# The structured Gerstenhaber problem (III)

## TL;DR

This paper investigates the maximum dimension of subspaces of nilpotent endomorphisms that are symmetric or alternating with respect to a symmetric bilinear form over a field of characteristic 2, relating it to various invariants.

## Contribution

It provides a formula for the maximum dimension of such subspaces, extending the structured Gerstenhaber problem to characteristic 2 and including a special case analysis.

## Key findings

- Derived the maximum dimension as a function of form invariants
- Extended Gerstenhaber problem to characteristic 2
- Identified an additional invariant in a special case

## Abstract

Let $b$ be a symmetric bilinear form on a finite-dimensional vector space over a field with characteristic $2$. Here, we determine the greatest possible dimension of a linear subspace of nilpotent $b$-symmetric or $b$-alternating endomorphisms of $V$, expressing it as a function of the dimension, the rank, the Witt index of $b$, and an additional invariant in a very special case.