1. Coxeter Cosets

Let us assume that \((W,\mathbb{S})\) is a Coxeter system.

Definition

We say \(\phi : W \to W\) is a Coxeter automorphism of \(W\) if \(\phi(\mathbb{S}) = \mathbb{S}\).

Note that such an automorphism implicitly depends upon our choice of generating set \(\mathbb{S}\). If \(W\) is finite then all such sets \(\mathbb{S}\) are conjugate, hence in this case any automorphism is a Coxeter automorphism for some choice of \(\mathbb{S}\).

Given such a Coxeter automorphism we may consider the semidirect product \(\widetilde{W} = W \rtimes \langle \phi \rangle \cong \langle W, \phi \rangle\), where \(\langle\phi\rangle\) is the cyclic subgroup of the automorphism group \(\mathrm{Aut}(W)\) generated by \(\phi\). Of particular importance to representation theory is the coset

\[W\phi = \{(w,\phi) \mid w \in W\} \subset \widetilde{W},\]

which we call a Coxeter coset. Below we will identify \(W\) with its natural image \(\{(w,1) \mid w \in W\}\) in \(\widetilde{W}\).

In CharLiePy we have implemented general algorithms for working with Coxeter cosets, which we describe in this section. Mainly these algorithms deal with conjugacy classes and irreducible characters.

Warning

For the moment these functions only support the case where \(W\) is a finite Coxeter group.

The following class implements Coxeter cosets in CharLiePy.