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Quadratic principal indecomposable modules and strongly real elements of finite groups

Authors: Rod Gow and John Murray
Journal: Proc. Amer. Math. Soc. 147 (2019), 2783-2796
MSC (2010): Primary 20C20; Secondary 20C05
Published electronically: April 3, 2019
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Abstract: If $ G$ is a finite group and $ k$ is an algebraically closed field of characteristic $ 2$, we show that the number of isomorphism classes of quadratic type principal indecomposable $ kG$-modules is equal to the number of conjugacy classes of strongly real odd order elements of $ G$.

If $ \varphi $ is a self-dual irreducible $ 2$-Brauer character of $ G$, we show that the corresponding principal indecomposable $ kG$-module has quadratic type if and only if $ \varphi (g)/2$ is not an algebraic integer for some strongly real odd order element $ g$ in $ G$.

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Rod Gow
Affiliation: School of Mathematics and Statistics, University College Dublin, Dublin, D14 YH57, Ireland

John Murray
Affiliation: Department of Mathematics and Statistics, National University of Ireland, Maynooth, County Kildare, Ireland

Received by editor(s): April 16, 2018
Received by editor(s) in revised form: August 16, 2018, September 11, 2018, and September 28, 2018
Published electronically: April 3, 2019
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2019 American Mathematical Society