Reflection Groups and Quiver Mutation: Diagrammatics

We extend Carter's notion of admissible diagrams and attach a Dynkin-like diagram to each reduced reflection factorization of an element in a finite Weyl group. We give a complete classification for the diagrams attached to reduced reflection factorizations. We show that such a diagram is cycli...

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Bibliographic Details
Published inThe Electronic journal of combinatorics Vol. 32; no. 2
Main Author Wegener, Patrick
Format Journal Article
LanguageEnglish
Published 25.04.2025
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Summary:We extend Carter's notion of admissible diagrams and attach a Dynkin-like diagram to each reduced reflection factorization of an element in a finite Weyl group. We give a complete classification for the diagrams attached to reduced reflection factorizations. We show that such a diagram is cyclically orientable if and only if it is isomorphic to the underlying graph of a quiver which is mutation-equivalent to a Dynkin quiver. Furthermore, these diagrams encode a natural presentation of the Weyl group as reflection group, as shown by Cameron-Seidel-Tsaranov (1994) as well as Barot-Marsh (2015).
ISSN:1077-8926
1077-8926
DOI:10.37236/13369