On this web page we present the computations of the basic
algebras of Schur algebras for some symmetric groups.
Let *G = Sym(r)* be the symmetric
group on *r* letters. Given a partition *L = [r _{1},
. . . , r_{t}]* of

It should be emphasized that no claim is being made for priority on this web page. Though some of our calculations appear to be new, it is likely that others of the calculation have been made before by other people. Nor are we claiming that this is the best method for computing Schur algebras. It is a method that is available. The information is offered as a service to anyone who might be interested.

The module *M* is the direct sum of permutation module on the
cosets of the Young subgroup *Y _{L}* where

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)/Rad ^{2} M)*, etc. For the
socle series, the

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*.

The smaller calculations were preformed on my laptop, a Dell Latitude 300, with approximately 1 GB. of RAM. Most of the rest of the calculation that are posted were performed on a SUN Blade 1000, (the sloth). The machine has 8 GB. of RAM and approximately 30 G. of hard drive. More recently, some larger calculations have been posted that were performed on a SUN with 32 G. of RAM. 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. The specific programs for computing the Schur algebras were developed at a meeting on "Cohomology and representation thoery for finite groups of Lie type", which took place at the American Institute of Mathematics in June of 2007. I want to thank AIM for support and thank several of the participants, especially Dave Hemmer, Brian Parshall, Leonard Scott and Lisa Townsley for encouragement and helpful discussions. J. F. Carlson, and G. Matthews, *Generators and relations for
matrix algebras*, J. Algebra 300(2006), 134-159.

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.