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Combinatorial Formula for the Hilbert Series of bigraded $S_n$-modules

Abstract : We introduce a combinatorial way of calculating the Hilbert series of bigraded $S_n$-modules as a weighted sum over standard Young tableaux in the hook shape case. This method is based on Macdonald formula for Hall-Littlewood polynomial and extends the result of $A$. Garsia and $C$. Procesi for the Hilbert series when $q=0$. Moreover, we give the way of associating the fillings giving the monomial terms of Macdonald polynomials to the standard Young tableaux.
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Meesue yoo. Combinatorial Formula for the Hilbert Series of bigraded $S_n$-modules. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.903-914, ⟨10.46298/dmtcs.2704⟩. ⟨hal-01185396⟩

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