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.