39 research outputs found

    A proof of the n!2\frac{n!}{2} conjecture for hook shapes

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    A well-known representation-theoretic model for the transformed Macdonald polynomial H~μ(Z;t,q)\widetilde{H}_\mu(Z;t,q), where μ\mu is an integer partition, is given by the Garsia-Haiman module Hμ\mathcal{H}_\mu. We study the n!k\frac{n!}{k} conjecture of Bergeron and Garsia, which concerns the behavior of certain kk-tuples of Garsia-Haiman modules under intersection. In the special case that μ\mu has hook shape, we use a basis for Hμ\mathcal{H}_\mu due to Adin, Remmel, and Roichman to resolve the n!2\frac{n!}{2} conjecture by constructing an explicit basis for the intersection of two Garsia-Haiman modules.Comment: 13 pages, 0 figure

    Combinatorics of Labelled Parallelogram polyominoes

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    We obtain explicit formulas for the enumeration of labelled parallelogram polyominoes. These are the polyominoes that are bounded, above and below, by north-east lattice paths going from the origin to a point (k,n). The numbers from 1 and n (the labels) are bijectively attached to the nn north steps of the above-bounding path, with the condition that they appear in increasing values along consecutive north steps. We calculate the Frobenius characteristic of the action of the symmetric group S_n on these labels. All these enumeration results are refined to take into account the area of these polyominoes. We make a connection between our enumeration results and the theory of operators for which the intergral Macdonald polynomials are joint eigenfunctions. We also explain how these same polyominoes can be used to explicitly construct a linear basis of a ring of SL_2-invariants.Comment: 25 pages, 9 figure
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