Cartan invariants and defect zero elements. (Q866823)

Let \(G\) be a finite group with irreducible Brauer characters \(\varphi_1,\dots,\varphi_l\) in characteristic 2, and let \(C=(c_{ij})\) denote the corresponding Cartan matrix. As a consequence of more general results, \textit{T. Breuer, L. Héthelyi, E. Horváth, B. Külshammer}, and \textit{J. Murray} [J. Algebra 296, No. 1, 177-195 (2006; Zbl 1121.20005)] have shown that \(G\) contains a real element of 2-defect zero if and only if \(c_{ii}\) is odd for some \(i\). Here the authors prove that in this case \(i\) can be chosen in such a way that \(\varphi_i\) is real-valued. Also, the existence of a real element of 2-defect zero is equivalent to the existence of an odd Cartan invariant \(c_{ij}\) with real-valued \(\varphi_i\) and \(\varphi_j\). Examples seem to indicate that there is no connection to the existence of a \(G\)-invariant quadratic form on the relevant simple modules. The authors show also that for a group of even order the first Cartan invariant \(c_{11}\) corresponding to the trivial Brauer character is always even.