W-graph ideals and biideals

TL;DR

This paper advances the theory of W-graph ideals by exploring subideals, induction, restriction, and introducing W-graph biideals, with a complete classification in rank 2 finite Coxeter groups.

Contribution

It introduces W-graph biideals and extends the understanding of W-graph ideals, including their classification in rank 2 finite Coxeter groups.

Findings

Classification of all W-graph ideals in rank 2 Coxeter groups

Introduction of W-graph biideals and their properties

Development of induction and restriction techniques for W-graph ideals

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.