5
$\begingroup$

What are the finite-dimensional irreducible representations of the quiver with one vertex and two loops?

$\endgroup$
4
  • 3
    $\begingroup$ Isn't this the same as classifying pairs of matrices up to simultaneous conjugation? This is a well-known "wild" classification problem (in fact, it is the prototype/definition of "wild") so it is generally thought to be hopeless... $\endgroup$ Jun 26, 2021 at 15:47
  • 8
    $\begingroup$ @SamHopkins they said irreducible representations. The module category can be wild and irreducible representations not wild. For example any finite acyclic quiver that is not Dynkin has wild representation type but finitely many irreducibles and are easy to describe but the indecomposables are wild. This question still seems hard though since you are clarifying up to simultaneous conjugacy pairs of matrices that generate the whole matrix algebra $\endgroup$ Jun 26, 2021 at 17:49
  • 2
    $\begingroup$ L. Le Bruyn, C. Procesi, Semisimple representations of quivers, Trans. Amer. Math. Soc. 317(1990) 585–598, Theorem 4 should be relevant. $\endgroup$ Jun 28, 2021 at 8:52
  • 1
    $\begingroup$ Sometimes the difference in reception of a question between MathSE and MO can be surreal. This question was deleted on MathSE for missing context $\endgroup$ Jul 6, 2021 at 15:26

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.