When is the algebra of multisymmetric polynomials generated by the elementary multisymmetric polynomials?

Abstract

Multisymmetric polynomials are the $r$-fold diagonal invariants of the symmetric group ${\mathfrak{S}}_n$. Elementary multisymmetric polynomials are analogues of the elementary symmetric polynomials, in the multisymmetric setting. In this paper, we give a necessary and sufficient condition on a ring $A$ for the algebra of multisymmetric polynomials with coefficients in $A$ to be generated by the elementary multisymmetric polynomials.