Journal of Combinatorics

Volume 2 (2011)

Number 2

The case $k=2$ of the Shuffle Conjecture

Pages: 193 – 229

DOI: http://dx.doi.org/10.4310/JOC.2011.v2.n2.a2

Authors

Adriano Garsia (Mathematics Department, University of California at San Diego)

Angela Hicks (Mathematics Department, University of California at San Diego)

Andrew Stout (Mathematics Department, University of California at San Diego)

Abstract

It was conjectured in [5] and proved by Mark Haiman in [13] that theFrobenius Characteristic of the $S_n$ Module of Diagonal Harmonics isnone other than $\nabla e_n$. Here $\nabla$ is the symmetric functionoperator introduced in [1] with eigen-functions the modified Macdonaldbasis $\{\TH_\mu\}_\mu$. The Shuffle Conjecture [12] expresses thescalar product $\LL \nabla e_n\scs h_{\mu_1} h_{\mu_2}\cdotsh_{\mu_k}\RR$ as a weighted sum of Parking Functions on the $n\times n$lattice square which are shuffles of $k$ increasing words. In [10]Jim Haglund succeeded in proving the $k=2$ case of this conjecture. Inthis paper we give a new and more direct proof of the combinatorialpart of Haglund’s argument and obtain a substantial reduction in thecomplexity of the symmetric function part.

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