Browsing by Author "Tarasov, V."

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    Completeness of the Bethe Ansatz for the Periodic Isotropic Heisenberg Model
    (World Scientific, 2018-09) Tarasov, V.; Mathematical Sciences, School of Science
    For the periodic isotropic Heisenberg model with arbitrary spins and inhomogeneities, we describe the system of algebraic equations whose solutions are in bijection with eigenvalues of the transfer-matrix. The system describes pairs of polynomials with the given discrete Wronskian (Casorati determinant) and additional divisibility conditions on discrete Wronskians with multiple steps. If the polynomial of the smaller degree in the pair is coprime with the Wronskian, this system turns into the standard Bethe ansatz equations. Moreover, if the transfer-matrix is diagonalizable, then its spectrum is necessarily simple modulo natural degeneration.
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    Elliptic and K-theoretic stable envelopes and Newton polytopes
    (Springer, 2019-03) Rimányi, R.; Tarasov, V.; Varchenko, A.; Mathematical Sciences, School of Science
    In this paper we consider the cotangent bundles of partial flag varieties. We construct the K -theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the K -theoretic stable envelopes and our elliptic stable envelopes. We show that the K -theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the gl2 case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the K -theoretic stable envelopes.