Duality for Bethe algebras acting on polynomials in anticommuting variables

Abstract

We consider actions of the current Lie algebras $\mathfrak{gl}_{n}[t]$ and $\mathfrak{gl}_{k}[t]$ on the space of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1}\dots z_{k})$ and $\bar{\alpha}=(\alpha_{1}\dots \alpha_{n})$, respectively. We show that the images of the Bethe algebras $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}\subset U(\mathfrak{gl}_{n}[t])$ and $\mathcal{B}_{\bar{z}}^{\langle k \rangle}\subset U(\mathfrak{gl}_{k}[t])$ under these actions coincide. To prove the statement, we use the Bethe ansatz description of eigenvalues of the actions of the Bethe algebras via spaces of quasi-exponentials and establish an explicit correspondence between these spaces for the actions of $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}$ and $\mathcal{B}_{\bar{z}}^{\langle k \rangle}$.