3.1. Type \(\mathrm{A}_n\)

Throughtout \((W,\mathbb{S})\) will denote an irreducible Coxeter system of type \(\mathrm{A}_n\). We will also denote by \(\Phi\) a root system whose associated reflection group is \(W\) and by \(\Delta \subset \Phi^+ \subset \Phi\) a set of simple and positive roots respectively.

3.1.1. Root System

One may realise the indecomposabe root system of type \(\mathrm{A}_n\) in the real vector space

\[V = \{(x_1,\dots,x_n) \mid x_1+\cdots+x_n = 0\} \subset \mathbb{R}^n.\]

In the following table we assume that \(\{e_1,\dots,e_n\}\) denotes the standard basis of \(\mathbb{R}^n\) (i.e. \(e_i = (0,\dots,0,1,\dots,0)\) with 1 occurring in the ith position).

\(\Phi\)	\(\{e_i - e_j \mid i \neq j, 1 \leqslant i,j \leqslant n+1\}\)
\(\Phi^+\)	\(\{e_i - e_j \mid 1 \leqslant i < j \leqslant n+1\}\)
\(\Delta\)	\(\{\alpha_1 = e_1 - e_2,\dots, \alpha_n = e_n - e_{n+1}\}\)

The Cartan matrix of the root system is

\[\begin{split}C = \begin{bmatrix} 2 & -1 & 0 & 0 & \cdots & 0\\ -1 & 2 & -1 & 0 & \cdots & 0\\ 0 & -1 & 2 & -1 & \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ 0 & \cdots & 0 & -1 & 2 & -1\\ 0 & \cdots & 0 & 0 & -1 & 2 \end{bmatrix}.\end{split}\]

3.1.2. Coxeter Group

In the following table we denote by \(s_i\) the reflection of the corresponding simple root \(\alpha_i\) for \(i \in \{1,\dots,n+1\}\).

\(W\)	\(\mathfrak{S}_{n+1}\)
\(s_i\)	\((i, i+1)\)
\(|W|\)	\((n+1)!\)

Here \(\mathfrak{S}_{n+1}\) denotes the symmetric group on \(\{1,\dots,n+1\}\) and \((i,i+1) \in \mathfrak{S}_{n+1}\) is a basic transposition. The Coxeter matrix of the Coxeter system is

\[\begin{split}M = \begin{bmatrix} 1 & 3 & 2 & 2 & \cdots & 2\\ 3 & 1 & 3 & 2 & \cdots & 2\\ 2 & 3 & 1 & 3 & \cdots & 2\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ 2 & \cdots & 2 & 3 & 1 & 3\\ 2 & \cdots & 2 & 2 & 3 & 1 \end{bmatrix}.\end{split}\]

3.1.3. Conjugacy Classes

The cycle type of any element of the symmetric group \(\mathfrak{S}_{n+1}\) gives a partition of \(n+1\) which uniquely determines the conjugacy class containing the element (see 1.2 of [JK81]). To each partition \(\lambda \vdash n+1\) we denote by \(w_{\lambda}\) a very good representative of the corresponding conjugacy class. This may be constructed in the following way (see 2.1 of [GM97]).

If \(\lambda = (\lambda_1,\dots,\lambda_r)\) then there is a corresponding standard parabolic subgroup \(W_{\lambda}\) of \(W\) isomorphic to \(\mathfrak{S}_{\lambda_1} \times \cdots \times \mathfrak{S}_{\lambda_r}\). The element \(w_{\lambda}\) may then simply be taken as a Coxeter element of \(W_{\lambda}\). In particular we have \(w_{\lambda} = w_1\cdots w_r\) where

\[w_i = s_{\lambda_1+\cdots+\lambda_{i-1}+1}\cdots s_{\lambda_1+\cdots+\lambda_i}\]

Let us now write the partition \(\lambda\) as \((1^{a_1},\dots,n^{a_n})\) for some non-negative integers \(a_i\), then we have the centraliser order is given by

\[|C_W(w_{\lambda})| = \prod_{i=1}^n i^{a_i}a_i.\]

(see 1.2.15 of [JK81]).

3.1.4. Irreducible Characters

There is a bijection between the set of partitions of \(n+1\) and the irreducible characters of \(W\) which may be constructed as follows. For each partition \(\lambda\) we consider the standard parabolic subgroup \(W_{\lambda^*}\) of \(W\) isomorphic to \(\mathfrak{S}_{\lambda_1^*} \times \cdots \times \mathfrak{S}_{\lambda_r^*}\), where \(\lambda^* = (\lambda_1^*, \dots, \lambda_r^*)\) is the dual partition of \(\lambda\). The j-induction of the sign character of \(W_{\lambda^*}\)

\[\chi_{\lambda} = j_{W_{\lambda^*}}^W(\mathrm{sgn})\]

is then an irreducible character of \(W\) (see 5.4.5 of [GP00]). The map \(\lambda \mapsto \chi_{\lambda}\) then gives the required bijection.