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15 February 2005 A combinatorial formula for the character of the diagonal coinvariants
J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, A. Ulyanov
Duke Math. J. 126(2): 195-232 (15 February 2005). DOI: 10.1215/S0012-7094-04-12621-1


Let Rn be the ring of coinvariants for the diagonal action of the symmetric group Sn. It is known that the character of Rn as a doubly graded Sn-module can be expressed using the Frobenius characteristic map as $\nabla e_{n}$, where en is the nth elementary symmetric function and $\nabla $ is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for $\nabla e_{n}$ and prove that it has many desirable properties that support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc, and Thibon. We also show that a variety of earlier conjectures and theorems on $\nabla e_{n}$ are special cases of our conjecture.

Finally, we extend our conjectures on $\nabla e_{n}$ and several of the results supporting them to higher powers $\nabla^{m}e_{n}$.


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J. Haglund. M. Haiman. N. Loehr. J. B. Remmel. A. Ulyanov. "A combinatorial formula for the character of the diagonal coinvariants." Duke Math. J. 126 (2) 195 - 232, 15 February 2005.


Published: 15 February 2005
First available in Project Euclid: 21 January 2005

zbMATH: 1069.05077
MathSciNet: MR2115257
Digital Object Identifier: 10.1215/S0012-7094-04-12621-1

Primary: 05E10
Secondary: 05A30 , 20C30

Rights: Copyright © 2005 Duke University Press


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Vol.126 • No. 2 • 15 February 2005
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