### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 091, 41 pages      arXiv:1505.07582      https://doi.org/10.3842/SIGMA.2015.091

### Populations of Solutions to Cyclotomic Bethe Equations

Alexander Varchenko a and Charles A.S. Young b
a) Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
b) School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK

Received June 17, 2014, in final form November 05, 2015; Published online November 14, 2015

Abstract
We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two interpretations of such solutions: as critical points of a cyclotomic master function, and as critical points with cyclotomic symmetry of a certain ''extended'' master function. In finite types, this yields a correspondence between the Bethe eigenvectors and eigenvalues of the cyclotomic Gaudin model and those of an ''extended'' non-cyclotomic Gaudin model. We proceed to define populations of solutions to the cyclotomic Bethe equations, in the sense of [Mukhin E., Varchenko A., Commun. Contemp. Math. 6 (2004), 111-163, math.QA/0209017], for diagram automorphisms of Kac-Moody Lie algebras. In the case of type A with the diagram automorphism, we associate to each population a vector space of quasi-polynomials with specified ramification conditions. This vector space is equipped with a ${\mathbb Z}_2$-gradation and a non-degenerate bilinear form which is (skew-)symmetric on the even (resp. odd) graded subspace. We show that the population of cyclotomic critical points is isomorphic to the variety of isotropic full flags in this space.

Key words: Bethe equations; cyclotomic symmetry.

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