1998/09/22 by François Bergeron, F. Bergeron, Bergeron, F. +2 · 6 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #math.CO #math.QA
paper · pdf · doi:10.48550/arxiv.math/9809128
47 pages, TeX
arxiv created 1998/09/22 · openalex publication_date 1998/09/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work studies the remarkable relationships that hold among certain m-tuples of the Garsia-Haiman modules \bf Mμ and corresponding elements of the Macdonald basis. We recall that \bf Mμ is defined for a partition μ\part n, as the linear span of derivatives of a certain bihomogeneous polynomial Δ_ μ(x,y) in the variables x1,x2,..., xn, y1,y2,..., yn. It has been conjectured by Garsia and Haiman that \bf Mμ has n! dimensions and that its bigraded Frobenius characteristic is given by the symmetric polynomial \widetildeHμ(x;q,t)=∑λ\part n Sλ(X) \widetildeKλμ(q,t) where the \widetildeKλμ(q,t) are related to the Macdonald q,t-Kostka coefficients Kλμ(q,t) by the identity \widetildeKλμ(q,t)=Kλμ(q,1/t)tn(μ) with n(μ) the x-degree of Δ_ μ(x;y). Computer data has suggested that as ν varies among the immediate predecessors of a partition μ, the spaces \bf Mν behave like a boolean lattice. We formulate a number of remarkable conjectures about the Macdonald polynomials. In particular we obtain a representation theoretical interpretation for some of the symmetries that can be found in the computed tables of q,t-Kostka coefficients.