Endomorphism Algebras and Representation Theory

Abstract

Endomorphism algebras figure prominently in group representation theory. For example, if G is a finite group of Lie type, the representation theory of the endomorphism algebra End_G( $\mathop {\left. \mathbb{C} \right|}\nolimits_B^G$)—sometimes known as the Hecke algebra over $\mathbb{C}$ of G—plays a central role in unraveling the complex unipotent characters on G [2, 7]. Another example arises in the modular representation theory of the finite general linear group G = GL _n (q) over an algebraically closed field k of characteristic p not dividing q. In this so-called non-describing characteristic representation theory, the Hecke algebras H over k associated with symmetric groups provide a link between the representation theory of G and that of quantum groups. In fact, if T denotes the direct sum of the various “transitive” q-permutation modules for H, then the endomorphism algebras End _H (T) are Morita equivalent to q-Schur algebras over k. In work of Dipper and James (see [8, 10, 11]) the decomposition numbers for kG are proved to be completely determined by the decomposition numbers for certain of these “quantized Schur algebras”. In turn, the representation theory of these latter algebras relates closely to that of the quantum linear group GL _n,q (k) over k.