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THE DISCRETE FUNDAMENTAL GROUP OF THE ASSOCIAHEDRON, AND THE EXCHANGE MODULE

HÉLÈNE BARCELO

Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720, USA

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CHRISTOPHER SEVERS

eBay Incorporated, 2065 Hamilton Avenue, San Jose, CA 95125, USA

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, and

JACOB A. WHITE

Texas A&M University, Department of Mathematics, Mailstop 3368, College Station, TX 77843-3368, USA

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https://doi.org/10.1142/S0218196713400079Cited by:0 (Source: Crossref)

Abstract

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is free abelian of rank . We also find a combinatorial description for a basis of this rank. We also introduce the exchange module of the type A_n cluster algebra, used to model the relations in the cluster algebra. We use the discrete fundamental group to the study of exchange module, and show that it is also free abelian of rank .

Keywords:

AMSC: 05E45

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