Vector invariants of symmetric groups in prime characteristic

Abstract

Let R be a commutative ring with the unit element 1 and Sn be the symmetric group of degree n 2::: 1. Let A~ denote the subalgebra of invariants of the polynomial algebra with respect to Sn, The classical result of H. Weyl implies that if every non-zero integer is invertible in R, then the algebra A~n is generated by the polarized elementary symmetric polynomials of degree at most n, no matter how large m is. As it was recently shown by D. Richman, this result remains true under the condition that ISnl = n! is invertible in R. On the other hand, if Risa field of prime characteristic p ~ n, D. Richman proved that every system of R-algebra generators of A~n contains a generator whose degree is no less than max { n, ( m + p - n) / (p - 1)}. The last result implies that the above Weyl bound on degrees of generators no longer holds when the characteristic p of R divides ISnl, In general, it is proved that, for an arbitrary commutative ring R, the algebra A~n is generated by the invariants of degree at most max{n,mn(n - 1)/2}. The purpose of this paper is to give a simple arithmetical proof of the first result of D. Richman and to sharpen his second result, again with the use of new arithmetical arguments. Independently, a similar refinement of Richman's lower bound was given by G. Kemper on the basis of completely different considerations. A recent result of P. Fleischmann shows that the lower bound obtained in the paper is sharp if m > 1 and n is a prime power, n = pa.