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Locally free representations of quivers over commutative Frobenius algebras

Tamas Hausel, Emmanuel Letellier, Fernando Rodriguez Villegas

Abstract

In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t^2). Using these results together with results of Geiss, Leclerc and Schroer we give, when k is algebraically closed, a classification of pairs (Q,R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally, when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra F_q[t]/(t^r). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.

Locally free representations of quivers over commutative Frobenius algebras

Abstract

In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t^2). Using these results together with results of Geiss, Leclerc and Schroer we give, when k is algebraically closed, a classification of pairs (Q,R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally, when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra F_q[t]/(t^r). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.

Paper Structure

This paper contains 25 sections, 49 theorems, 192 equations, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Frobenius algebras
  4. Fourier transform
  5. Fourier transform for group representations
  6. Moment maps associated to group scheme actions
  7. Fourier transforms
  8. A counter-example
  9. Representations of quivers
  10. The finite field case
  11. The algebraically closed fields case
  12. The algebraic closure of a finite field
  13. Arbitrary algebraically closed field
  14. Finite locally free representation type
  15. Proof of Proposition \ref{['d4']}.
  16. ...and 10 more sections

Key Result

Theorem 1.1

Assume that ${\rm R}$ is a Frobenius algebra over a finite or algebraically closed field ${\rm k}$. Let $Q$ be a quiver and let $Q'$ be another quiver obtained from $Q$ by changing the orientation of some of the arrows. Then If moreover ${\rm R}$ is local, split and if ${\rm k}$ contains a primitive $|\alpha|$-th root of unity then

Theorems & Definitions (98)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Example 2.2
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Proposition 3.4
  • proof
  • ...and 88 more