Homology representations arising from the half cube, II

In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes of the $n$-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least $k$, and we proved that the homology of such a subcomplex is concentrated in degree $k-1$. This homology group supports a natural action of the Coxeter group $W(D_n)$ of type $D$. In this paper, we explicitly determine the characters (over ${\Bbb C}$) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group $S_n$ by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of \sym_n agree (over ${\Bbb C}$) with the representations of \sym_n on the $(k-2)$-nd homology of the complement of the $k$-equal real hyperplane arrangement.Comment: 19 pages AMSTeX. One figure. The Conjecture in the previous version is now a Theorem. This research was supported by NSF grant DMS-090576

On rank functions for heaps

Motivated by work of Stembridge, we study rank functions for Viennot's heaps of pieces. We produce a simple and sufficient criterion for a heap to be a ranked poset and apply the results to the heaps arising from fully commutative words in Coxeter groups.Comment: 18 pages AMSTeX, 3 figure

On planar algebras arising from hypergroups

Let $A$ be an associative algebra with identity and with trace. We study the family of planar algebras on 1-boxes that arise from $A$ in the work of Jones, but with the added assumption that the labels on the 1-boxes come from a discrete hypergroup in the sense of Sunder. This construction equips the algebra $P_n^A$ with a canonical basis, \BB_n^A, which turns out to be a ``tabular'' basis. We examine special cases of this construction to exhibit a close connection between such bases and Kazhdan--Lusztig bases of Hecke algebras of types $A$, $B$, $H$ or $I$.Comment: AMSTeX, approx. 32 pages, 11 figure