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

TL;DR

This paper explores the spectrum of inhomogeneous Gaudin algebras acting on tensor representations of gl_r, connecting the Robinson-Schensted-Knuth correspondence, Calogero-Moser phase space, and Kazhdan-Lusztig cells to confirm a conjecture for symmetric groups.

Contribution

It establishes a link between Gaudin algebra spectra, RSK correspondence, Calogero-Moser cells, and Kazhdan-Lusztig cells, confirming a conjecture for symmetric groups.

Findings

RSK correspondence describes spectrum behavior along special paths.

Calogero-Moser phase space relates to rational Cherednik algebra spectrum.

Confirmed conjecture equating Calogero-Moser and Kazhdan-Lusztig cells for symmetric groups.

Abstract

We study the spectrum of a family of algebras, the inhomogeneous Gaudin algebras, acting on the $n$-fold tensor representation $\mathbb{C}[x_1, \ldots, x_r]^{\otimes n}$ of the Lie algebra $\mathfrak{gl}_r$. 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 $\mathfrak{S}_n$. As a result, we confirm for symmetric groups a conjecture of Bonnaf\'e-Rouquier which proposes an equality between the Calogero-Moser cells they defined and the well-known Kazhdan-Lusztig cells.