The structured Gerstenhaber problem (II)

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 [10], 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 is 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 is large enough).