Spectra of Bethe subalgebras of $Y(\mathfrak{gl}_n)$ in tame representations

TL;DR

This paper investigates the eigenstructure of Bethe subalgebras within the Yangian $Y(rak{gl}_n)$ acting on tame, finite-dimensional representations, establishing cyclicity and simplicity of spectrum for generic parameters, with implications for crystal structures.

Contribution

It proves that Bethe subalgebras act with cyclic vectors and simple spectra on tensor products of skew modules in tame representations, extending understanding of their spectral properties.

Findings

Bethe subalgebras act with a cyclic vector on tensor products of skew modules.

For certain real forms, Bethe subalgebras have simple spectra on Kirillov-Reshetikhin modules.

Results set the stage for defining KR-crystal structures on spectra of Bethe subalgebras.

Abstract

We study the eigenproblem for Bethe subalgebras of the Yangian $Y(\mathfrak{gl}_n)$ in tame representations, i.e. in finite dimensional representations which admit Gelfand-Tsetlin bases. Namely, we prove that for any tensor product of skew modules $V=\otimes_{i=1}^k V_{\lambda_i \setminus \mu_i}(z_i)$ over the Yangian $Y(\mathfrak{gl}_n)$ with generic $z_i$'s, the family of Bethe subalgebras $B(X)$ with $X$ being a regular element of the maximal torus of $GL_n$ (or, more generally, with $X \in \overline{M_{0,n+2}}$) acts with a cyclic vector on $V$. Moreover, for $X$ in the real form of $\overline{M_{0,n+2}}$ which is the closure of regular unitary diagonal matrices we show, that the family of subalgebras $B(X)$ acts with simple spectrum on $\otimes_{i=1}^k V_{\lambda_i \setminus \mu_i}(z_i)$ for generic $z_i$'s where all $V_{\lambda_i \setminus \mu_i}(z_i)$ are Kirillov-Reshetikhin modules. In the subsequent paper we will use this to define a KR-crystal structure on the spectrum of a Bethe subalgebra on $V$.