Algebraic and Topological Combinatorics

Special Session at the Fall Eastern Section Meeting
of the American Mathematical Society

Williams College, Williamstown, MA
October 13-14, 2001

Ira M. Gessel
(Brandeis University)

Variations of hyperoctahedral descent algebras

Abstract [ps] [pdf]: A descent of a permutation $a_1\cdots a_n$ of $\{1,2,\dots,n\}$ is an $i$ such that $a_i>a_{i+1}$. We define descents of signed permutations similarly, but we also count 0 as a descent if $a_1<0$. The descent set of a (signed) permutation is the set of its descents, and the set of all permutations with a given descent set is a descent class. For both ordinary permutations (the symmetric group) and signed permutations (the octahedral group), the set of sums of descent classes spans a subalgebra of the group algebra, called the descent algebra. Moreover, the set of sums of permutations with a given number of descents spans a commutative subalgebra of the descent algebra, in which the multiplication constants are given by a simple generating function. I will discuss some variations of these results when the definition of descents for signed permutations is modified so that $n$ is considered a descent if $a_n>0$.

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last updated: August 8, 2001