Tales of the descent algebra

The descent algebra (introduced by Solomon in 1976) is a remarkable $2^{n-1}$-dimensional subalgebra of the group algebra of the symmetric group $S_n$, spanned by sums of all permutations with a given descent set. In some aspects, it is similar to the symmetric group algebra, while in others, it is entirely opposite, such as its heavy non-semisimplicity. (It also makes for a natural example of an algebra not isomorphic to its opposite.)

In this talk, I will present two new developments on the descent algebra:

1. A new basis of the symmetric group algebra represents the elements of the descent algebra as triangular matrices with combinatorially meaningful diagonal entries. This gives a new approach to Bidigare’s eigenvalue formulas for descent-related shuffles.
This is joint work with Ekaterina A. Vassilieva.

2. Certain generators of the descent algebra generate right ideals of the group algebra that are Gelfand models of $S_n$: representations that contain each irreducible exactly once. This leads to a new view of the classical Gelfand model of Adin, Postnikov and Roichman as well as a new proof of one of the Reiner-Saliola-Welker commutativities; we also resolve recent questions of Lafrenière.
This is joint work with Sarah Brauner, Patricia Commins and Franco Saliola.

For information about the Richard P. Stanley Seminar in Combinatorics, visit… https://math.mit.edu/combin/