Exercise 5, § 6

Let $K$ be a commutative field and $f \in K[X_1, \ldots, X_n, Y_1, \ldots, Y_n]$; for every permutation $\sigma \in S_n$ denote by $\sigma f$ the rational fraction

Then $f$ is said to be symmetric with respect to the pairs $(X_i, Y_i)$ if $u f = f$ for all $\sigma \in S_n$. a) For every element $\nu = (A, \ldots, A, \mu_1, \ldots, \mu_n)$ of $\mathbf{N}^{2n}$ and every permutation $\sigma \in S_n$ write

so that $\mathfrak{S}_n$ in this way becomes a group of operators on $\mathbf{N}^{2^n}$. Let $\gamma$ be any orbit of $\mathfrak{S}_n$ in $\mathbf{N}^{2^n}$, and denote by $u_\gamma$ the symmetric polynomial

where $(\lambda_1, \ldots, \lambda_n, \mu_1, \ldots, \mu_n)$ ranges over $\gamma$. Show that the polynomials $u_\gamma$ form a basis over $K$ of the vector space of all polynomials that are symmetric in the $(X_i, Y_i)$.

\* b) Suppose that $K$ has characteristic 0. Show that every polynomial $u_\gamma$ is equal to a polynomial with rational coefficients in the symmetric functions

(argue by induction on the number of pairs $(A_i, \mu_i)$ which are not $(0, 0)$ in $\nu = (\lambda_1, \ldots, \lambda_n, \mu_1, \ldots, \mu_n)$ in considering the products $u, v_{\lambda \mu}$).

c) By elementary symmetric functions of the $(X_i, Y_i)$ we shall understand the $n(n+3)/2$ polynomials $w_{hk} = u$, corresponding to the orbits $\gamma$ of elements $(A_i, \ldots, A_i, \mu_1, \ldots, \mu_n)$ other than $(0, \ldots, 0)$ and such that h of the pairs $(\lambda_i, \mu_i)$ are equal to $(1, 0)$, k others are $(0, 1)$ and the remaining $n - h - k$ are $(0, 0)$. Show that if $K$ is of characteristic 0, then every symmetric polynomial in the $(X_i, Y_i)$ is equal to a polynomial in the elementary symmetric functions with coefficients in $K$ (consider the sums $\sum_{i=1}^n (UX_i + VY_i)^k$, where U and V are two indeterminates, and use b)).

d) If $K$ is a field of characteristic 3, and $n \geq 4$, show that $\sum_{i=1}^n X_i^2 Y_i^2$ is not equal to any polynomial in the symmetric functions $w_{hk}$ with coefficients in $K$. \*