## Descent Algebras

Descent algebras are associative but not semi-simple algebras associated
with Coxeter groups. The easiest one to describe shows what their fascination
is. We partition the permutations of *S*_{n} into
*2*^{n-1}
subsets according to the up-down shape of each permutation (e.g.
3 4 5 1 2 has shape 'up','up', 'down', 'up'). Now add
up the elements of each class (working in the group algebra). Miraculously,
you get the basis of an associative algebra with integer structure
coefficients!

The study of these algebras began in the 1970s and is still very active.
My part in it was some contributions towards the representation theory
in characteristic zero, and in characteristic *p*.

### Papers on descent algebras

- A new proof of a theorem of Solomon, Bull. London Math. Soc. 18 (1986), 351-354.
- Solomon's descent algebra revisited, Bull. London Math. Soc. 24 (1992), 545-551.
- The p-modular descent algebra of the symmetric group (with S. van
Willigenburg). Bull. London Math. Soc. 29 (1997), 407-414. (Postscript,
pdf)
- The p-modular descent algebras, Algebras and Representation
Theory 5 (2002), 101-113 (with S. van Willigenburg, G. Pfeiffer).
(Postscript,
pdf)