Additive diameters of group representations

Abstract

We explore the concept of additive diameters in the context of group representations, unifying various noncommutative Waring-type problems. Given a finite-dimensional representation $\ddot{I}:G\to \mathrm{GL}(V)$ and a subspace $U\le V$ that generates $V$ as a $G$-module, we define the $G$-additive diameter of $V$ with respect to $U$ as the minimal number of translates of $U$ under the representation $\ddot{I}$ needed to cover $V$. We demonstrate that every irreducible representation of ${\mathrm{SL}}_2(\mathbf{C})$ exhibits optimal additive diameters and establish sharp bounds for the conjugation representation of ${\mathrm{SL}}_n(\mathbf{C})$ on its Lie algebra ${\mathfrak{sl}}_n(\mathbf{C})$. Additionally, we investigate analogous notions for additive diameters in Lie representations. We provide applications to additive diameters with respect to images of equivariant algebraic morphisms, linking them to the corresponding $G$-additive diameters of images of their differentials.