W-graph ideals and biideals

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.