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Rational quiver representations: tame and wild

Fabian Januszewski

TL;DR

The paper analyzes $K$-rational quivers over characteristic $0$ fields and their indecomposable representations through Galois descent, establishing an étale $K$-species framework that preserves representations. It generalizes Gabriel-type results: for underlying ADE quivers, $K$-rational quivers are representation-finite and indecomposables over $K$ correspond to $ ext{Gal}(L/K)$-orbits on the positive roots, with no Brauer descent obstruction in this case. It then presents a $D_4$-type triality example, showing a $G_2$-type associated species with six indecomposables aligned with root-orbit data. Finally, it exhibits a wild case via a two-loop quiver where arbitrary Brauer obstructions can be realized, linking endomorphism rings to a central division algebra $D$ and the class $[D]\

Abstract

We study representation finite $K$-rational quivers over fields of characteristic $0$ and their indecomposable representations, exploiting that all Brauer obstructions for descent of representations are trivial in this case. Contrasting the tame case, we give an example of a simple quiver of wild representation type, where we realize every possible Brauer obstruction of a given Galois extension $L/K$ in the category of quiver representations over $L$.

Rational quiver representations: tame and wild

TL;DR

The paper analyzes -rational quivers over characteristic fields and their indecomposable representations through Galois descent, establishing an étale -species framework that preserves representations. It generalizes Gabriel-type results: for underlying ADE quivers, -rational quivers are representation-finite and indecomposables over correspond to -orbits on the positive roots, with no Brauer descent obstruction in this case. It then presents a -type triality example, showing a -type associated species with six indecomposables aligned with root-orbit data. Finally, it exhibits a wild case via a two-loop quiver where arbitrary Brauer obstructions can be realized, linking endomorphism rings to a central division algebra and the class $[D]\

Abstract

We study representation finite -rational quivers over fields of characteristic and their indecomposable representations, exploiting that all Brauer obstructions for descent of representations are trivial in this case. Contrasting the tame case, we give an example of a simple quiver of wild representation type, where we realize every possible Brauer obstruction of a given Galois extension in the category of quiver representations over .

Paper Structure

This paper contains 4 sections, 6 theorems, 6 equations.

Table of Contents

  1. Rational quiver representations
  2. $K$-rational quivers of finite representation type
  3. A rational quiver of type $D_4$ and triality
  4. The wild case: Realizing arbitrary Brauer obstructions

Key Result

Theorem 2.1

Let $\Gamma$ be a connected quiver of type $A$, $D$, or $E$. Then the indecomposable representaitons of $\Gamma$ over any field $L$ of charactereristic $0$ are in canonical bijection with the positive roots of the underyling root system $\Delta$. The endomorphism rings of each indecomposable represe

Theorems & Definitions (17)

  • Definition 1.1: $K$-rational quiver, januszewskirationalquivers
  • Definition 1.2: Representation of a rational quiver, januszewskirationalquivers
  • Definition 1.3: $K$-species
  • Definition 1.4: Étale $K$-species, januszewskirationalquivers
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • ...and 7 more