Year: 2025
Type: preprint
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 $ρ\colon G \to \mathrm{GL}(V)$ and a subspace $U \leq V$ 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 $ρ$ 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.
Source: arXiv (Cornell University)
Authors: Jezernik, Urban, Špenko, Špela
Institution: Université Libre de Bruxelles
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6910701558
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