Tight quotients and double quotients in the Bruhat order.

John R. Stembridge

The electronic journal of combinatorics, The Stanley Festschrift volume, Tome 11 (2004) no. 2

Résumé

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., "double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are "tight" in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups.

Détail

DOI : 10.37236/1871

Classification : 20F55, 06A07
Mots-clés : Bruhat order, Coxeter groups, maximal parabolic subgroups, reflection representations, root systems, order dimensions, affine Weyl groups