W-graph ideals and biideals

Robert B. Howlett and Van Minh Nguyen

Abstract

We further develop the theory of \(W\!\)-graph ideals, first introduced by the authors in reference [6]. We discuss \(W\!\)-graph subideals, and induction and restriction of \(W\!\)-graph ideals for parabolic subgroups. We introduce \(W\!\)-graph biideals: those \(W\!\)-graph ideals that yield \((W\times W^{\mathrm o})\)-graphs, where \(W^{\mathrm o}\) is the group opposite to \(W\). We determine all \(W\!\)-graph ideals and biideals in finite Coxeter groups of rank 2.

Keywords: Coxeter groups, Hecke algebras, W-graphs, Kazhdan–Lusztig polynomials, cells.

AMS Subject Classification: Primary 20C08.