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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In this paper we show how a summation identity of A. Garsia and M. Zabrocki [“Some calculations for the outer hookwalk, unpublished note (2005)] for Macdonald polynomial Pieri coefficients can be used to transform Haiman’s formula for the Hilbert series into an explicit polynomial in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.

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
Full Text: DOI
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