Exercise 23, § 21

Keep the notation of the previous exercise. Let $\mathbf{p}= (p_i)$ and $\mathbf{q}= (q_j)$ be two elements of $\mathscr{S}_n$.

a) Suppose that we have $\mathbf{p}>\mathbf{q}$ for the lexicographical order. Prove that we have $a_{\mathbf{p}}xb_{\mathbf{q}}= 0$ for every $x\in A$. (Reduce to the case $x= 1$. Observe that if $(P_i),(P'_j)$ denote the partitions associated with $\mathbf{p}$ and $(Q_k),(Q'_l)$ those associated with $\mathbf{q}$, then, denoting by $s$ the smallest integer $n$ such that $p_n\not=q_n$, we have Card(P$_s\cap Q'_1$)$\geqslant 2$, and construct a transposition $\tau$ such that $a_{\mathbf{p}}\tau =a_{\mathbf{p}}$ and $\tau b_{\mathbf{q}}=-b_{\mathbf{q}}.$)

b) Deduce that if $\mathbf{p}\not=\mathbf{q}$, then the A-modules $V_{\mathbf{p}}$ and $V_{\mathbf{q}}$ are not isomorphic (apply a), using the antiautomorphism of A that sends $e_{\sigma}$ to $e_{\sigma^{-1}})$.

c) Prove that the mapping that sends a sequence $\mathbf{p}$ to the representation $(V_{\mathbf{p}})_{(K)}$ defines a bijection from $\mathscr{S}_n$ to the set $\mathscr{S}_K(\mathfrak{S}_n)$ of classes of simple $K[\mathfrak{S}_n$]-modules. d) For $\mathbf{p}\in \mathscr{S}_n$, denote by $\overline{\mathbf{p}}$ the sequence defined by $\overline{p}_j=$ Card(P$'_j$) (Exercise 22). Prove that the mapping $\mathbf{p}\mapsto \overline{\mathbf{p}}$ is a involution of $\mathscr{S}_n$; the A-module $V_{\overline{\mathbf{p}}}$ is isomorphic to $V_{\mathbf{p}}\otimes_{\mathbf{Q}}\mathbf{Q}_{\varepsilon}$, where $\mathbf{Q}_{\varepsilon}$ denotes the 1-dimensional representation over $\mathbf{Q}$ associated with the signature.

e) Describe the representations corresponding to the sequences $(1, . . . ,1)$ and $(n)$, as well as to the sequence $(n-1,1)$.