Witt vectors, modular Lie powers and the Solomon descent algebra
Manfred Schocker (University of Wales, Swansea)
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
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
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.
modules are defined implicitly by a description of the ghost components of
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
and algebraic skills onto the subject. Accordingly, the lecture will be as
self-contained as possible.
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