 
Summary: The pmodular Descent Algebra of the
Symmetric Group
S.J. van Willigenburg \Lambda
M.D. Atkinson
School of Mathematical and Computational Sciences
North Haugh, St Andrews KY16 9SS, UK
October 1, 1996
Abstract
The descent algebra of the symmetric group, over a field of nonzero
characteristic p, is studied. A homomorphism into the algebra of general
ised pmodular characters of the symmetric group is defined. This is then
used to determine the radical, and its nilpotency index. It also allows the
irreducible representations of the descent algebra to be described.
1 Introduction
In 1976, Louis Solomon defined a family of algebras associated with Coxeter
groups [6]. In the case of symmetric groups their definition can be expressed as
follows:
If oe is any permutation in the symmetric group Sn written in image form
(e.g. [1342]) then the signature of oe is the sequence of signs fx i g n\Gamma1
i=1 where
