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.
Erickson et al. (Wed,) studied this question.