Spectra of Bethe subalgebras of Y(gl_n) in tame representations

Letters in Mathematical Physics. 2022. Vol. 112. No. 5. Article 99.

Aleksei Ilin, Mashanova-Golikova I., Leonid Rybnikov

We study the eigenproblem for Bethe subalgebras of the Yangian Y({{\mathfrak {g}}}{{\mathfrak {l}}}_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 {g}}}{{\mathfrak {l}}}_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.

Keywords: Bethe subalgebra, Yangian, tame representation