Indecomposable symplectic $k(C_2\times C_2)$--modules and their quadratic forms

TL;DR

This paper classifies all indecomposable symplectic modules over the Klein-Four Group in characteristic two and explores their associated quadratic forms, providing a comprehensive understanding of their structure and isometry classes.

Contribution

It provides a complete classification of indecomposable symplectic modules and their quadratic forms over the Klein-Four Group in characteristic two, including isometry classes.

Findings

All indecomposable symplectic $kG$-modules are classified.

The associated quadratic forms and their isometry classes are determined.

The results are specific to the Klein-Four Group in characteristic two.

Abstract

For the Klein-Four Group $G$ and a perfect field $k$ of characteristic two we determine all indecomposable symplectic $kG$-modules, that is, $kG$-modules with a symplectic, $G$-invariant form which do not decompose into smaller such modules, and classify them up to isometry. Also we determine all quadratic forms that have one of the above symplectic forms as their associated bilinear form and describe their isometry classes.