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

SIGMA 16 (2020), 035, 10 pages      arXiv:1904.07309      https://doi.org/10.3842/SIGMA.2020.035
Contribution to the Special Issue on Representation Theory and Integrable Systems in honor of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday

Duality for Knizhnik-Zamolodchikov and Dynamical Operators

Vitaly Tarasov ^ab and Filipp Uvarov ^a
a)  Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA
^b)  St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia

Received February 25, 2020, in final form April 10, 2020; Published online April 25, 2020

Abstract
We consider the Knizhnik-Zamolodchikov and dynamical operators, both differential and difference, in the context of the $(\mathfrak{gl}_{k}, \mathfrak{gl}_{n})$-duality for the space of polynomials in $kn$ anticommuting variables. We show that the Knizhnik-Zamolodchikov and dynamical operators naturally exchange under the duality.

Key words: Knizhnik-Zamolodchikov operators; dynamical operators; the $(\mathfrak{gl}_{k}, \mathfrak{gl}_{n})$-duality.

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