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Published March 8, 2018 | Submitted + Published
Journal Article Open

Multivariate quadratic transformations and the interpolation kernel

Abstract

We prove a number of quadratic transformations of elliptic Selberg integrals (conjectured in an earlier paper of the author), as well as studying in depth the ''interpolation kernel'', an analytic continuation of the author's elliptic interpolation functions which plays a major role in the proof as well as acting as the kernel for a Fourier transform on certain elliptic double affine Hecke algebras (discussed in a later paper). In the process, we give a number of examples of a new approach to proving elliptic hypergeometric integral identities, by reduction to a Zariski dense subset of a formal neighborhood of the trigonometric limit.

Additional Information

© 2018 The Author(s). The authors retain the copyright for their papers published in SIGMA under the terms of the Creative Commons Attribution-ShareAlike License. Received September 12, 2017, in final form February 27, 2018; Published online March 08, 2018. This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications. The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html The author would particularly like to thank P. Etingof for an initial suggestion that taking p to be a formal variable might allow one to extend the W(E7) symmetry of the order 1 elliptic Selberg to W(E8); this turned out not to work (some symmetries are, indeed, gained, but at the expense of others), but led the author to a more general study of the formal limit. In addition, the author would like to thank D. Betea, M. Wheeler, and P. Zinn-Justin for discussions relating to Izergin-Korepin determinants and their elliptic analogues, and especially for discussions relating to Conjecture 1 of [1] (which led the author to consider the general case of the Littlewood kernel below). The author would also like to thank O. Warnaar for additional discussions related to the Macdonald polynomial limit. The author would finally like to thank H. Rosengren for providing extra motivation to finish writing the present work, as well as some helpful pointers to the vertex model literature. The author was partially supported by the National Science Foundation (grant number DMS-1001645).

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Published - sigma18-019.pdf

Submitted - 1408.0305.pdf

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August 21, 2023
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