Bethe subalgebras of the group algebra of the symmetric group

Abstract

We introduce families \( \mathcal{B}_n^S\left( {{z_1},\ldots,{z_n}} \right) \) and \( \mathcal{B}_{{n,\hbar}}^S\left( {{z_1},\ldots,{z_n}} \right) \) of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra \( \mathbb{C}\left[ {\mathfrak{S}n} \right] \) of the symmetric group. Bethe subalgebras are deformations of the Gelfand−Zetlin subalgebra of \( \mathbb{C}\left[ {\mathfrak{S}n} \right] \). We describe various properties of Bethe subalgebras.