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Rectangular representations and $λ$-independence of algebraic monodromy groups

Chun-Yin Hui, Wonwoong Lee

Abstract

Let $\mathfrak g$ be a complex semisimple Lie algebra. We define what it means for a finite dimensional representation of $\mathfrak g$ to be rectangular and completely classify faithful rectangular representations. As an application, we obtain new $λ$-independence results on the algebraic monodromy groups of compatible systems of $λ$-adic Galois representations of number fields.

Rectangular representations and $λ$-independence of algebraic monodromy groups

Abstract

Let be a complex semisimple Lie algebra. We define what it means for a finite dimensional representation of to be rectangular and completely classify faithful rectangular representations. As an application, we obtain new -independence results on the algebraic monodromy groups of compatible systems of -adic Galois representations of number fields.

Paper Structure

This paper contains 25 sections, 39 theorems, 66 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Rectangular representations
  3. Application to Galois representations
  4. Organization of the article
  5. Classification of rectangular representations
  6. Root system and Weyl group of $B_n$
  7. Rectangular and hypercubic subsets
  8. Rectangular and hypercubic representations
  9. Reduction to the hypercubic case
  10. Extracting $A_1$-factors and excluding $A_2$, $A_r$ ($r\geq 4$), and exceptional factors
  11. Proof of Theorem \ref{['main1']}(i)
  12. Proof of Theorem \ref{['main1']}(ii)
  13. Proof of Theorem \ref{['main1']}(iii)
  14. Proofs of Theorem \ref{['main1']}(iv),(v)
  15. $\lambda$-independence of algebraic monodromy groups
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\psi$ be a faithful rectangular representation of a complex semisimple Lie algebra $\mathfrak{g}$. Fix a decomposition $\mathfrak{g}=\mathfrak{g}_1\times\mathfrak{g}_2\times\cdots\times \mathfrak{g}_k$, where $\mathfrak{g}_1$ denotes the product of $A_1$-factors and $\mathfrak{g}_2,\ldots,\math

Figures (5)

  • Figure 1: $(A_1, \mathrm{Sym}^{4}(\mathrm{Std}) \oplus \mathrm{Sym}^{3}(\mathrm{Std}))$ and $(A_1 \times A_1, (\mathrm{Std} \otimes \mathbb 1) \oplus (\mathbb 1 \otimes \mathrm{Std}))$
  • Figure 2: $(B_2, \mathrm{Std}\oplus \mathrm{Spin})$ and $(A_3,\mathrm{Std}\oplus\mathrm{Std}^{\vee})$
  • Figure 3: $\Phi_{B_2}$ (left) and $\Phi_{B_3}$ (right, and $\Phi_{D_3}$ consists of long roots in $\Phi_{B_3}$)
  • Figure 4: \ref{['sq']} for $d=4$
  • Figure 5: $f_1,f_2,f_3,f_4$ in $[-1/2,1/2]^3$

Theorems & Definitions (78)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Corollary 1.10
  • ...and 68 more