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Almost irreducible tensor squares. (English) Zbl 0931.20009
Let \(G\) be a covering group of a finite almost simple group. The author determines faithful irreducible complex characters \(\chi\) of \(G\) for which \(\chi\otimes\chi^*-1\) is irreducible, where \(\chi^*\) is the character of the dual of the \(\mathbb{C} G\)-module which affords \(\chi\). This gives a classification of the quasisimple absolutely irreducible subgroups of \(\text{GL}_n(q)\) of order prime to \(q\) which act irreducibly on the Lie algebra of type \(A_{n-1}\) via the adjoint representation. The proof uses Lusztig’s description of the degrees of irreducible characters of reductive groups and the determination of Brauer trees by Fong and Srinivasan to handle the case of groups of Lie type.

MSC:
20C15 Ordinary representations and characters
20C33 Representations of finite groups of Lie type
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
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