Mark Haiman

Conjectures on the Quotient Ring by Diagonal Invariants

AbstractWe formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring

A combinatorial formula for Macdonald polynomials

\mathbb{Q}[x_1 , \ldots ,x_n ,y_1 , \ldots ,y_n ]

Multigraded Hilbert schemes

in two sets of variables by the ideal generated by all Sn invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = {x1, ..., xn} is classical. Introducing the second set of variables leads to a ring about which little is yet understood, but for which there is strong evidence of deep connections with many fundamental results of enumerative combinatorics, as well as with algebraic geometry and Lie theory.

Dual equivalence with applications, including a conjecture of Proctor

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.

Schubert polynomials for the classical groups

Abstract: We prove a combinatorial formula for the Macdonald polynomial

On mixed insertion, symmetry, and shifted young tableaux

\tilde{H}_{\mu }(x;q,t)

A simple and relatively efficient triangulation of the n -cube

which had been conjectured by Haglund. Corollaries to our main theorem include the expansion of

Proof theory for linear lattices

\tilde{H}_{\mu }(x;q,t)