Toric braids and $(m,n)$-parking functions

Abstract

The Dyck path algebra construction of Carlsson and Mellit from arXiv:1508.06239 is interpreted as a representation of "the positive part" of the group of toric braids. Then certain sums over $(m,n)$-parking functions are related to evaluations of this representation on some special braids. The compositional $(km,kn)$-shuffle conjecture of Bergeron, Garsia, Leven and Xin from arXiv:1404.4616 is then shown to be a corollary of this relation.

References in corpus (5)

• Torus knots and the rational DAHA
• The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra
• DAHA and iterated torus knots
• Sweeping up Zeta
• Some remarkable new Plethystic Operators in the Theory of Macdonald Polynomials

Cited by in corpus (22)

• The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link
• Homology of torus knots
• A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
• Hopf dreams and diagonal harmonics
• Torus link homology
• Affine Schubert calculus and double coinvariants
• The Delta Conjecture at $q=1$
• The cohomology ring of certain compactified Jacobians
• Elliptic and $q$-Analogs of the Fibonomial Numbers
• Tautological classes and symmetry in Khovanov-Rozansky homology
• The Stable Limit DAHA and the Double Dyck Path Algebra
• (GL_k x S_n)-modules and nabla of hook-indexed Schur functions
• $(G{L}_k\times S_n)$-Modules of Multivariate Diagonal Harmonics
• Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions
• Chain Decompositions of $q,t$-Catalan Numbers via Local Chains
• Dinv, Area, and Bounce for $\vec{k}$-Dyck paths
• q,t-Catalan measures
• Toward a Schurification of Parking Function Formulas via bijections with Young Tableaux
• Homology of Hilbert schemes of reducible locally planar curves
• A combinatorial formula for the nabla operator
• Explicit Combinatorial Formulas for Some Irreducible Characters of the $G{L}_k\times {\mathbb{S}}_n$-module of multivariate diagonal harmonics
• Algebra and geometry of link homology