Witt vectors, modular Lie powers and the Solomon descent algebra

Manfred Schocker (University of Wales, Swansea)

Abstract: The descent algebra of the symmetric group S_n, discovered by Solomon in the wider context of finite Coxeter groups, is a subalgebra of the integral group algebra of S_n. It is intimately related to the free algebra L(V) over a finite-dimensional vector space V, due to the work of Garsia and Reutenauer.
The graded component L^n(V) of L(V) is the nth Lie power of V. In the classical case (over a field of characteristic zero) the descent algebra has proven to be of tremendous help for the study of L^n(V) as a module for the general linear group GL(V). In fact, the structure of L^n(V) is reasonably well understood in this case.
The structure of modular Lie powers (over a field of prime characteristic p) has largely been a mystery. In recent collaboration with Bryant and Erdmann, we employed the descent algebra in this case as well. We could reduce the general problem to the case where n is a power of p. As a by-product, there is a set of canonical modules B_n which are determined by their characters and which can therefore be studied using the theory of symmetric functions. These modules are defined implicitly by a description of the ghost components of the Witt vector (B_1,B_2,B_3,...).
So far, we have not been able to derive an explicit description of B_n from this, and we would like to invite the audience to introduce their combinatorial and algebraic skills onto the subject. Accordingly, the lecture will be as self-contained as possible.

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