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1 December 2021 Toric braids and (m,n)-parking functions
Anton Mellit
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Duke Math. J. 170(18): 4123-4169 (1 December 2021). DOI: 10.1215/00127094-2021-0011

Abstract

We find a geometric interpretation of the Aq,t algebra, the algebra which appeared in the previous work of Erik Carlsson and the author on the proof of the shuffle conjecture. This allows us to construct 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 is then shown to be a corollary of this relation.

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Anton Mellit. "Toric braids and (m,n)-parking functions." Duke Math. J. 170 (18) 4123 - 4169, 1 December 2021. https://doi.org/10.1215/00127094-2021-0011

Information

Received: 19 October 2018; Revised: 14 November 2020; Published: 1 December 2021
First available in Project Euclid: 22 November 2021

Digital Object Identifier: 10.1215/00127094-2021-0011

Subjects:
Primary: 05A15
Secondary: 05E05 , 20C08 , 20F36

Keywords: Dyck paths , parking functions , shuffle conjecture , symmetric functions , toric braids

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 18 • 1 December 2021
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