Stable graded multiplicities for harmonics on a cyclic quiver

Frohmader, Andrew; Heaton, Alexander

doi:10.5802/alco.391

Frohmader, Andrew ¹; Heaton, Alexander ²

¹ University of Wisconsin - Milwaukee Department of Mathematics 3200 N. Cramer St. Milwaukee WI 53211 (USA)
² Lawrence University Department of Mathematics, Computer Science, and Statistics 711 E. John St. Appleton WI 54911 (USA)

Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1603-1614.

Abstract

We consider Vinberg $θ$ -groups associated to a cyclic quiver on $k$ nodes. Let $K$ be the product of the general linear groups associated to each node. Then $K$ acts naturally on $\oplus Hom (V_{i}, V_{i + 1})$ and by Vinberg’s theory the polynomials are free over the invariants. We therefore consider the harmonics as a representation of $K$ , and give a combinatorial formula for the stable graded multiplicity of each $K$ -type. A key lemma provides a combinatorial separation of variables that allows us to cancel the invariants and obtain generalized exponents for the harmonics.

Received: 2024-04-06
Accepted: 2024-07-12
Published online: 2025-01-07

DOI: 10.5802/alco.391

Classification: 22E47, 20G05, 05E10
Mots-clés : Vinberg $\theta $-group, cyclic quiver, harmonic polynomials, graded multiplicity, crystal base

Author's affiliations:

Frohmader, Andrew ¹; Heaton, Alexander ²

¹ University of Wisconsin - Milwaukee Department of Mathematics 3200 N. Cramer St. Milwaukee WI 53211 (USA)
² Lawrence University Department of Mathematics, Computer Science, and Statistics 711 E. John St. Appleton WI 54911 (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Frohmader, Andrew; Heaton, Alexander. Stable graded multiplicities for harmonics on a cyclic quiver. Algebraic Combinatorics, Volume 7 (2024) no. 6, pp. 1603-1614. doi : 10.5802/alco.391. https://alco.centre-mersenne.org/articles/10.5802/alco.391/

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