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Locally free representations of quivers over commutative Frobenius algebras

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Abstract

In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra \(\textrm{R}\) 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 \(\textrm{R}\) equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over \(\textrm{R}[t]/(t^2)\). Using these results together with results of Geiss, Leclerc and Schröer we give, when \(\textrm{k}\) is algebraically closed, a classification of pairs \((Q,\textrm{R})\) such that the set of isomorphism classes of indecomposable locally free representations of Q over \(\textrm{R}\) 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 \(\mathbb {F}_q[t]/(t^r)\). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.

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Notes

  1. We thank Bernard Leclerc for pointing out to us this argument from [18].

  2. Tameness in exact subcategories of module categories is not well established, here we simply mean that their constructible set of 1-dimensional.

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Acknowledgements

Special thanks go to Christof Geiss, Bernard Leclerc and Jan Schröer for explaining their work but also for sharing some unpublished results with us. We also thank the referee for many useful suggestions. We would like to thank Tommaso Scognamiglio for pointing out a mistake in the proof of Proposition 5.17 in an earlier version of the paper. We would like also to thank Alexander Beilinson, Bill Crawley-Boevey, Joel Kamnitzer, and Peng Shan for useful discussions.

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Authors and Affiliations

  1. IST Austria, Klosterneuburg, Austria

    Tamás Hausel

  2. Université Paris Cité, IMJ-PRG, CNRS, Paris, France

    Emmanuel Letellier

  3. ICTP Trieste, Trieste, Italy

    Fernando Rodriguez-Villegas

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  1. Tamás Hausel
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  2. Emmanuel Letellier
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Correspondence to Emmanuel Letellier.

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Hausel, T., Letellier, E. & Rodriguez-Villegas, F. Locally free representations of quivers over commutative Frobenius algebras. Sel. Math. New Ser. 30, 20 (2024). https://doi.org/10.1007/s00029-023-00914-2

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