Stanley decompositions of rings of invariants and certain highest weight Harish-Chandra modules

TL;DR

This paper develops a combinatorial approach using lattice paths to analyze polynomial invariants and modules of covariants for classical groups and highest weight modules, providing new formulas for Hilbert series and decompositions.

Contribution

It introduces a systematic combinatorial framework for Stanley decompositions and Hilbert series of invariants and modules, extending to non-Cohen-Macaulay cases and beyond classical groups.

Findings

Graphical description of graded linear bases

Explicit Hilbert series formulas via lattice paths

Decomposition of covariant modules using combinatorial methods

Abstract

The first half of this paper is largely expository, wherein we present a systematic combinatorial approach to the theory of polynomial (semi)invariants and multilinear invariants of several vectors and covectors, for the classical groups. This culminates in a graphical description of graded linear bases. By applying well-known results of lattice path combinatorics to Weyl's fundamental theorems of classical invariant theory, we write down Stanley decompositions and Hilbert-Poincare series in terms of families of non-intersecting lattice paths, enumerated with respect to certain corners. In the second half of the paper, we revisit the (semi)invariants in the first half as a special case of a much broader phenomenon. On one hand, polynomial invariants of a group $H$ can be generalized to modules of covariants, i.e., $H$-equivariant polynomial functions between $H$-modules. On the other hand, from the perspective of Roger Howe's theory of dual pairs, these modules of covariants can be viewed as infinite-dimensional simple $(\mathfrak{g}, K)$-modules. This suggests an expanded program in which our goal is to apply combinatorial techniques involving lattice paths in order to write down Hilbert series for arbitrary unitarizable highest-weight $(\mathfrak{g},K)$-modules. As a preview of future work in this program, we present examples showing how modules of covariants -- even those which are not Cohen-Macaulay, and therefore which we would not expect to be combinatorially nice -- can be decomposed in terms of lattice paths. We also extend these methods beyond the classical groups.