A combinatorial formula for the character of the diagonal coinvariants

By James Haglund, Mark D. Haiman, Nicholas A. Loehr, Jeffrey B. Remmel, A. Ulyanov. 2003

326 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c7d7b19a0f787d37e36850f30d2f5ad46867b809

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Hopf algebras and dendriform structures arising from parking functions (2005, background): “These identities are easily visualized on the encoding of parking functions with skew Young diagrams as in [30] or in [9].”
Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture (2016, background): “General facts about Gessel fundamental quasisymmetric functions and the superization of symmetric functions [16], [14, pp.”
Toric braids and (m,n)-parking functions (2016, background): “tional (km,kn)-shuﬄe conjecture of [BGLX15] is then shown to be a corollary of this relation. 1. Introduction in [CM15] Erik Carlsson and the author proved a generalization of the shuﬄe conjecture of [HHLRU05]. This conjecture gave a combinatorial interpretation of the Frobenius character of the alg”
Affine permutations and rational slope parking functions (2014, background): “= (m−1)(n−1) 2 −QAω(i), dinv(ω) ∶= (m−1)(n−1) 2 −QPSω(i). For the case m = n +1 Armstrong showed in [2] (in slightly different terms) that area and dinv statistic agrees with the statistics deﬁned in [19] as a part of “Shufﬂe Conjec ture”. Conjecture 1.8. The combinatorial Hilbert series Hm~n(q,t) ∶”
Refined knot invariants and Hilbert schemes (2013, methodology): “whereQDes(F ) is the Gessel quasisymmetric function [25, 40] associated w ith the set Des (F ).”
Affine Springer fibers of type A and combinatorics of diagonal coinvariants (2012, background): “Just as in [8], we can prove the following lemma by virtue of the notion of negative Young tableau.”
The Delta Conjecture (2015, background): “Building on this work, Haiman, Loehr, Ulyanov, and the first two authors proposed a connection between ∇en and parking functions which has come to be known as the Shuffle Conjecture [HHL05b].”
A proof of the q,t-Schröder conjecture (2004, unknown): “We discuss how our formulas for 〈∇en, en−dhd〉 and 〈∇en, hn−dhd〉 prove two special cases of a recent conjecture of Haglund, Haiman, Loehr, Remmel and Ulyanov.”
Rational Parking Functions and Catalan Numbers (2014, background): “Conjecture 17 (Shuffle Conjecture [19]).”
Higher Trivariate Diagonal Harmonics via generalized Tamari Posets (2011, background): “The above formulas, which clearly generalize (2) and (4), are both encompassed in the following generalization of a Conjecture of [15], in conjunction with a second conjecture below.”