Stability theorems for multiplicities in graded $S_n$-modules

In this paper, we prove several stability theorems for multiplicities of naturally defined representations of symmetric groups. The first such theorem states that if we consider the diagonal action of the symmetric group $S_{m+r}$ on $k$ sets of $m+r$ variables, then the dimension of the invariants of degree $m$ is the same as the dimension of the invariants of degree $m$ for $S_{m}$ acting on $k$ sets of $m$ variables. The second type of stability result is for Weyl modules. We prove that the dimension of the $S_{n+r}$ invariants for a Weyl module, ${}_{m+r}F^{lambda}$ (the Schur-Weyl dual of the $S_{|lambda|}$ module $V^{lambda}$) with $leftvert lambda rightvert leq m$ is of the same dimension as the space of $S_{m}$ invariants for ${}_{m}F^{lambda}$. Multigrad