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June 2015 Schur Coefficients of the Integral Form Macdonald Polynomials
Meesue YOO
Tokyo J. Math. 38(1): 153-173 (June 2015). DOI: 10.3836/tjm/1437506242

Abstract

In this paper, we consider the combinatorial formula for the Schur coefficients of the integral form of the Macdonald polynomials. As an attempt to prove Haglund's conjecture that $\Biggl<\frac{J_{\mu}(X;q,q^k)}{(1-q)^{|\mu|}},s_{\lambda}(X)\Biggr> \in \mathbb{N}[q]$, we have found explicit combinatorial formulas for the Schur coefficients in one row case, two column case and certain hook shape cases [Yoo12]. A result of Egge-Loehr-Warrington [ELW] gives a combinatorial way of getting Schur expansion of symmetric functions when the expansion of the function in terms of Gessel's fundamental quasi symmetric functions is known. We apply this result to the combinatorial formula for the integral form Macdonald polynomials of Haglund [Hag] in quasi symmetric functions to prove the Haglund's conjecture in more general cases.

Citation

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Meesue YOO. "Schur Coefficients of the Integral Form Macdonald Polynomials." Tokyo J. Math. 38 (1) 153 - 173, June 2015. https://doi.org/10.3836/tjm/1437506242

Information

Published: June 2015
First available in Project Euclid: 21 July 2015

zbMATH: 1256.05250
MathSciNet: MR3374619
Digital Object Identifier: 10.3836/tjm/1437506242

Subjects:
Primary: 05A10

Rights: Copyright © 2015 Publication Committee for the Tokyo Journal of Mathematics

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Vol.38 • No. 1 • June 2015
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