Exercise 21, § 21

Let $n\in \mathbf{N}$. Denote by $\mathscr{D}_n$ the subset of $\mathbf{N}^n$ consisting of the elements $(p_1, . . . , p_n)$ such that $p_1\geqslant \cdots \geqslant p_n$; endow it with the lexicographical order. For $\mathbf{p}\in \mathscr{D}_n$, denote by $\mathbf{p}'$ the element of $\mathscr{D}_n$ defined by $p'_i=p_i+n-i$. Let $\mathbf{X}= (X_1, . . . ,X_n)$ be a sequence of variables; set $\Delta (\mathbf{X}) =\prod_{i<j}(X_i-X_j)$.

a) Let $P\in \mathbf{Z}[\mathbf{X}]$ be an antisymmetric polynomial, that is, a polynomial satisfying the relation $P(X_{\sigma(1)}, . . . ,X_{\sigma(n)}) =\varepsilon (\sigma )P(X_1, . . . ,X_n)$ for every element $\sigma \in \mathfrak{S}_n$. Prove that there exists a symmetric polynomial $Q\in \mathbf{Z}[\mathbf{X}]$ such that P = ΔQ.

b) For every $\mathbf{p}\in \mathscr{D}_n$, set $S_{\mathbf{p}}(\mathbf{X}) = \Delta (\mathbf{X})^{-1}$ det(X$^{p'_i}_j$)$_{1\leqslant i,j\leqslant n}$. Prove that $S_{\mathbf{p}}$ is a homogeneous symmetric polynomial of degree $\sum p_i$ with integer coefficients.

c) If $\mathbf{q}= (q_i)_{1\leqslant i\leqslant n}$ is an element of $\mathscr{D}_n$ and P is an element of $\mathbf{Q}[\mathbf{X}]$, then we denote by $[P]_{\mathbf{q}}$ the coefficient of $\mathbf{x}^{\mathbf{q}}$ in P. Set $K_{\mathbf{p}\mathbf{q}}= [S_{\mathbf{p}}]_{\mathbf{q}}$. Prove that we have $K_{\mathbf{p}\mathbf{p}}= 1$ and $K_{\mathbf{p}\mathbf{q}}= 0$ for $\mathbf{q}>\mathbf{p}$ (expand the equality det(X$_j^{p'_i}$) $= S_{\mathbf{p}}(\mathbf{X}) \Delta (\mathbf{X})$ in

the lexicographical order). Deduce that the polynomials $S_p$ for $p \in \mathscr{D}_n$ form a basis of the $\mathbf{Z}$-submodule of $\mathbf{Z}[\mathbf{X}]$ consisting of the symmetric polynomials. d) Let $\mathbf{Y} = (Y_1, \ldots, Y_n)$ be a sequence of variables. Prove the equality of formal power series

(use the equality $\det((1 - X_i Y_j)^{-1})_{1 \leq i, j \leq n} = \Delta(\mathbf{X}) \Delta(\mathbf{Y}) \prod_{1 \leq i, j \leq n} (1 - X_i Y_j)^{-1}$ (cf. III, §8, p. 639, Exercise 5) by calculating the left-hand side using the expansion of $(1 - T)^{-1}$ as a formal power series). e) Let $P \in \mathbf{Q}[\mathbf{X}]$ be a symmetric polynomial. For every $p \in \mathscr{D}_n$, set $\omega_p(P) = [P \Delta]_{p'}$. Prove the equality $\omega_p(S_q) = \delta_{pq}$. Deduce that for every $q \in \mathscr{D}_n$, we have f) For $\alpha \in \mathbf{N}^n$, set $f^\alpha(\mathbf{X}) = \prod_{j=1}^n (\sum_i X_i^j)^{\alpha_j}$ and $c_\alpha = \prod_{j=1}^n \frac{1}{j^{\alpha_j} \alpha_j!}$. Prove that we have $\sum_{\alpha \in \mathbf{N}^n} c_\alpha \omega_p(f^\alpha) \omega_q(f^\alpha) = \delta_{pq}$ for $p, q$ in $\mathscr{D}_n$ (observe that we have