The structured Gerstenhaber problem. III. (Q2187388)

The author starts with an overview of the background motivating his research. Let \(\mathbb{F}\) be a field and let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\), equipped with a bilinear form \(b\) which is either symmetric, i.e., \(\forall(x, y) \in V^2, b(x, y) = b(y, x)\), or alternating, i.e., \(\forall x \in V\), \(b(x,x) = 0\). An endomorphism \(u\) of \(V\) is called \(b\)-symmetric (respectively, \(b\)-alternating) whenever the bilinear form \((x, y) \in V^2\mapsto b(x, u(y))\) is symmetric (respectively, alternating). The set of all \(b\)-symmetric endomorphisms is denoted by \(\mathcal{S}_b\), and the set of all \(b\)-alternating ones is denoted by \(\mathcal{A}_b\). Moreover, \(\mathcal{A}_b \subset \mathcal{S}_b\) if \(\mathbb{F}\) has characteristic 2. The main result that the author obtains is the following theorem. Let \(\mathbb{F}\) be a field of characteristic \(2\). Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\) and \(b\) be a non-degenerate symmetric bilinear form on \(V,\) with corresponding quadratic form \(\mathrm{Q}\). Denote by \(\nu\) the Witt index of \(b\), and set \(n := \dim V\). (a) The greatest dimension of a nilpotent subspace of \(\mathcal{S}_b\) is \(\nu(n-\nu)\). (b) If \(n\neq 2\nu+1\), then the greatest dimension of a nilpotent subspace of \(\mathcal{A}_b\) is \(\nu(n-\nu-1).\) (c) If \(n = 2\nu + 1\), then the greatest dimension of a nilpotent subspace of \(\mathcal{A}_b\) is \(\nu(n-\nu)-\dim\mathrm{SKer} \, \mathrm{Q}\). Here the notation SKer stands for super-kernel. For Part I and II see [the author, ibid. 567, 263--298 (2019; Zbl 1415.15018); ibid. 569, 113--145 (2019; Zbl 1415.15024)].