Stanley decompositions of modules of covariants

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 φ : W ⟶ U . In this paper, we take H to be one of the classical groups GL ( V ) , O ( V ) , or 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 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 SL ( V ) and SO ( V ) . Our proofs 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.