Isomorphisms, automorphisms, and generalized involution models of projective reflection groups
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11856_2013_Article_44.pdf
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Author(s)
Caselli, Fabrizio
Marberg, Eric Paul
Date Issued
August 2013
Journal
Israel Journal of Mathematics
Publisher
The Hebrew University Magnes Press
Citation
Caselli, Fabrizio, and Eric Marberg. “Isomorphisms, Automorphisms, and Generalized Involution Models of Projective Reflection Groups.” Israel Journal of Mathematics 199.1 (2014): 433–483. © 2016 Springer International Publishing AG
Version
Final published version
Abstract
We investigate the generalized involution models of the projective reflection groups G(r, p, q, n). This family of groups parametrizes all quotients of the complex reflection groups G(r, p, n) by scalar subgroups. Our classification is ultimately incomplete, but we provide several necessary and sufficient conditions for generalized involution models to exist in various cases. In the process we solve several intermediate problems concerning the structure of projective reflection groups. We derive a simple criterion for determining whether two groups G(r, p, q, n) and G(r, p′, q′, n) are isomorphic. We also describe explicitly the form of all automorphisms of G(r, p, q, n), outside a finite list of exceptional cases. Building on prior work, this allows us to prove that G(r, p, 1, n) has a generalized involution model if and only if G(r, p, 1, n) ≌ G(r, 1, p, n). We also classify which groups G(r, p, q, n) have generalized involution models when n = 2, or q is odd, or n is odd.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s11856-013-0044-5
