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 ${\mathcal{B}}_{\bar{\alpha }}^{⟨n⟩}\subset U({\mathfrak{gl}}_n[t])$ and ${\mathcal{B}}_{\bar{z}}^{⟨k⟩}\subset U({\mathfrak{gl}}_k[t])$ 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 ${\mathcal{B}}_{\bar{\alpha }}^{⟨n⟩}$ and ${\mathcal{B}}_{\bar{z}}^{⟨k⟩}$.

21 page, 1 figure