A combinatorial formula for the character of the diagonal coinvariants
Let \(R_n={\mathbb C}[{\mathbf x},{\mathbf y}]/I\) be the algebra of coinvariants for the diagonal action of the symmetric group. Here \({\mathbb C}[{\mathbf x},{\mathbf y}]={\mathbb C}[x_1,y_1,\ldots,x_n,y_n]\) and \(I\) is the ideal generated by the \(S_n\)-invariant polynomials without constant term. The algebra \(R_n\) is a doubly graded \(S_n\)-module. Its graded character is encoded by means of its Frobenius series \({\mathcal F}_{R_n}(z;q,t)\). A celebrated conjecture of Garsia and Haiman confirmed by Haiman expresses \({\mathcal F}_{R_n}(z;q,t)\) as \(\nabla e_n(z)\), where \(\nabla\) is a linear operator defined in terms of modified Macdonald symmetric functions and \(e_n(z)\) is the \(n\)th elementary symmetric function. Numerious recent results suggest that one should understand the rather mysterious quantity \(\nabla e_n(z)\) in more combinatorial terms. In the paper under review the authors conjecture a combinatorial formula for the full expansion of \(\nabla e_n(z)\) in terms of monomials, generalizing the Garsia-Haglund formula \(C_n(q,t)=\langle\nabla e_n,e_n\rangle\) for the \(q,t\)-analogue \(C_n(q,t)\) of the Catalan numbers, the Haglund-Loehr conjecture \({\mathcal H}_n(q,t)=\langle\nabla e_n,e_1^n\rangle\) for the Hilbert series \({\mathcal H}_n(q,t)\) of \(R_n\), and a conjecture of Egge, Haglund, Killpatrick, and Kremer (Zbl 1011.05006) expressing \(\langle\nabla e_n,h_de_{n-d}\rangle\) in terms of Schröder paths. The authors prove that their formula has many properties supporting the conjecture. In particular, it is a symmetric function which is Schur positive. Finally, the authors extend their considerations to higher powers \(\nabla^me_n(z)\), giving corresponding conjectured formulas and examining their properties. The results obtained make use of the theory of ribbon tableau generating functions of Lascoux-Leclerc-Thibon.
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