A Combinatorial Formula for the Character of the Diagonal Coinvariants

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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 ∇ eₙ, where eₙ is the n-th elementary symmetric function, and ∇ is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for ∇ eₙ and prove that it has many desirable properties which 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 ∇ eₙ are special cases of our conjecture. Finally, we extend our conjectures on ∇ eₙ and several of the results supporting them to higher powers ∇ᵐ eₙ.