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Finite groups of matrices over quadratic number fields

Daniil Yurshevich

TL;DR

The paper develops a practical algorithm to classify all finite matrix groups over a number field by combining representation-theoretic tools (irreducibles, characters, and the canonical decomposition) with two key devices: Weil restriction to realize representations over a base field and Schur index constraints to control realizability. It leverages Schur bounds to bound orders, reduces problems to finite fields, and reconstructs faithful representations from character data, with an explicit Magma implementation. The authors apply the method to quadratic fields with class number one, delivering explicit classifications for GL_3 and SL_3 over $K=\mathbb{Q}(\sqrt{-19})$ and providing a detailed computational account. The work provides a computational framework that supports explicit classification of finite subgroups of GL_n(K) and has applications in algebraic geometry, notably in constructing Calabi–Yau quotients via crepant resolutions.

Abstract

In this paper we give an algorithm to determine all finite matrix groups over a number field. Our algorithm is based on the representation theory of finite groups.

Finite groups of matrices over quadratic number fields

TL;DR

The paper develops a practical algorithm to classify all finite matrix groups over a number field by combining representation-theoretic tools (irreducibles, characters, and the canonical decomposition) with two key devices: Weil restriction to realize representations over a base field and Schur index constraints to control realizability. It leverages Schur bounds to bound orders, reduces problems to finite fields, and reconstructs faithful representations from character data, with an explicit Magma implementation. The authors apply the method to quadratic fields with class number one, delivering explicit classifications for GL_3 and SL_3 over and providing a detailed computational account. The work provides a computational framework that supports explicit classification of finite subgroups of GL_n(K) and has applications in algebraic geometry, notably in constructing Calabi–Yau quotients via crepant resolutions.

Abstract

In this paper we give an algorithm to determine all finite matrix groups over a number field. Our algorithm is based on the representation theory of finite groups.

Paper Structure

This paper contains 7 sections, 27 theorems, 53 equations.

Table of Contents

  1. Preliminaries
  2. Weil Restriction of scalars
  3. Schur index
  4. Schur bound
  5. Algorithm
  6. Finite subgroups of $\operatorname{GL}(3,\mathds{Q}{\sqrt{-19}})$ and $\operatorname{SL}(3,\mathds{Q}{\sqrt{-19}})$
  7. MAGMA

Key Result

Lemma 1.5

If $\rho$ and $\rho'$ are two representations of $G$ in $V$ and $W$ resp. then $\langle \chi_\rho, \chi_{\rho'}\rangle = \dim(\operatorname{Hom}_G(V, W))$, where $\operatorname{Hom}_G(V, W)$ are $G$-linear maps between $V$ and $W$.

Theorems & Definitions (47)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5
  • Definition 1.6
  • Theorem 1.7: Schur's lemma, Serre
  • Theorem 1.8: Maschke Theorem, CR
  • Corollary 1.9
  • Corollary 1.10
  • ...and 37 more