Quadratic principal indecomposable modules and strongly real elements of finite groups

Gow, Rod; Murray, John

doi:10.1090/proc/14441

Quadratic principal indecomposable modules and strongly real elements of finite groups
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by Rod Gow and John Murray PDF

Proc. Amer. Math. Soc. 147 (2019), 2783-2796 Request permission

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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