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

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 $G{L}_n$ (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 $V$.

References in corpus (3)

• Crystals and monodromy of Bethe vectors
• The $R$-matrix presentation for the Yangian of a simple Lie algebra
• Bethe subalgebras in Yangians and the wonderful compactification