Exercise 24, § 21

Keep the notation of Exercises 21 through 23. Let $\mathbf{p}= (p_i)_{1\leqslant i\leqslant d}\in \mathscr{S}_n$; denote by $\mathbf{p}$ the element $(p_1, . . . , p_d,0, . . . ,0)$ of $\mathscr{D}_n$. If $\boldsymbol{\alpha }= (\alpha_1, . . . , \alpha_n)$ is an element of $\mathbf{N}^n$ such that $\sum_ii\alpha_i=n$, then we denote by $C_{\boldsymbol{\alpha }}$ the conjugacy class of $\mathfrak{S}_n$ consisting of the elements that are the product of a family of cycles with disjoint support consisting of $\alpha_2$ cycles of order $2,\alpha_3$ cycles of order 3, etc.

a) Prove that the representation $\rho_{\mathbf{p}}$ of $\mathfrak{S}_n$ in the A-module $Aa_{\mathbf{p}}$ is the representation induced by the trivial representation of $\mathscr{P}$. Use Exercise 20 to deduce the formula $\chi_{\rho_{\mathbf{p}}}(C_{\boldsymbol{\alpha }}) = [f^{\boldsymbol{\alpha }}]_{\mathbf{p}}$.

b) For any $\mathbf{q}\in \mathscr{S}_n$, denote by $\lambda_{\mathbf{q}}$ the representation of $\mathfrak{S}_n$ in the A-module $V_{\mathbf{q}}$. Prove that there exist positive integers $n_{\mathbf{p}\mathbf{q}}$ such that we have $\chi_{\rho_{\mathbf{p}}}=\sum_{\mathbf{q}\in\mathscr{S}_n}n_{\mathbf{p}\mathbf{q}}\chi_{\lambda_{\mathbf{q}}}$ with $n_{\mathbf{p}\mathbf{p}}>0$.

c) Let $\eta_{\mathbf{p}}$ be the central function on $\mathfrak{S}_n$ with value $\omega_{\mathbf{p}}(f^{\boldsymbol{\alpha }})$ on the elements of the class $C_{\boldsymbol{\alpha }}$. Prove that $\eta_{\mathbf{p}}$ is a linear combination of the characters $\chi_{\lambda_{\mathbf{q}}}$ for $\mathbf{q}\in \mathscr{S}_n$ with integer coefficients (use Exercise 21, c) and e)). Deduce that $\eta_{\mathbf{p}}$ is equal, up to a sign, to one of the characters $\chi_{\lambda_{\mathbf{q}}}$. (Calculate $\langle \eta_{\mathbf{p}}, \eta_{\mathbf{p}}\rangle$ using Exercise 21, f).) d) Deduce the equality $\chi_{\lambda_{\mathbf{p}}}(C_{\boldsymbol{\alpha }}) =$ [Δ$f^{\boldsymbol{\alpha }}]_{\mathbf{p}'}$ (“Frobenius formula”).

e) Prove that the multiplicity of the irreducible representation $\lambda_{\mathbf{p}}$ in the representation $\rho_{\mathbf{q}}$ is equal to $K_{\mathbf{p}\mathbf{q}}$.

f ) Prove that the degree of $\lambda_{\mathbf{p}}$ is $\frac{n!}{\mathbf{p}!}$Δ($\mathbf{p}'$) (expand the polynomial Δ($\mathbf{X}$)$(\sum X_i)^n$).