 

Joel Brewster Lewis and Victor Reiner
Circuits and Hurwitz action in finite root systems view print


Published: 
December 8, 2016 
Keywords: 
Root system, reflection group, factorization, Hurwitz action, Coxeter element, reflection, acuteness, Gram matrix, circuit, matroid 
Subject: 
20F55, 51F15, 05Exx 


Abstract
In a finite real reflection group, two factorizations of a Coxeter element
into an arbitrary number of reflections are shown to 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.


Acknowledgements
This work was partially supported by NSF grants DMS1148634 and DMS1401792.


Author information
Joel Brewster Lewis:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
jblewis@umn.edu
Victor Reiner:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
reiner@math.umn.edu

