Mutations of reflections and existence of pseudo-acyclic orderings for\n type A_n

2021/08/06 by Tucker J. Ervin, Blake Jackson, Ervin, Tucker J. +5
Computer Science · Mathematics · #13F60 (Primary) #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2108.03309

openalex publication_date 2021/08/06 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

In a recent paper by K.-H. Lee, K. Lee and M. Mills, a mutation of\nreflections in the universal Coxeter group is defined in association with a\nmutation of a quiver. A matrix representation of these reflections is\ndetermined by a linear ordering on the set of vertices of the quiver. It was\nconjectured that there exists an ordering (called a pseudo-acyclic ordering in\nthis paper) such that whenever two mutation sequences of a quiver lead to the\nsame labeled seed, the representations of the associated reflections also\ncoincide. In this paper, we prove this conjecture for every quiver\nmutation-equivalent to an orientation of a type A_n Dynkin diagram by\ndecomposing a mutation sequence into a product of elementary swaps and checking\nrelations studied by Barot and Marsh.\n

Citations

• Quivers with potentials and their representations I: Mutations

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