Additive diameters of group representations
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908311450812416 |
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| author | Jezernik, Urban Špenko, Špela |
| author_facet | Jezernik, Urban Špenko, Špela |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07573 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Additive diameters of group representations Jezernik, Urban Špenko, Špela Representation Theory Algebraic Geometry Group Theory Rings and Algebras 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. |
| title | Additive diameters of group representations |
| topic | Representation Theory Algebraic Geometry Group Theory Rings and Algebras |
| url | https://arxiv.org/abs/2504.07573 |