Gaudin Algebras, RSK and Calogero-Moser Cells in Type A

2020/12/18 by Adrien Brochier, Iain J. Gordon, Brochier, Adrien +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2012.10177

openalex publication_date 2020/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the spectrum of a family of algebras, the inhomogeneous Gaudin algebras, acting on the n-fold tensor representation ℂ[x₁, …, x_r]^{⊗ n} of the Lie algebra \mathfrakgl_r. We use the work of Halacheva-Kamnitzer-Rybnikov-Weekes to demonstrate that the Robinson-Schensted-Knuth correspondence describes the behaviour of the spectrum as we move along special paths in the family. We apply the work of Mukhin-Tarasov-Varchenko, which proves that the rational Calogero-Moser phase space can be realised as a part of this spectrum, to relate this to behaviour at t=0 of rational Cherednik algebras of \mathfrakS_n. As a result, we confirm for symmetric groups a conjecture of Bonnafé-Rouquier which proposes an equality between the Calogero-Moser cells they defined and the well-known Kazhdan-Lusztig cells.

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