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. Building on this stability, the last section looks at the Hilbert series of coinvariants of the polynomial ring in $k$ sets of $m$ variables. We address a conjecture that the Hilbert series, in degrees no more than $m$, can be computed by a truncated power series expression. Using some auxiliary results and manipulations of power series, we show that if this holds for $k$ and $m$, then the truncation gives the correct Hilbert series up to degree $m$ for $k$ sets of $n \geq m$ variables. This shows the validity of the conjecture up to certain degrees. We also provide a new equivalent conjecture regarding Gr\"{o}bner bases. The second type of stability result is for Weyl modules. We prove that the dimension of the $S_{m+r}$ invariants for a Weyl module ${}_{m+r}F^{\lambda}$ (the Schur-Weyl dual of the $S_{|\lambda|}$ module $V^{\lambda}$) with $\left\vert \lambda \right\vert \leq m$ is of the same dimension as the space of $S_{m}$ invariants for ${}_{m}F^{\lambda}$. Multigraded versions of the first type of result are given, as are multigraded generalizations to non-trivial modules of symmetric groups.