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Self-dual modules in characteristic two and normal subgroups. (English) Zbl 07290688
The paper under review is concerned with special features of the Clifford theory for finite group representations over an algebraically closed field \(F\) of characteristic \(2\). Thus let \(N\) be a normal subgroup of a finite group \(G\).
Suppose first that \(W\) is a simple \(FN\)-module. The authors prove that \(W\) is conjugate to its dual if and only if there is a self-dual simple \(FG\)-module \(V\) such that \(W\) occurs in the restriction of \(V\) with odd multiplicity. Moreover, if \(W\) is self-dual then \(V\) is unique, and the multiplicity is \(1\).
Suppose next that \(V\) is a self-dual simple \(FG\)-module not of quadratic type. The authors show that the restriction of \(V\) to \(FN\) is a sum of distinct self-dual simple \(FN\)-modules, none of which have quadratic type. Furthermore, all non-trivial self-dual simple \(FG\)-modules are of quadratic type if and only if the same is true for \(FN\) and \(F[G/N]\).
The authors also prove that, for any real block \(b\) of \(FN\), there is a unique real block of \(FG\) which covers \(b\) and is weakly regular.
MSC:
20C20 Modular representations and characters
20C15 Ordinary representations and characters
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
Software:
GAP
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