Abstract:

The absolute order is a natural partial order that can be defined on a Coxeter group W. Its Hasse diagram can be obtained from the Cayley graph of W, with respect to the set of all reflections of W, by directing its edges away from the identity. Reiner has asked whether the absolute order and whether its order ideal generated by the Coxeter elements are Cohen-Macaulay. We will discuss joint work with M. Kallipoliti, in which a positive answer is given for the case of the symmetric group, by use of a notion of constructibility for posets. We will also report on recent work of Kallipoliti which gives positive answers to these questions for the case of the hyperoctahedral group, by use of other methods.