The case $k=2$ of the Shuffle Conjecture
Adriano M. Garsia, Angela Hicks, Andrew R. Stout · Journal of Combinatorics · 2011
It was conjectured in [5] and proved by Mark Haiman in [13] that the Frobenius Characteristic of the S n Module of Diagonal Harmonics is none other than ∇e n .Here ∇ is the symmetric function operator introduced in [1] with eigen-functions the modified Macdonald basis { Hμ } μ .The Shuffle Conjecture [12] expresses the scalar product ∇e n , h μ1 h μ2 • • • h μ k as a weighted sum of Parking Functions on the n × n lattice square which are shuffles of k increasing words.In [10] Jim Haglund succeeded in proving the k = 2 case of this conjecture.In this paper we give a new and more direct proof of the combinatorial part of Haglund's argument and obtain a substantial reduction in the complexity of the symmetric function part.