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

Abstract The symmetric group S n acts on the polynomial ring Q [ x n ] = Q [ x 1 , … , x n ] by variable permutation. The invariant ideal I n is the ideal generated by all S n -invariant polynomials with vanishing constant term. The quotient R n = Q [ x n ] I n is called the coinvariant algebra . The coinvariant algebra R n has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization I n , k ⊆ Q [ x n ] of the ideal I n indexed by two positive integers k ≤ n . The corresponding quotient R n , k : = Q [ x n ] I n , k carries a graded action of S n and specializes to R n when k = n . We generalize many of the nice properties of R n to R n , k . In particular, we describe the Hilbert series of R n , k , give extensions of the Artin and Garsia–Stanton monomial bases of R n to R n , k , determine the reduced Grobner basis for I n , k with respect to the lexicographic monomial order, and describe the graded Frobenius series of R n , k . Just as the combinatorics of R n are controlled by permutations in S n , we will show that the combinatorics of R n , 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 R n , 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 S n -module V n , k whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module R n , k solves this problem in the specialization t = 0 .