Breadcrumb
Cartan invariants of generalised blocks of symmetric groups
Wed 27 February 2013, 14:30
Anton Evseev
University of Birmingham
Organisers: Tony Skyner, Neil Saunders
ABSTRACT
For a finite group G and a prime l, an important invariant of l-modular representation theory G is the Cartan matrix, which is the Gram matrix formed by the scalar products of projective indecomposable characters of G. Kulshammer, Olsson, and Robinson generalized many character-theoretic aspects of this theory to the case when G is a symmetric group but l is no longer assumed to be a prime. Their KOR conjecture gave a possible combinatorial description of the invariant factors of the Cartan matrix in this context (and also for Iwahori-Hecke algebras at l-th roots of unity). In papers of Hill and Bessenrodt-Hill, the KOR conjecture was reduced to finding the invariant factors of a certain matrix X, with rows and columns indexed by partitions of a fixed number w. In the talk, I will define X and discuss further reasons for studying it. The talk will go on to explore some aspects of a recent proof that the invariant factors of X are exactly those conjectured by Hill, and hence of the KOR conjecture.