Schur Algebras

# Schur Algebras, S(n,r)

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 = [r1, . . . , rt] of r, the Young subgroup corresponding to L is the direct product Sym(r1) x ... x Sym(rt). Let k be the field with p elements where p is the given characteristic. We compute an algebra which is Morita equivalent to the Schur algebra S(n,r) by computing the endomorphism algebra of the module M which is the direct sum of the permutation modules with point stabilizers being the Young subgroups corresponding to partitions having at most n parts.

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.

The module M is the direct sum of permutation module on the cosets of the Young subgroup YL where L runs over the partitions of r with at most n parts. The coefficients of M lie 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 action algebra A.

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 basic algebra, S, of the Schur algebra S(n,r).

First we take the kG-endomorphism ring of the module M. That is, it is the algebra of all matrices that commute with the algebra A. What we actually compute is the commuting ring of the condensed algebra eAe. Because eAe is Morita equivalent to A, the two have isomorphic commuting rings. The algebra S is the condensation of the commuting ring. We calculate the structure of S as well as its Cartan matrix, and the Loewy and socle series for its projective modules. S is Morita equivalent to the Schur algebra.

## The Calculations

#### Equipment

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.

#### Programs

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.

## References

J. F. Carlson, and G. Matthews, Generators and relations for matrix algebras, J. Algebra 300(2006), 134-159.

## Acknowledgement

Thanks are due to NSF for support of the project in both time and equipment.

## Manditory Disclaimer

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