Gaudin algebras, RSK and Calogero–Moser cells in Type A

We study the spectrum of a family algebras, inhomogeneous Gaudin acting on n $n$ -fold tensor representation C [ x 1 , … r ] ? ${\mathbb{C}}[x_1, \ldots x_r]^{\otimes n}$ Lie algebra gl $\mathfrak {gl}_r$ . use work Halacheva–Kamnitzer–Rybnikov–Weekes to demonstrate that Robinson–Schensted–Knuth correspondence describes behaviour as we move along special paths in family. apply Mukhin–Tarasov–Varchenko, which proves rational Calogero–Moser phase space can be realised part this spectrum, relate at t = 0 $t=0$ Cherednik algebras S {S}_n$ As result, confirm for symmetric groups conjecture Bonnafé–Rouquier proposes an equality between cells they defined and well-known Kazhdan–Lusztig cells.