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Abstract
We give an explicit combinatorial formula for the Schur expansion of Macdonald polynomials indexed by partitions with second part at most two. This gives a uniform formula for both hook and two column partitions. The proof comes as a corollary to the result that generalized dual equivalence classes of permutations are in explicit bijection with unions of standard dual equivalence classes of permutations for certain cases, establishing an earlier conjecture of the author, and suggesting that this result might be generalized to arbitrary partitions.
Highlights
The transformed Macdonald polynomials, Hμ(X; q, t), a transformation of the polynomials introduced by Macdonald [Mac88] in 1988, are the simultaneous generalization of Hall–Littlewood and Jack symmetric functions with two parameters, q and t
Following an idea outlined by Procesi, Haiman [Hai01] proved this conjecture by analyzing the algebraic geometry of the isospectral Hilbert scheme of n points in the plane, thereby establishing Macdonald Positivity
We extend the combinatorial formula for two column Macdonald polynomials to partitions with second part at most 2
Summary
The transformed Macdonald polynomials, Hμ(X; q, t), a transformation of the polynomials introduced by Macdonald [Mac88] in 1988, are the simultaneous generalization of Hall–Littlewood and Jack symmetric functions with two parameters, q and t. Following an idea outlined by Procesi, Haiman [Hai01] proved this conjecture by analyzing the algebraic geometry of the isospectral Hilbert scheme of n points in the plane, thereby establishing Macdonald Positivity This proof, is purely geometric and does not offer a combinatorial interpretation for Kλ,μ(q, t). The proof is purely combinatorial and combines the bijective proofs of the two column and single row Macdonald polynomials in [Ass08] utilizing the structure of dual equivalence classes in [Ass15]. It establishes [Ass15](Conjecture 5.6) for μ2 ≤ 2.
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