Ulam Seminar - Math 8815 - Fall 2012
This is the class web page for Math 8815 in Fall 2012 at CU Boulder.
Instructor Matt Douglass
This Fall's Ulam seminar will be on a topic in algebraic combinatorics. Below are a title and abstract with more specific information. One feature of algebraic combinatorics is that frequently not a lot of prior knowledge is required to get started. For this seminar, a good undergraduate algebra course could be sufficient background. A standard beginning graduate algebra class is more than sufficient.
Fancy title: Toward a non-commutative character theory of the hyperoctahedral group
Plain title: An analog of Solomon's descent algebra for signed permutations
Abstract: The "descent set" of a permutation w of 1, ..., n, is the set of i such that w(i)<w(i+1). The set of permutations of 1, ..., n with a given descent set is called a descent class. The formal sum of the elements in a descent class is an element in the group algebra of the symmetric group on n letters. The set of such elements for all possible descent sets form a subalgebra of the group algebra known as Solomon's descent algebra or just the descent algebra.
Descent algebras were first studied by Solomon in 1976. In the late 1980s and early 1990s there was a flurry of activity when it was discovered that the descent algebra could be used to study a canonical decomposition of the free Lie algebra (Bergeron, Garsia, Reutenaeur) and a Hodge-type decomposition in cyclic homology (Hanlon). In the mid-1990s Malvenuto and Reutenaeur showed that the direct sum of all descent algebras is a Hopf algebra that is dual to the Hopf algebra of quasi-symmetric polynomials. In 2005, Blessenohl and Schocker pushed the Hopf algebra approach further and developed what they call "non-commutative character theory of the symmetric group." In the past several years, several authors have studied the finer structure of descent algebras (Loewy series, quiver presentations,...) In another direction, descent algebras arise as rings of invariants in semi-group algebras. In this guise, they have been used by Brown, Diaconis, and others to analyze random walks induced by different methods of shuffling a deck of cards (notice that shuffling of a deck of n cards is just a permutation of 1, ..., n).
In this seminar we will investigate an analog of the descent algebra when permutations of 1, ..., n are replaced by signed permutations of 1, ..., n, with the goal of making some progress on understanding a recent conjecture of Bonnafé that has applications to computing the (singular) cohomology of hyperplane complements associated with classical (algebraic or Lie) groups.
Course Notes ulam-final.pdf