A Note on Additive Diameter

RESUMO

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 $0<\varepsilon<1/3$, let $n>9/\varepsilon^2$, and let $U\leq\mathfrak{sl}_n(\mathbb{C})$ with $\dim U>\varepsilon n^2$. Then $\operatorname{diam}^{\mathrm{SL}_n(\mathbb{C})}_{+}(\mathfrak{sl}_n(\mathbb{C}),U)\leq 32/\varepsilon+8$. 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, $U\leq V$, and $\mathfrak g=\operatorname{Lie}(G)$, then $\operatorname{diam}^{G}_{+}(V,U)\leq\operatorname{diam}^{\mathfrak g}_{+}(V,U)$. No irreducibility or connectedness assumption is needed.