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Haglund’s conjecture on 3-column Macdonald polynomials

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Abstract

We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund (Proc Natl Acad Sci USA 101(46):16127–16131, 2004). The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam’s (Eur J Combin 29(1):343–359, 2008) algebra of ribbon Schur operators. Combining this result with the expression of Haglund et al. (J Am Math Soc 18(3):735–761, 2005) for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Macdonald polynomials indexed by a shape with 3 columns.

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Notes

  1. There is an algebra antiautomorphism from \(\mathbb {Q}({\hat{q}})\otimes _\mathbf {A}\mathcal {U}/{{I_{\mathrm{L}, k}}}\) to the algebra in [15], defined by sending \(u_i \mapsto u_i\) and setting \(q = {\hat{q}}^{-2}\), where \({\hat{q}}\) denotes the q from [15].

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Acknowledgments

I am extremely grateful to Sergey Fomin for his valuable insights and guidance on this project and to John Stembridge for his generous advice and many helpful discussions. I thank Thomas Lam and Sami Assaf for several valuable discussions and Xun Zhu and Caleb Springer for help typing and typesetting figures.

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Authors and Affiliations

  1. Department of Mathematics, Drexel University, Philadelphia, PA, 19104, USA

    Jonah Blasiak

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  1. Jonah Blasiak
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Correspondence to Jonah Blasiak.

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This work was supported by NSF Grant DMS-14071174.

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Blasiak, J. Haglund’s conjecture on 3-column Macdonald polynomials. Math. Z. 283, 601–628 (2016). https://doi.org/10.1007/s00209-015-1612-7

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