A combinatorial formula for the character of the diagonal coinvariants

Abstract

Let R_n be the ring of coinvariants for the diagonal action of the symmetric group S_n. It is known that the character of R_n as a doubly graded S_n-module can be expressed using the Frobenius characteristic map as $\nabla e_{n}$, where e_n 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 $\nabla^{m}e_{n}$.