Quotients of Coxeter complexes and 𝑃-partitions

Abstract

Let $W$ be a finite reflection group acting on R$\sp n$. As $W$ preserves the unit sphere S$\sp{n-1}$, for any subgroup $G\ \subseteq\ \ W$, there is a quotient S$\sp{n-1} /G$ of this sphere under the action of $G$. We study the combinatorial (and topological) structure of these quotients as certain kinds of cell complexes (balanced simplicial posets). In particular, we give sufficient conditions on $G$ for the quotient to be Cohen-Macaulay or Gorenstein over a field $k$, and a simple characterization of those $G$ for which the quotient is a pseudomanifold, and when it is orientable as a pseudomanifold. We then look at quotients for particular classes of subgroups $G$, namely reflection subgroups, alternating subgroups of reflection subgroups, and their diagonal embeddings in the product groups $W\sp r$. For these groups, we show that the quotient is always partitionable, that in some cases it is shellable, and when shellable it is either a sphere or a disk. For all of these groups, the partitioning yields combinatorial interpretations for certain non-negative integers $\beta\sb J$ associated to the quotient known as the type-selected Mobius invariants. Applications to calculating invariant polynomials of permutation groups and their Hilbert series (as developed by Garsia and Stanton (GS)) are discussed. Our methods require an extension of some of the theory of P-partitions, and multi-partite P-partitions from the symmetric group $S\sb n$ to other finite reflection groups. In particular, for the hyperoctahedral group $B\sb n$, we work out analogues to almost all of the standard $P$-partition results. This yields hyperoctahedral analogues for the connection between posets and distributive lattices. These methods also suggest a new approach and generalization to the Neggers-Stanley conjecture. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690).