Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups

TL;DR

This paper investigates bilinear and quadratic forms on rational modules of split reductive groups over fields of prime characteristic, extending classical results from complex representation theory to positive characteristic cases, including characteristic 2.

Contribution

It provides new results on the existence of symplectic or orthogonal structures for modules over fields of positive characteristic, especially addressing the case of characteristic 2.

Findings

Complete classification for Weyl modules over characteristic 2

Extension of classical results to positive characteristic fields

Analysis of bilinear and quadratic forms on various modules

Abstract

The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the question of whether a given representation is symplectic or orthogonal has been solved over the complex numbers since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from 2 hold by using similar proofs. This paper considers analogues of these results for simple, induced and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.