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# A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants (condensed version)

Abstract : A special case of Haiman's identity [Invent. Math. 149 (2002), pp. 371–407] 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 $q,t$. In this paper we show how a summation identity of Garsia and Zabrocki 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.
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Cited literature [13 references]

https://hal.inria.fr/hal-01215103
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### Citation

J. Haglund. A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants (condensed version). 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.445-456, ⟨10.46298/dmtcs.2924⟩. ⟨hal-01215103⟩

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