Adriano M. Garsia

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We define doubly graded Sn modules Rmu for which we conjecture that the multiplicities of irreducible representations in various bi-degrees are given by the Macdonald coefficients Klambdamu. Assuming one fundamental conjecture, the modules Rmu can be given several equivalent definitions, which we discuss. We prove the conjectures in various special cases.
We outline here a proof that a certain rational function C(n)(q, t), which has come to be known as the "q, t-Catalan," is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because C(n)(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the(More)
We construct for each μ ` n a bigraded Sn-module Hμ and conjecture that its Frobenius characteristic Cμ(x; q, t) yields the Macdonald coefficients Kλμ(q, t). To be precise, we conjecture that the expansion of Cμ(x; q, t) in terms of the Schur basis yields coefficients Cλμ(q, t) which are related to the Kλμ(q, t) by the identity Cλμ(q, t) = Kλμ(q, 1/t)t. The(More)
F. Bergeron , N. Bergeron , A. M. Garsiay, M. Haimany, and G. Teslery Abstract. The lattice cell in the i+ 1 row and j + 1 column of the positive quadrant of the plane is denoted (i; j). If is a partition of n + 1, we denote by =ij the diagram obtained by removing the cell (i; j) from the (French) Ferrers diagram of . We set =ij = det kx pj i y qj i k n(More)
We present here a proof that a certain rational function Cn(q, t) which has come to be known as the “q, t-Catalan” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. The precise form of the conjecture is given in the J. Algebraic Combin. 5 (1996), no. 3, 191–244, where it is further conjectured that Cn(q,(More)