W-graph ideals and biideals

We further develop the theory of $W\text{\hspace{0.17em}⁣}$-graph ideals, first introduced by the authors in reference [6]. We discuss $W\text{\hspace{0.17em}⁣}$-graph subideals, and induction and restriction of $W\text{\hspace{0.17em}⁣}$-graph ideals for parabolic subgroups. We introduce $W\text{\hspace{0.17em}⁣}$-graph biideals: those $W\text{\hspace{0.17em}⁣}$-graph ideals that yield $(W\times W^{\mathrm o})$-graphs,where$W^{\mathrm o}$isthegroupoppositeto$W$. We determine all $W\text{\hspace{0.17em}⁣}$-graph ideals and biideals in finite Coxeter groups of rank 2.