3.2. Type \({}^2\mathrm{A}_n\)

3.2.1. Coxeter Coset

There is a unique non-trivial automorphism \(\phi : W \to W\) stabilising the set of Coxeter generators \(\mathbb{S}\). It may be realised as conjugation by the longest element \(w_0 \in W\).

3.2.2. \(\phi\)-Conjugacy Classes

As \(\phi\) can be realised as an inner automorphism the \(\phi\)-conjugacy classes of \(W\) are simply the conjugacy classes of \(W\). In particular, the map \(w \mapsto ww_0\) defines a bijection between the conjugacy classes and \(\phi\)-conjugacy classes of \(W\) (see 2.9 of [GKP00]). The map

\[\lambda \mapsto w_{\lambda} \mapsto w_{\lambda}w_0\]

gives a bijection between the partitions of \(n+1\) and the \(\phi\)-conjugacy classes of \(W\). Here \(w_{\lambda}\) is as in Conjugacy Classes.

To construct minimal length representatives of the \(\phi\)-conjugacy classes we will need to work in the symmetric group \(\mathfrak{S}_{n+1}\). Let \(\lambda\) be a partition of \(n+1\). We will denote by \(\mu = (\mu_1,\dots,\mu_r)\) a maximal composition obtained from \(\lambda\) by rearranging the entries. In other words, there exists \(k\) \((0 \leqslant k \leqslant r)\) such that \(\mu_1,\dots,\mu_k\) are even numbers (in any order) and \(\mu_{k+1},\dots,\mu_r\) are odd numbers such that \(\mu_{k+1} \geqslant \cdots \geqslant \mu_r\).

Now, to \(\mu\) we define an element \(\sigma_{\mu} \in \mathfrak{S}_{n+1}\) as follows. Consider the list \((a_1,\dots,a_{n+1})\) with \(a_{2i-1} = i\) and \(a_{2i} = n+1 - (i-1)\) for all \(1 \leqslant i \leqslant n+1\) then we set

\[\sigma_{\mu} = \sigma_{\mu_1}\sigma_{\mu_2}\cdots\sigma_{\mu_r}\]

where \(\sigma_{\mu_i}\) is the \(\mu_i\)-cycle given by

\[\sigma_{\mu_i} = (a_{\mu_1+\cdots+\mu_{i-1}+1}, a_{\mu_1+\cdots+\mu_{i-1}+2},\dots, a_{\mu_1+\cdots+\mu_{i-1}+\mu_i}).\]

By Theorem 3.3 of [GKP00] we have \(\sigma_{\mu}w_0\) is an element of minimal length in its \(\phi\)-conjugacy class. From this we obtain a reduced expression for this element by applying Algorithm A from pg. 9 of [GP00].

Warning

It is an open problem to determine whether these minimal length representatives are good in the sense of Definition 5.3 of [GKP00].

3.2.3. Irreducible Characters

As \(\phi\) can be realised as an inner automorphism we have all irreducible characters of \(W\) are \(\phi\)-stable. Thus the irreducible characters of the coset \(W\phi\) are labelled by the partitions of \(n+1\). Each such irreducible character is obtained by restricting an extension \(\tilde{\chi}_{\lambda}\) of \(\widetilde{W} = W \rtimes \langle \phi \rangle\) to the coset \(W\phi\). Here we choose Lusztig’s preferred extension. In particular, if \(E\) is the simple module affording \(\chi_{\lambda}\) then \(\phi\) acts on \(E\) as \((-1)^{a_{\lambda}}w_0\) where \(a_{\lambda}\) is the a-invariant of \(\chi_{\lambda}\).