Exercise 22, § 21

Let $n$ be a natural number. Denote by $\mathscr{I}_n$ the set of decreasing finite sequences of strictly positive integers with sum $n$. Let $p = (p_i)_{1 \leq i \leq d}$ be an element of $\mathscr{I}_n$. For $1 \leq i \leq d$, denote by $P_i$ the set of integers $r$ such that $p_1 + \cdots + p_{i-1} < r \leq p_1 + \cdots + p_i$, and for $1 \leq j \leq p_1$, denote by $P'_j$ the set of integers of the form $p_1 + \cdots + p_{i-1} + j$ with $1 \leq i \leq d$ and $p_i \geq j$. The subsets $P_i$, on the one hand, and the subsets $P'_j$, on the other, form a partition of the interval $[1, n]$ in $\mathbf{N}$.

(We often represent this situation by a diagram, called a Young diagram: in the example below, we have $n = 9$ and $p = (4, 2, 2, 1)$. The set $P_i$ consists of the elements of the $i$-th row, and $P'_j$ of those of the $j$-th column.)

<table> <tr> <th>1</th> <th>2</th> <th>3</th> <th>4</th> </tr> <tr> <td>5</td> <td>6</td> <td></td> <td></td> </tr> <tr> <td>7</td> <td>8</td> <td></td> <td></td> </tr> <tr> <td>9</td> <td></td> <td></td> <td></td> </tr> </table>

Let $\mathscr{P}$ (resp. $\mathscr{P}'$) be the subgroup of $\mathfrak{S}_n$ consisting of the permutations $\sigma$ such that $\sigma(P_i) \subset P_i$ for every $i$ (resp. $\sigma(P'_j) \subset P'_j$ for every $j$). a) Prove that we have $\mathscr{P} \cap \mathscr{P}' = \{1\}$. b) Let $\sigma \in \mathfrak{S}_n$. Show that if $\operatorname{Card}(P_i \cap \sigma(P'_j)) \leq 1$ for every $i$ and $j$, then $\sigma$ belongs to $\mathscr{P} \mathscr{P}'$. Deduce that if $\sigma \notin \mathscr{P} \mathscr{P}'$, then the subgroup $\mathscr{P} \cap \sigma \mathscr{P}' \sigma^{-1}$ contains a transposition. c) In the group algebra $\mathbf{Z}[\mathfrak{S}_n]$, set

Prove that $c_{\mathbf{p}}$ satisfies $e_{\sigma}c_{\mathbf{p}}e_{\sigma'}=\varepsilon (\sigma ')c_{\mathbf{p}}$ for every $\sigma \in \mathscr{P}$ and $\sigma '\in \mathscr{P}'$ and that every element that has this property belongs to $\mathbf{Z}c_{\mathbf{p}}$ (write such a element as $\sum n_{\sigma}e_{\sigma}$, and deduce from b) that $n_{\sigma}$ is zero for $\sigma \notin \mathscr{P}\mathscr{P}')$. Deduce that for every element $x$ of $\mathbf{Z}[\mathfrak{S}_n]$, the element $c_{\mathbf{p}}xc_{\mathbf{p}}$ belongs to $\mathbf{Z}c_{\mathbf{p}}$.

d) Denote by A the $\mathbf{Q}$-algebra $\mathbf{Q}[\mathfrak{S}_n]$. Prove that the ideal $V_{\mathbf{p}}= Ac_{\mathbf{p}}$ of A is an absolutely simple A-module. (Observe that we have $c_{\mathbf{p}}V_{\mathbf{p}}\subset \mathbf{Q}c_{\mathbf{p}}$ by c). Prove that if $\mathfrak{a}$ is a left ideal strictly contained in $V_{\mathbf{p}}$, then we have $c_{\mathbf{p}}\mathfrak{a}= 0$, and therefore $\mathfrak{a}^2= 0$ and finally $\mathfrak{a}= 0.)$

e) Prove that we have $c^2_{\mathbf{p}}= (n$!$/[V_{\mathbf{p}}:\mathbf{Q}])c_{\mathbf{p}}$ (calculate the trace of the endomorphism $x\mapsto xc_{\mathbf{p}}$ of A).