Duke Mathematical Journal

A combinatorial formula for the character of the diagonal coinvariants

J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, and A. Ulyanov

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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 $\nabla^{m}e_{n}$.

Article information

Duke Math. J. Volume 126, Number 2 (2005), 195-232.

First available in Project Euclid: 21 January 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05E10: Combinatorial aspects of representation theory [See also 20C30]
Secondary: 05A30: $q$-calculus and related topics [See also 33Dxx] 20C30: Representations of finite symmetric groups


Haglund, J.; Haiman, M.; Loehr, N.; Remmel, J. B.; Ulyanov, A. A combinatorial formula for the character of the diagonal coinvariants. Duke Math. J. 126 (2005), no. 2, 195--232. doi:10.1215/S0012-7094-04-12621-1. http://projecteuclid.org/euclid.dmj/1106332719.

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