Quadratic representations for groups of Lie type over fields of characteristic two.
Quadratic representations for simple groups have a long history. Here we will consider simple groups \(G\) acting faithfully on a vector-space \(V\) over a field in characteristic \(p\). A subgroup \(A\) of \(G\) is said to act quadratically if \([V,A,A]=0\). The problem is to determine all pairs \((G,V)\) for \(A\) with \(|A|>2\). If \(| A|\) is odd this has been done by \textit{J. G. Thompson} [Actes Congr. Internat. Math. 1970, 1, 375-376 (1971; Zbl 0236.20024)], \textit{A. A. Premet} and \textit{I. D. Suprunenko} [Math. Nachr. 110, 65-96 (1983; Zbl 0522.20027)] and \textit{A. Chermak} [J. Algebra (to appear)]. If \(|A|=4\) this has been done for alternating groups, sporadic groups and groups of Lie type in odd characteristic by \textit{U. Meierfrankenfeld} and the reviewer [Arch. Math. 55, No. 2, 105-110 (1990; Zbl 0719.20008); Commun. Algebra 18, No. 7, 2099-2139 (1990; Zbl 0707.20009)]. If \(G\) is a group of Lie type in characteristic two and \(A\) is not in a root group but intersects a root group nontrivially this has been done by the reviewer [Isr. J. Math. 79, No. 2-3, 257-279 (1992; Zbl 0803.20014)]. In this paper the author extends the last result by classifying those modules for Lie groups in characteristic two where \(A\) is a linearly independent acting quadratic fours group. Here a fours group \(A=\langle a,b\rangle\) in said to act linearly dependent on a \(K\)-vector-space \(V\) if there is some \(\gamma\in K\) such that \([v,a]=\gamma[v,b]\) for all \(v\in V\). Otherwise it acts linearly independent. In fact if \(K\) is algebraically closed, \(G\) a group of Lie type over \(K\) and \(A\) a quadratic fours group which acts linearly dependent then also the modules are classified. In that case \(A\) mostly is contained in a root group. The list of outcomming modules is the one one would expect, like fundamental modules for classical groups \(V(\lambda_1)\), \(V(\lambda_n)\) for \(E_6\), \(F_4\), \(V(\lambda_7)\) for \(E_7\) and \(V(\lambda_2)\) for \(G_2\). The main problem in proving the results is to get rid of the tensor products.
scientific article; zbMATH DE number 1995424
scientific article