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Joel Brewster Lewis and Victor Reiner
Circuits and Hurwitz action in finite root systems view print
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| Published: |
December 8, 2016 |
| Keywords: |
Root system, reflection group, factorization, Hurwitz action, Coxeter element, reflection, acuteness, Gram matrix, circuit, matroid |
| Subject: |
20F55, 51F15, 05Exx |
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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.
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| Acknowledgements
This work was partially supported by NSF grants DMS-1148634 and DMS-1401792.
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| 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
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