On this web page we present the computations of the Hecke algebras of permutation modules with point stabilizers being normalizers of Sylow subgroups.
This information is posted for the use and edification of anyone who is interested. It is information that has interested me throughout my career and I am happy to be able to privide as a service to the representation theory community. We state at the beginning that no claim is being made for priority on this web page. It is known that some of these calculation have been made before by other people. Anyone who wishes to quote a particular calculation should consult the literature for a proper reference. Also, we are not claiming that this is the best method for computing Hecke algebras. It is a method that is available.
The reader may notice that there are a few gaps in the calculation. This is frankly because some of the calculation did not work. This is a continuing effort and we hope to fill gaps as time passes. With the current equipment, the limits of the calculation are approximately dimension of 3000 on the module M in characteristic 2, and somewhat smaller dimensions in other characteristics. The current record on dimensions is for the group PSU(3,5) in characteristic 2, where the module has dimension 7875. Of course, the limits also depends on the degree of complication of the algebras.
In each of the calculation, we let G be a finite simple group and let k be the prime field with p elements where p is a prime dividing the order of G. Let H = NG(S) be the normalizer of a Sylow p-subgroup S of G.
The module M is the permutation module on the cosets of the Young subgroup H with coefficients in the prime field of characteristic p. We compute its composition factors and its indecomposable components. The list of dimensions of the nonisomorphic simple modules occurring as composition factors is given. The simple modules are numbered as in that list. In the displays of the Loewy series and socle series for M the numbers refer to the simple modules in that list.
The displays of the Loewy series and socle series both go from top to bottom. That is, in the Loewy series for M, the first line lists the simple modules that are in M/(Rad M), the second line lists the simple modules in (Rad M)/Rad2 M), etc. For the socle series, the last line is Soc M, while the next to last line is (Soc2 M)/(Soc M).
The algebra A is the image of the group algebra of G in the endomorphism ring of M. Hence it is isomorphic to the quotient of the group algebra kG by the annihilator in kG of the module M. The simple modules for A are precisely the simple composition factors of M. We compute the Cartan matrix of A and the structure of the projective modules for A. Note that these projective modules are not, in general, projective over the group algebra kG. In the actual computation, the structure of these modules is made at the level of the condensed algebra eAe where e is a sum of primitive idempotents in A, one for each simple A-module. The algebra eAe is Morita equivalent to the algebra A.
Most of the calculation that are posted were performed on an SUN Blade 1000, (the sloth). The machine has 8 GB. of RAM and approximately 30 G. of hard drive. I want to thank the National Science Foundation and University of Georgia Research Foundation for providing me with both the equipment and the time to work on this project.
All of the programs are written in MAGMA code and run on the MAGMA platform. The programs for computing the generators and relations for algebras and for finding condensed algebras were developed and written by myself and Graham Matthews.
Thanks are due to the people of the MAGMA project in Sydney, particularly John Cannon and Allan Steel, for numerous instances of help with the tools to make the programs work and for their enthusiastic support.
Do you have your own example to compute? The program which runs the computations in MAGMA and creates the html files is available by clicking here. Simply download the program and follow the instructions. The program requires a relatively recent version of MAGMA as well as a computer with a reasonable amount of RAM, depending on the size of the example that you wish to calculate.
This program offered here is a superior version to the one originally offered. The replacement was made in January of 2007.
J. F. Carlson, and G. Matthews, Generators and relations for matrix algebras, J. Algebra, 300(2006), 134-159.
Thanks are due to NSF for support of the project in both time and equipment.
The content and opinions expressed on this Web page do not necessarily reflect the views of nor are they endorsed by the University of Georgia or the University System of Georgia.