Koszulity, supersolvability and Stirling Representations

Almousa, Ayah; Reiner, Victor; Sundaram, Sheila

doi:10.5802/art.23

Almousa, Ayah ¹; Reiner, Victor ²; Sundaram, Sheila ²

¹ University of South Carolina Department of Mathematics Columbia, SC, USA
² University of Minnesota – Twin Cities School of Mathematics, Minneapolis, MN, USA

Annals of Representation Theory, Volume 2 (2025) no. 2, pp. 173-247.

Abstract

Supersolvable hyperplane arrangements and matroids are known to give rise to certain Koszul algebras, namely their Orlik–Solomon algebras and graded Varchenko–Gel’fand algebras. We explore how this interacts with group actions, particularly for the braid arrangement and the action of the symmetric group, where the Hilbert functions of the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively. The corresponding symmetric group representations exhibit branching rules that interpret Stirling number recurrences, which are shown to apply to all supersolvable arrangements. They also enjoy representation stability properties that follow from Koszul duality.

Received: 2024-08-18
Revised: 2024-12-11
Accepted: 2025-01-22
Published online: 2025-04-01

DOI: 10.5802/art.23

Classification: 16S37, 05B35
Keywords: Stirling number, Koszul algebra, Orlik–Solomon, Varchenko–Gelfand, quadratic algebra, Groebner basis, branching, holonomy Lie algebra, Drinfeld–Kohno, infinitesimal braid, chord diagrams

Author's affiliations:

Almousa, Ayah ¹; Reiner, Victor ²; Sundaram, Sheila ²

¹ University of South Carolina Department of Mathematics Columbia, SC, USA
² University of Minnesota – Twin Cities School of Mathematics, Minneapolis, MN, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Almousa, Ayah; Reiner, Victor; Sundaram, Sheila. Koszulity, supersolvability and Stirling Representations. Annals of Representation Theory, Volume 2 (2025) no. 2, pp. 173-247. doi : 10.5802/art.23. https://art.centre-mersenne.org/articles/10.5802/art.23/

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