Descent Algebra

This story starts with Louis Solomon’s paper

• A Mackey formula in the group ring of a Coxeter group, Journal of Algebra 41 (1976) 255-268.

where he proves that the linear span of the sums

$\displaystyle y_T:=\sum_{{\mathrm{Des}}(w)=T}\ w,\qquad {\rm with}\qquad T\subseteq S,$

forms a subalgebra of the group algebra of a Coxeter group $W,$ with generator set $S$. Here ${\mathrm{Des}}(w)$ stands for the descent set of $w\in W,$ which is to say

${\mathrm{Des}}(w):=\{s\in S\ |\ \ell(w\,s)<\ell(w) \}.$

As usual, one denotes by $\ell(w)$ the length of a reduced expression of $w$ in terms of products of elements in $S$. The resulting algebra plays a role in many interesting situations. A closely related algebra, called the shuffle algebra, is obtained by applying the anti-endomorphism of the group algebra that corresponds to inverting all elements of the group. Properties of this last algebra has been exploited by Persi Diaconis in the study of card shuffling. This is nicely discussed and generalized in a paper by J. Fulman.