Stanley decompositions of modules of covariants

Abstract

Let $H$ be a complex reductive group, with finite-dimensional representations $W$ and $U$. The module of covariants for $W$ of type $U$ is the space of all $H$-equivariant polynomial maps $\varphi: W \longrightarrow U$. In this paper, we take $H$ to be one of the classical groups $\operatorname{GL}(V)$, $\operatorname{O}(V)$, or $\operatorname{Sp}(V)$, where $W$ is a direct sum of copies of $V$ and $V^*$, and $U$ is an arbitrary rational representation (with $U$ restricted to exterior powers of $V$ in the $H= \operatorname{O}(V)$ case). Our main result gives uniform Stanley decompositions of these modules of covariants, with Stanley spaces parametrized by combinatorial objects we call jellyfish. As a corollary, we write down the Hilbert series as a finite sum of rational functions, each with a combinatorial interpretation in terms of lattice paths. Notably, these results do not rely on the module being Cohen-Macaulay. We further apply our methods to invariant rings for $\operatorname{SL}(V)$ and $\operatorname{SO}(V)$. Our proofs (for $H = \operatorname{GL}(V)$ and $\operatorname{Sp}(V)$) rely on previous work by Jackson on standard monomial theory for dual reductive pairs, since classical modules of covariants can be viewed via Howe duality as Harish-Chandra modules of unitary highest weight representations of a certain real reductive group. As a first step toward extending this program to arbitrary unitary highest weight representations (including those of the exceptional groups), we establish analogous results uniformly for the Wallach representations of type ADE.