The Cartan Matrix of the Schur Algebra

ANNE E HENKE, UNIVERSIT AT KASSEL Abstra t. Let k be an in nite eld of prime hara teristi and let r be a positive integer. Using admissible de ompositions, we determine expli itly the entries of the de omposition matrix of the S hur algebra S(2; r) over k and prove that any two blo ks with the same number of simple modules have the the same de omposition matrix and hen e the same Cartan matrix. 1. Introdu tion 1.1 The General Setting. Let k be an in nite eld of prime hara teristi p and let n; r be positive integers. The S hur algebra S(n; r) over k is a nite-dimensional, asso iative algebra whose module ategory is equivalent to the ategory of r-homogeneous polynomial representations of the general linear group GL(n; k). S hur algebras and their representations are losely onne ted with representations of the group algebra of the symmetri groups. Let r be the symmetri group on r symbols. Then the S hur algebra S(n; r) an be de ned as the endomorphism ring Endk r(E r), where E is an n-dimensional ve tor spa e over k, and where the symmetri group a ts on E r by pla e permutations. This is the rst in a series of papers on the representation theory and ombinatori s of S hur algebras S(2; r) and of group algebras of symmetri groups, and we des ribe now the overall goal. 1.2 The Symmetri Group. The Spe ht modules of the group group algebra k r are parametrized by partitions of r, we denote the Spe ht module orresponding to the partition by S . The simple k r-modules an be parametrized by p-regular partitions of r, and the simple module orresponding to is denoted by D ; here D is the unique simple quotient of S . The de omposition matrix of the group algebra k r is the matrix re ording the multipli ities [S : D ℄ of the simple k r-module D as a omposition fa tor of the Spe ht module S . For details on these results see the monograph by James [10℄. A lassi al example for a soalled Sierpinski gasket is given by the triangle of binomial oeÆ ients onsidered modulo a prime p. This fra tal stru ture is geometri ally built up from a