Orbit Sums and Modular Vector Invariants

Abstract

Let m, n positive integers, R a commutative ring with the unit element 1, and

$$ A_{mn} = R\left[ {x_{11} , \ldots ,x_{m1} ; \ldots ;x_{1n} , \ldots ,x_{mn} } \right] $$

the algebra of polynomials in mn variables x_ij over R. The symmetric group S _n operates on the algebra A _mn as a group of R-automorphisms by the rule: σ(x_ij) = x_i,σ(j), σ ∈ G. Denote by $A_{mn}^{S_n }$ the subalgebra of invariants of the algebra A _mn with respect to S _n and define polarized elementary symmetric polynomials $u_{r_1 , \ldots ,r_m } \in A_{mn}^{S_n }$ in n vector variables (x₁₁,..., x_m1),..., (x_1n,..., x_mn) by means of the following formal identity

$$ \prod\limits_{j = 1}^n {\left( {1 + x_1 jz_1 + \cdots + x_{mj} z_m } \right) = 1 + \sum\limits_{1 \leqslant r_1 + \cdots + r_m \leqslant n} {u_{r1, \ldots rm} z_1^{r_1 } \cdots z_m^{r_m } .} } $$