Séminaire Lotharingien de Combinatoire, B87b (2023), 20 pp.
Aritra Bhattacharya
Haglund's Positivity Conjecture for Multiplicity One Pairs
Abstract.
Haglund's conjecture states that
⟨Jλ(q,qk),sμ⟩
/ (1-q)|λ|
∈ Z≥0[q] for all partitions
λ,μ and all non-negative integers k, where
Jλ is the integral form Macdonald symmetric
function and sμ is the Schur function. This paper
proves Haglund's conjecture in the cases when the pair
(λ,μ) satisfies Kλ,μ=1 or
Kμ',λ'=1 where K denotes the Kostka
number. We also obtain some general results about the
transition matrix between Macdonald symmetric
functions and Schur functions.
Received: July 1, 2022.
Revised: January 30, 2023.
Accepted: February 5, 2023.
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