Stanley decompositions of modules of covariants
William Q. Erickson, Markus Hunziker
TL;DR
The paper provides a uniform, combinatorial description of Stanley decompositions for classical modules of covariants $M_\sigma$ associated with $H \in \{\mathrm{GL}(V), \mathrm{O}(V), \mathrm{Sp}(V)\}$, parametrized by jellyfish, a lattice-path–based data set. It derives explicit Hilbert–Poincaré series as finite sums of rational functions, independent of Cohen–Macaulayness, via Stanley–Reisner theory and Howe duality; the method passes through standard monomial theory and shellings to handle GL and Sp, with careful treatment for $O$-cases. It further extends these decompositions to invariant rings for $\mathrm{SO}(V)$ and $\mathrm{SL}(V)$, and develops arc-diagram weight bases that connect to unitary highest weight representations in the Howe duality picture. As a step toward broader applicability, the authors generalize the approach to Wallach representations of simply laced type ADE, expressing their Hilbert series in terms of nonintersecting lattice paths on the noncompact root posets. The work links classical invariant theory, combinatorial topology, and representation theory, with potential implications for equivariant machine learning and structured symmetry-aware models.
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.