The indecomposable symplectic and quadratic modules of the Klein-four group
Let \(G\) be the Klein \(4\)-group and \(k\) be a perfect field of characteristic \(2\). Since \(G\) has tame representation type, \(kG\) has only finitely many indecomposable modules of each dimension and these have been classified in [\textit{S. B. Conlon}, J. Aust. Math. Soc. 10, 363--366 (1969; Zbl 0186.04603)]. Using the notation of the latter paper, the \(kG\)-modules which give rise to indecomposable symplectic modules are: \((k_{G})^{2}\), \(kG\), \((kG)^{2}\), \(A_{n}\oplus B_{n}\), \(C_{n}(\pi)\), \(C_{n}(\pi)^{2}\), \(C_{n}(\infty)\) and \(C_{n}(\infty)^{2}\) where \(\pi\) is a monic irreducible polynomial over \(k\). For each of these \(kG\)-modules, the authors describe all symplectic forms, give representatives of the isometry classes of each form, and determine the \(G\)-invariant quadratic forms (if any) which polarize to the symplectic forms.
indecomposable symplectic and quadratic modules
scientific article; zbMATH DE number 6893261