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Factorization problems in complex reflection groups

Part of: Algebraic combinatorics Special aspects of infinite or finite groups Combinatorics

Published online by Cambridge University Press:  02 April 2020

Joel Brewster Lewis  and
Alejandro H. Morales
Joel Brewster Lewis
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA
Alejandro H. Morales*
Affiliation:
Department of Mathematics, George Washington University, Washington, DC e-mail: jblewis@gwu.edu
*
e-mail: ahmorales@math.umass.edu
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Abstract

We enumerate factorizations of a Coxeter element in a well-generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of W-noncrossing partitions.

Keywords

factorizationscomplex reflection groupswreath productpermutations

MSC classification

Primary: 05A15: Exact enumeration problems, generating functions
Secondary: 05E10: Combinatorial aspects of representation theory 20F55: Reflection and Coxeter groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 4 , August 2021 , pp. 899 - 946
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

Dedicated to David M. Jackson in recognition of his 75th birthday

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