The structured Gerstenhaber problem (I)

TL;DR

This paper determines the maximum dimension of nilpotent subspaces of bilinear form-preserving endomorphisms, generalizing previous results over complex fields to broader algebraic settings.

Contribution

It extends the known bounds for nilpotent subspaces of symmetric, alternating, and Hermitian endomorphisms to arbitrary fields and forms, including characterizations of maximal subspaces.

Findings

Maximum dimension formulas depend on form rank, dimension, and Witt index.

Characterization of subspaces with maximal dimension in specific cases.

Generalization of classical results to broader algebraic contexts.

Abstract

Let $b$ be a symmetric or alternating bilinear form on a finite-dimensional vector space $V$. When the characteristic of the underlying field is not $2$, we determine the greatest dimension for a linear subspace of nilpotent $b$-symmetric or $b$-alternating endomorphisms of $V$, expressing it as a function of the dimension, the rank, and the Witt index of $b$. Similar results are obtained for subspaces of nilpotent $b$-Hermitian endomorphisms when $b$ is a Hermitian form with respect to a non-identity involution. In three situations ($b$-symmetric endomorphisms when $b$ is symmetric, $b$-alternating endomorphisms when $b$ is alternating, and $b$-Hermitian endomorphisms when $b$ is Hermitian and the underlying field has more than $2$ elements), we also characterize the linear subspaces with the maximal dimension. Our results are wide generalizations of results of Meshulam and Radwan, who tackled the case of a non-degenerate symmetric bilinear form over the field of complex numbers, and recent results of Bukov\v{s}ek and Omladi\v{c}, in which the spaces with maximal dimension were determined when the underlying field is the one of complex numbers, the bilinear form $b$ is symmetric and non-degenerate, and one considers $b$-symmetric endomorphisms.