*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.