and found complete agreement.
Scott et al. have also been able to compute the left hand side in the SL5, p = 7 case, which also confirmed Lusztigs prediction.
The original modular Lusztig conjecture carries the Jantzen condition, which forces the prime to be > 8 in the SL5-case to cover every restricted weight. The conjecture alluded to in Andersen-Jantzen-Soergel (Asterisque 220) is for restricted weights and p greater than or equal to the Coxeter number. This is what we are checking here (in the extremal case p = h = 5). It seems that Kato was the first to explicitly conjecture this (we thank Scott for this remark).
The software developed for this project is available for downloading. It is written in C, and known to run under Linux and HP-UX. It will probably work under any UNIX type system. We recommend using GCC and GNU MAKE to compile it.
Download the software: dynkin.tar.gz
To install the program, use the unix command:
% gzip -cd dynkin.tar.gz | tar xvf -
This will create the directory dynkin with all necessary
files. You can compile the programs by issuing the make
command in the directory:
% cd dynkin
% make
Note: On some systems, GNU's make program is called
gmake.
A brief manual is available in HTML.
Kazhdan-Lusztig polynomials
The relevant Kazhdan-Lusztig polynomials for Lusztig's conjecture have been calculated for the root systems:
Dominant weight multiplicities in restricted irreducibles
A2:
p=3
p=5
p=7
p=11
A3:
p=5
p=7
p=11
A4:
p=5
B2:
p=5
p=7
p=11
G2:
p=7
p=11
Below we have put some poset diagrams of Weyl groups and alcoves for different
root systems. All are produced with the program psbruhat. All
the files are postscript files.
Ordinary Weyl groups (poset diagram):
A2
A3
B2
B3
C3
G2
Restricted alcoves below top restricted alcove (poset diagram):
A2:
reduced words in aff Weyl group
p=3
p=5
p=7
A3:
reduced words in aff Weyl group
p=5
p=7
A4:
reduced words in aff Weyl group
p=5
p=7
B2:
reduced words in aff Weyl group
p=5
p=7
B3:
reduced words in aff Weyl group
p=7
C3:
reduced words in aff Weyl group
p=7
D4:
reduced words in aff Weyl group
p=7
G2:
reduced words in aff Weyl group
p=7
Dominant alcoves below top restricted alcove (poset diagram):
A2:
reduced words in aff Weyl group
p=3
p=5
p=7
A3:
reduced words in aff Weyl group
p=5
p=7
A4:
reduced words in aff Weyl group
p=5
p=7
B2:
reduced words in aff Weyl group
p=5
p=7
B3:
reduced words in aff Weyl group
p=7
C3:
reduced words in aff Weyl group
p=7
D4:
reduced words in aff Weyl group
p=7
G2:
reduced words in aff Weyl group
p=7