The structured Gerstenhaber problem (II)

TL;DR

This paper classifies maximal dimension spaces of nilpotent endomorphisms that are symmetric or alternating with respect to a bilinear form, extending previous results and introducing a new technique for reducibility of nilpotent operator spaces.

Contribution

It provides a classification of maximal dimension nilpotent subspaces related to bilinear forms under certain field conditions, and introduces a new reducibility criterion for nilpotent operator spaces.

Findings

Classified spaces with maximal dimension of nilpotent endomorphisms respecting bilinear forms.

Developed a new sufficient condition for reducibility of nilpotent operator spaces.

Provided a new proof of Gerstenhaber's theorem using the new technique.

Abstract

Let $b$ be a non-degenerate symmetric (respectively, alternating) bilinear form on a finite-dimensional vector space $V$, over a field with characteristic different from $2$. In a previous work, we have determined the maximal possible dimension for a linear subspace of $b$-alternating (respectively, $b$-symmetric) nilpotent endomorphisms of $V$. Here, provided that the cardinality of the underlying field be large enough with respect to the Witt index of $b$, we classify the spaces that have the maximal possible dimension. Our proof is based on a new sufficient condition for the reducibility of a vector space of nilpotent linear operators. To illustrate the power of that new technique, we use it to give a short new proof of the classical Gerstenhaber theorem on large vector spaces of nilpotent matrices (provided, again, that the cardinality of the underlying field be large enough).