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Circuits and Hurwitz action in finite root systems
by
Joel Brewster Lewis
GWU
Coauthors: Victor Reiner (UMN)
There is a natural braid group action on tuples of elements in a group, called the Hurwitz action. In a finite real reflection group, all factorizations of a Coxeter element into a minimal number of reflections lie in the same orbit of this action. We extend this result to factorizations of arbitrary length, showing that two reflection factorizations of a Coxeter element lie in the same orbit under the Hurwitz action if and only if they use the same multiset of conjugacy classes. The proof makes use of a surprising lemma, derived from a classification of the minimal linear dependences (matroid circuits) in finite root systems: any set of roots forming a minimal linear dependence with positive coefficients has a disconnected graph of pairwise acuteness.
Date received: November 28, 2017
Copyright © 2017 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cboj-17.