Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture

Abstract

The symmetric group Sn acts on the polynomial ring Q[xn]=Q[x1,…,xn] by variable permutation. The invariant ideal In is the ideal generated by all Sn-invariant polynomials with vanishing constant term. The quotient Rn=Q[xn]In is called the coinvariant algebra. The coinvariant algebra Rn has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization In,k⊆Q[xn] of the ideal In indexed by two positive integers k≤n. The corresponding quotient Rn,k:=Q[xn]In,k carries a graded action of Sn and specializes to Rn when k=n. We generalize many of the nice properties of Rn to Rn,k. In particular, we describe the Hilbert series of Rn,k, give extensions of the Artin and Garsia–Stanton monomial bases of Rn to Rn,k, determine the reduced Gröbner basis for In,k with respect to the lexicographic monomial order, and describe the graded Frobenius series of Rn,k. Just as the combinatorics of Rn are controlled by permutations in Sn, we will show that the combinatorics of Rn,k are controlled by ordered set partitions of {1,2,…,n} with k blocks. The Delta Conjecture of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of Rn,k is (up to a minor twist) the t=0 specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded Sn-module Vn,k whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module Rn,k solves this problem in the specialization t=0.