A Note on Additive Diameter

Abstract

We record two results on additive diameters of group representations. First we give a corrected version of Proposition 5.3 from the first arXiv version of arXiv:2504.07573. Let $0<\epsilon <1/3$, let $n>9/{\epsilon }^2$, and let $U\le {\mathfrak{sl}}_n(\mathbb{C})$ with $\dim U>\epsilon n^2$. Then ${\mathrm{diam}}_{+}^{{\mathrm{SL}}_n(\mathbb{C})}({\mathfrak{sl}}_n(\mathbb{C}),U)\le 32/\epsilon +8$. The proof uses an averaging argument for the action of the symmetric group on the off-diagonal matrix positions. This argument was subsequently developed further in arXiv:2609.03882. We also prove the right-hand inequality in Question 6.7 of arXiv:2504.07573. If $G$ is a complex algebraic group, $V$ is a finite-dimensional $G$-module, $U\le V$, and $\mathfrak{g}=\mathrm{Lie}(G)$, then ${\mathrm{diam}}_{+}^G(V,U)\le {\mathrm{diam}}_{+}^{\mathfrak{g}}(V,U)$. No irreducibility or connectedness assumption is needed.

7 pages