Séminaire Lotharingien de Combinatoire, 89B.78 (2023), 10 pp.
Sam Armon
Foata-Like Bijections and science Fiction
Abstract.
A central open problem in algebraic combinatorics is to find a combinatorial formula for the Kostka-Macdonald polynomials
K~λμ(q,t), which describe the expansion of the Macdonald polynomial
H~μ(Z;q,t) in the Schur basis. Haiman proved that
K~λμ(q,t)
has nonnegative integer coefficients by proving that the dimension of the Garsia-Haiman module Hμ equals n!, demonstrating an intricate relationship between the Kostka-Macdonald polynomials and this module. This relationship was further expounded upon by Bergeron and Garsia, whose "science fiction" heuristics conjecture certain intersection properties of Garsia-Haiman modules which mirror observed symmetries in the Kostka-Macdonald coefficients. The most potent of these heuristics is the n!/k conjecture, which asserts that the dimension of the intersection of k Garsia-Haiman modules should have dimension n!/k. We solve the special case of the n!/k conjecture where the indexing partitions have hook shape by constructing an explicit basis for the intersection, using two maps in the spirit of Foata's.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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