Sur les invariants du groupe symétrique dans certaines représentations. (On the invariants of the symmetric group in certain representations) (Q1084500)

Denote by \(W_ n\) the ordinary irreducible representation of \(G=S_{2n+1}\), the symmetric group, which corresponds to the hook shaped Young diagram \([n+1,1,...,1]\). Let \(\mu_ n\) indicate the minimal number of generators of the algebra \(C[W_ n]^ G\), consisting of the G- invariant polynomial functions on \(W_ n\). Then it is proved that, for even n and prime numbers \(2n+1\), the fraction \(\mu_ n/\dim W_ n\) and even the fraction log \(\mu\) \({}_ n/\log \dim W_ n\) tends to \(+\infty\).