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

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arxiv 1604.07456 v1 pith:E6U6NOWF submitted 2016-04-25 math.CO math.QAmath.RT

Anton Mellit This is my paper · ORCID
classification math.COmath.QAmath.RT
keywords braidsarxivfunctionsparkingrepresentationthentoricalgebra
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Torus link homology

    math.GT 2019-09 conditional novelty 6.0 of 10

    Positive torus links T(m,n) and Sym^l-colored torus knots have triply graded Khovanov-Rozansky homology equal to an explicitly defined family of polynomials p(v,w).

  2. Recursions for rational q,t-Catalan numbers

    math.CO 2019-08 conditional novelty 6.0 of 10

    A fixed-length binary-sequence recursion computes rational q,t-Catalan power series and matches the Hogancamp-Mellit recursion, confirming the link to Khovanov-Rozansky homology.

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