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Ordered set partitions, generalized coinvariant algebras, and the delta conjecture. (English) Zbl 1384.05043

Summary: The symmetric group \(\mathfrak{S}_n\) acts on the polynomial ring \(\mathbb{Q} [\mathbf{x}_n] = \mathbb{Q} [x_1, \dots, x_n]\) by variable permutation. The invariant ideal \(I_n\) is the ideal generated by all \(\mathfrak{S}_n\)-invariant polynomials with vanishing constant term. The quotient \(R_n = \frac{\mathbb{Q} [\mathbf{x}_n]}{I_n}\) 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 \(I_{n, k} \subseteq \mathbb{Q} [\mathbf{x}_n]\) of the ideal \(I_n\) indexed by two positive integers \(k \leq n\). The corresponding quotient \(R_{n, k} : = \frac{\mathbb{Q} [\mathbf{x}_n]}{I_{n, k}}\) carries a graded action of \(\mathfrak{S}_n\) 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 Gröbner 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 \(\mathfrak{S}_n\), we will show that the combinatorics of \(R_{n, k}\) are controlled by ordered set partitions of \(\{1, 2, \dots, n \}\) with \(k\) blocks. The Delta Conjecture of J. Haglund et al. [Trans. Am. Math. Soc. 370, No. 6, 4029–4057 (2018; Zbl 1383.05308)] 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 \(\mathfrak{S}_n\)-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 \(t = 0\).

MSC:

05A18 Partitions of sets
05E05 Symmetric functions and generalizations

Citations:

Zbl 1383.05308
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References:

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