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A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants. (English) Zbl 1258.13020
Summary: A special case of M. Haiman’s identity [Invent. Math. 149, No. 2, 371–407 (2002; Zbl 1053.14005)] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in $$q,\l,t$$. In this paper we show how a summation identity of A. Garsia and M. Zabrocki [“Some calculations for the outer hookwalk, unpublished note (2005)] for Macdonald polynomial Pieri coefficients can be used to transform Haiman’s formula for the Hilbert series into an explicit polynomial in $$q,\,t$$ with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series.

##### MSC:
 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 05A15 Exact enumeration problems, generating functions 05E40 Combinatorial aspects of commutative algebra 05E05 Symmetric functions and generalizations
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##### References:
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