Hecke Algebras for Symmetric Groups

HECKE ALGEBRAS FOR SYMMETRIC GROUPS

for Young subgroups

On this web page we present the computations of the Hecke algebras of permutation modules with point stabilizers being Young subgroups. The basic setup is the following. Let G = Sym(n) be the symmetric group on n letters. Given a partition P = [n1, . . . , nt], the Young subgroup H corresponding to P is the direct product Sym(n1) x ... x Sym(nt). Let k be the field with p elements where p is the given characteristic.

It should be emphasized that no claim is being made for priority on this web page. It is almost certain that some of these calculation have been made before by other people. Nor are we claiming that this is the best method for computing Hecke 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 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 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 Hecke algebra H.

The Hecke algebra is 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. We calculate the structure of H as well as its Cartan matrix, and the Loewy and socle series for its projective modules.

The Calculations


Symmetric Group on 8 letters in characteristic 2

Symmetric Group on 8 letters in characteristic 3

Symmetric Group on 8 letters in characteristic 5

Symmetric Group on 9 letters in characteristic 2

Symmetric Group on 9 letters in characteristic 3

Symmetric Group on 10 letters in characteristic 2

Symmetric Group on 11 letters in characteristic 2