Vector invariants of symmetric groups in the case of a field of prime characteristic

S. A. Stepanov

Diskretnaya Matematika, Tome 12 (2000) no. 4, pp. 25-38

Résumé

Let $R$ be a commutative ring with the unit element $1$ and $S_n$ be the symmetric group of degree $n \geq 1$. Let $A_{mn}^{S_n}$ denote the subalgebra of invariants of the polynomial algebra $$ A_{mn}=R[x_{11},\dots,x_{1n};\dots;x_{m1},\dots,x_{mn}] $$ with respect to $S_n$. The classical result of H. Weyl implies that if every non-zero integer is invertible in $R$, then the algebra $A_{mn}^{S_n}$ 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 $|S_n|=n!$ is invertible in $R$. On the other hand, if $R$ is a field of prime characteristic $p \leq n$, D. Richman proved that every system of $R$-algebra generators of $A_{mn}^{S_n}$ 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 $|S_n|$. In general, it is proved that, for an arbitrary commutative ring $R$, the algebra $A_{mn}^{S_n}$ 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, $n=p^\alpha$.