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

We study the isospectral Hilbert scheme X n X_{n} , defined as the reduced fiber product of ( C 2 ) n (\mathbb {C}^{2})^{n} with the Hilbert scheme H n H_{n} of points in the plane C 2 \mathbb {C}^{2} , over the symmetric power S n C 2 = ( C 2 ) n / S n S^{n}\mathbb {C}^{2} = (\mathbb {C}^{2})^{n}/S_{n} . By a theorem of Fogarty, H n H_{n} is smooth. We prove that X n X_{n} is normal, Cohen-Macaulay and Gorenstein, and hence flat over H n H_{n} . We derive two important consequences. (1) We prove the strong form of the n ! n! conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients K λ μ ( q , t ) K_{\lambda \mu }(q,t) . This establishes the Macdonald positivity conjecture, namely that K λ μ ( q , t ) ∈ N [ q , t ] K_{\lambda \mu }(q,t)\in {\mathbb N} [q,t] . (2) We show that the Hilbert scheme H n H_{n} is isomorphic to the G G -Hilbert scheme ( C 2 ) n / / S n (\mathbb {C}^{2})^{n}{/\!\!/}S_n of Nakamura, in such a way that X n X_{n} is identified with the universal family over ( C 2 ) n / / S n ({\mathbb C}^2)^n{/\!\!/}S_n . From this point of view, K λ μ ( q , t ) K_{\lambda \mu }(q,t) describes the fiber of a character sheaf C λ C_{\lambda } at a torus-fixed point of ( C 2 ) n / / S n ({\mathbb C}^2)^n{/\!\!/}S_n corresponding to μ \mu . The proofs rely on a study of certain subspace arrangements Z ( n , l ) ⊆ ( C 2 ) n + l Z(n,l)\subseteq (\mathbb {C}^{2})^{n+l} , called polygraphs, whose coordinate rings R ( n , l ) R(n,l) carry geometric information about X n X_{n} . The key result is that R ( n , l ) R(n,l) is a free module over the polynomial ring in one set of coordinates on ( C 2 ) n (\mathbb {C}^{2})^{n} . This is proven by an intricate inductive argument based on elementary commutative algebra.

We prove a combinatorial formula for the Macdonald polynomial H ~ μ ( x ; q , t ) \tilde {H}_{\mu }(x;q,t) which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of H ~ μ ( x ; q , t ) \tilde {H}_{\mu }(x;q,t) in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schützenberger for Hall-Littlewood polynomials, a new proof of Knop and Sahi’s combinatorial formula for Jack polynomials as well as a lifting of their formula to integral form Macdonald polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficients K ~ λ μ ( q , t ) \tilde {K}_{\lambda \mu }(q,t) in the case that μ \mu is a partition with parts ≤ 2 \leq 2 .

We introduce a rational function Cn (q, t) and conjecture that it always evaluates to a polynomial in q, t with non-negative integer coefficients summing to the familiar Catalan number n-~l (~)" We give supporting evidence by computing the specializations Dn(q) = Ca(q, l/q)q(~) and Ca(q) = Ca(q, 1) = Cn(1, q). We show that, in fact, Dn(q) q-counts Dyek words by the major index and Cn(q) q-coants Dyck paths by area. We also show that Ca (q, t) is the coefficient of the elementary symmetric function en in a symmetric polynomial DHn (x; q, t) which is the conjectured Frobenius characteristic of the module of diagonal harmonic polynomials. On the validity of certain conjectures this yields that Ca(q, t) is the Hilbert series of the diagonal harmonic altemants. It develops that the specialization DHn (x; q, 1) yields a novel and combinatorial way of expressing the solution of the q-Lagrange inversion problem studied by Andrews [2], Garsia [5] and Gessel [11]. Our proofs involve manipulations with the Macdonald basis {Pu(x; q. t)}u which are best dealt with in A-ring notation. In particular we derive here the A-ring version of several symmetric function identities.

Abstract. In an earlier paper [14], we showed that the Hilbert scheme of points in the plane Hn = Hilb n (C 2 ) can be identified with the Hilbert scheme of regular orbits C 2n //Sn. Using this result, together with a recent theorem of Bridgeland, King and Reid [4] on the generalized McKay correspondence, we prove vanishing theorems for tensor powers of tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish (among other things) the character formula for diagonal harmonics conjectured by Garsia and the author in [9]. In particular we prove that the dimension of the space of diagonal harmonics is equal to (n + 1) n−1 .