A combinatorial formula for the character of the diagonal coinvariants

Author(s): Haglund, J; Haiman, M; Loehr, N; Remmel, J B; Ulyanov, A | Abstract: 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 S-module can be expressed using the Frobenius characteristic map as nabla en, where en is the n-th elementary symmetric function and nabla is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for nabla en 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 en are special cases of our conjecture. Finally, we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en.