Populations of solutions to cyclotomic Bethe equations (Q903690)

The present paper addresses the construction of Bethe ansatz solutions for cyclotomic Gaudin models, first introduced by \textit{B. Vicedo} and \textit{C. A. S. Young} [``Cyclotomic Gaudin models: construction and Bethe ansatz'', Preprint, \\url{arXiv:1409.6937}] and followed by the study of vertex Lie algebras and cyclotomic coinvariants [``Vertex Lie algebras and cyclotomic coinvariants'', Preprint, \\url{arXiv:1410.7664}]. The focus is on populations of solutions of cyclotomic Bethe equations, whose specialized version is known to reduce to constraint equations for critical points of the master functions that appear in the integral expression for hypergeometric functions of the Knizhnik-Zamolodchikov (KZ) equation. The authors identify the cyclotomic Bethe equations as critical point equations for a cyclotomic master function. The main goal of the paper is to define and classify populations of solutions to these (Bethe) equations, as formulated in [\textit{E. Mukhin} and \textit{A. Varchenko}, Commun. Contemp. Math. 6, No. 1, 111--163 (2004; Zbl 1050.17022)], i.e., for diagram automorphisms of suitable Kac-Moody algebras. The graded vector space is constructed and the populations of cyclotomic critical points are proven to be isomorphic to the variety of isotropic full flags in this space.