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Additive diameters of group representations

Urban Jezernik, Špela Špenko

TL;DR

The paper introduces and develops the notion of G-additive diameters for finite-dimensional G-representations, unifying noncommutative Waring-type problems. It proves that all irreducible SL_2(C) representations have optimal diameters and derives sharp bounds for the conjugation action of SL_n(C) on sl_n(C), including precise diameter behavior for large versus moderately large subspaces. A parallel Lie-additive framework is developed, with results showing optimal monomial diameters in sl_2(C) and small-diameter behavior for large subspaces in Lie settings, along with insightful distinctions from the group case. Applications to equivariant morphisms connect diameters of im f with diameters of derivatives, yielding bounds that recover and extend known matrix-Waring-type results and linking to Terracini-type tangent-space considerations. Overall, the work provides a cohesive, representation-theoretic approach to additive decompositions, offering concrete diameter bounds, structural descriptions of stable subspaces, and a toolkit for analyzing images of equivariant maps in both group and Lie settings.

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.

Additive diameters of group representations

TL;DR

The paper introduces and develops the notion of G-additive diameters for finite-dimensional G-representations, unifying noncommutative Waring-type problems. It proves that all irreducible SL_2(C) representations have optimal diameters and derives sharp bounds for the conjugation action of SL_n(C) on sl_n(C), including precise diameter behavior for large versus moderately large subspaces. A parallel Lie-additive framework is developed, with results showing optimal monomial diameters in sl_2(C) and small-diameter behavior for large subspaces in Lie settings, along with insightful distinctions from the group case. Applications to equivariant morphisms connect diameters of im f with diameters of derivatives, yielding bounds that recover and extend known matrix-Waring-type results and linking to Terracini-type tangent-space considerations. Overall, the work provides a cohesive, representation-theoretic approach to additive decompositions, offering concrete diameter bounds, structural descriptions of stable subspaces, and a toolkit for analyzing images of equivariant maps in both group and Lie settings.

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 and a subspace that generates as a -module, we define the -additive diameter of with respect to as the minimal number of translates of under the representation needed to cover . We demonstrate that every irreducible representation of exhibits optimal additive diameters and establish sharp bounds for the conjugation representation of on its Lie algebra . 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 -additive diameters of images of their differentials.

Paper Structure

This paper contains 34 sections, 30 theorems, 75 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Additive diameters
  3. Diameters in finite groups
  4. Diameters in infinite groups
  5. Additive diameters in algebras
  6. Contributions -- Additive diameters in representations
  7. Definition and examples
  8. Dimension expanders
  9. Completing the covering
  10. Lie-additive diameters
  11. Applications -- Additive diameters in equivariant morphisms
  12. Reader's Guide
  13. Acknowledgements
  14. Borel Stable Subspaces
  15. Strategy for bounding diameters
  16. ...and 19 more sections

Key Result

Theorem 1.2

Let $G$ be a finite simple group of Lie type of bounded rank. There exists $\epsilon > 0$, depending only on the rank of $G$, such that for any generating set $X \subseteq G$, we have

Figures (6)

  • Figure 1: The upper closed subspace $\langle e_3, e_4, \dots, e_8 \rangle \leq \mathbf{C}[X,Y]_8$.
  • Figure 2: The upper right block subspace $B_{53}$.
  • Figure 3: The spaces $\mathfrak{b}_{\widehat{11}}$, $\mathfrak{b}_{\widehat{nn}}$, $B_{21} + B_{n3}$ and $B_{n,n-1} + B_{n-2,1}$. Note that $E_{22} - E_{33} \notin B_{21} + B_{n3}$, as represented by the red hatching.
  • Figure 4: The space $\operatorname{span}_{\mathbf{C}} \langle E_{11}-E_{nn} \rangle + B_{n-1,2}$.
  • Figure 5: The subspaces $B_{mm}$ ($n$ odd) and $B_{mm} + B_{m+1,m+1}$ ($n$ even). Note that both are contained in $\sl_n(\mathbf{C})$, so the diagonal is zero.
  • ...and 1 more figures

Theorems & Definitions (67)

  • Example 1.1
  • Theorem 1.2: Product theorem, helfgott2008growthPyberSzaboBreuillardGreenTao
  • Example 1.3
  • Theorem 1.4
  • Theorem 1.5: Dimension expanders, Proposition 2.1 in LubotzkyZelmanov
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • proof
  • ...and 57 more