Generalized quotients in Coxeter groups

Abstract:

For ($W$, $S$) a Coxeter group, we study sets of the form \[ W/V = \{ w \in W|l(wv) = l(w) + l(v)\;{\text {for all}}\;v \in V\} ,\] , where $V \subseteq W$. Such sets $W/V$, here called generalized quotients, are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when $V \subseteq S$ (i.e., for minimal coset representatives modulo a parabolic subgroup). We show that Bruhat intervals in $W/V$, for general $V \subseteq W$, are lexicographically shellable. The Möbius function on $W/V$ under Bruhat order takes values in $\{ - 1, 0, + 1\}$. For finite groups $W$, generalized quotients are the same thing as lower intervals in the weak order. This is, however, in general not true. Connections with the weak order are explored and it is shown that $W/V$ is always a complete meet-semilattice and a convex order ideal as a subset of $W$ under weak order. Descent classes ${D_I} = \{ w \in W|l(ws) < l(w) \Leftrightarrow s \in I,\;{\text {for all}}\;s \in S\}$, $I \subseteq S$, are also analyzed using generalized quotients. It is shown that each descent class, as a poset under Bruhat order or weak order, is isomorphic to a generalized quotient under the corresponding ordering. The latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics. For instance, the set of standard Young tableaux of a fixed shape or the set of linear extensions of a rooted forest, suitably interpreted, form generalized quotients. We prove a factorization result for the quotients that come from rooted forests, which shows that algebraically these quotients behave as a system of minimal "coset" representatives of a subset which is in general not a subgroup. We also study the rank generating function for certain quotients in the symmetric group.