Free Coxeter Groups

Abstract

This chapter, mostly taken from an article of Shi and Yang, contains the full description of cells and the proof of Lusztig’s Conjectures for free Coxeter groups (often called universal Coxeter groups).

Appendices

Notes

The Kazhdan–Lusztig theory for free Coxeter groups (unequal parameters, partition into cells, Lusztig’s Conjectures) is due to Shi and Yang [ShYa1]: this chapter is mostly taken from their paper, except that we have made use of the Guilhot induction process in order to simplify some of their arguments.

The particular case of infinite dihedral groups (see Section 24.7), and particularly the description of the asymptotic algebras in terms of representation rings of ${\mathbf S}{\mathbf L}_2({\mathbb {C}})$ and ${\mathbf P}{\mathbf G}{\mathbf L}_2({\mathbb {C}})$ is due to Lusztig [Lus20, §7, §17, §18]. In the equal parameter case, this is a special case of the description of the asymptotic algebra of the lowest two-sided cell of an affine Weyl group thanks to the representation ring of the corresponding adjoint group over ${\mathbb {C}}$ (see [Lus6, Corollary 8.7], [Xi2] or [Lus20, §18.20]¹).

Exercises

Exercise 24.1.

Prove Theorem 24.6.1(c).

Exercise 24.2.

Assume here that $|S|=2$ and that $\varphi$ is constant. Prove that there is no non-trivial left cellular map.

Exercise 24.3.

Assume here that $|S|=2$ and that $\varphi$ is not constant. Prove that there is only one non-trivial bijective left cellular map (the one given by Proposition 24.7.17).