Stanley decompositions of modules of covariants
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914391561076736 |
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| author | Erickson, William Q. Hunziker, Markus |
| author_facet | Erickson, William Q. Hunziker, Markus |
| contents | 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 $φ: W \longrightarrow U$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_16749 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stanley decompositions of modules of covariants Erickson, William Q. Hunziker, Markus Combinatorics Representation Theory 05E10 (Primary) 13A50, 22E47, 17B10 (Secondary) 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 $φ: W \longrightarrow U$. 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. |
| title | Stanley decompositions of modules of covariants |
| topic | Combinatorics Representation Theory 05E10 (Primary) 13A50, 22E47, 17B10 (Secondary) |
| url | https://arxiv.org/abs/2312.16749 |