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arXiv · 2608.02810

Formulas for Koornwinder polynomials

Laura Colmenarejo·Lucas Gagnon·Arun Ram
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Abstract

This paper provides formulas for Koornwinder polynomials in analogy with the creation formula, the alcove walk formula and the non-attacking fillings formula for the type $GL_n$ Macdonald polynomials. We state the creation formula in terms of the divided-difference operators used in Schubert calculus, and we use a box-greedy reduced word to reformulate the alcove walk formula in terms of uncompressed set-valued tableaux. Then two types of compression, ``around-the-end compression'' and ``across-the-$0$-gap compression'', are used to derive a formula for Koornwinder polynomials in terms of compressed set-valued tableaux. Throughout we work in the full generality of relative Koornwinder polynomials, which are the analogues of the permuted basement Macdonald polynomials used in the type $GL_n$ case.

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Laura Colmenarejo, Lucas Gagnon, Arun Ram. 2026-08-03. Formulas for Koornwinder polynomials. https://arxiv.org/abs/2608.02810

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