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Strongly primitive species with potentials I: mutations

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Abstract

Motivated by the mutation theory of quivers with potentials developed by Derksen–Weyman–Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials for species that arise from skew-symmetrizable matrices that admit a skew-symmetrizer with pairwise coprime diagonal entries. The class of skew-symmetrizable matrices covered by the mutation theory proposed here contains a class of matrices that do not admit global unfoldings, that is, unfoldings compatible with all possible sequences of mutations.

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Notes

  1. Which also go under the name of modulations of valued quivers.

  2. This material has been added following a suggestion of an anonymous referee.

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Acknowledgments

The collaborations [6, 7] of Harm Derksen and Jerzy Weyman with A. Zelevinsky have served as an inspiring and guiding source throughout the development of the present manuscript. We have made a conscious effort to follow [6] as much as possible, both in the structure of the paper, and in the form and proofs of the results. We thank Laurent Demonet, Claus Michael Ringel and Dylan Rupel for helpful discussions. We are particularly grateful to David Speyer for his collaboration [27] with A. Zelevinsky that showed the existence of skew-symmetrizable matrices without global unfoldings. D. Labardini-Fragoso thanks the Representation theory group of the National Autonomous University of Mexico, especially Michael Barot and Christof Geiss, for their hospitality and financial support during his several visits to UNAM’s Institute of Mathematics in Mexico City. Parts of the work presented here were completed in these visits. This project started several years ago, when D. Labardini-Fragoso was a Ph.D. student of A. Zelevinsky at Northeastern University’s Department of Mathematics (Boston, MA, USA). Unfortunately, Professor Andrei Zelevinsky passed away during the last stage of the preparation of this paper. D. Labardini-Fragoso is sincerely grateful to him for his enthusiasm and deep insights and contributions throughout the collaboration that led to the present paper. It is needless to say that any possible inaccuracies or mistakes should be attributed to D. Labardini-Fragoso alone. D. Labardini-Fragoso wishes to thank an anonymous referee for his/her thorough reading and a number of remarks and suggestions.

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Author notes

  1. Daniel Labardini-Fragoso

    Present address: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico, Mexico

Authors and Affiliations

  1. Mathematisches Institut, Universität Bonn, 53115, Bonn, Germany

    Daniel Labardini-Fragoso

  2. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Andrei Zelevinsky

Authors
  1. Daniel Labardini-Fragoso
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  2. Andrei Zelevinsky
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Correspondence to Daniel Labardini-Fragoso.

Additional information

A. Zelevinsky passed away on April 10, 2013.

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Labardini-Fragoso, D., Zelevinsky, A. Strongly primitive species with potentials I: mutations. Bol. Soc. Mat. Mex. 22, 47–115 (2016). https://doi.org/10.1007/s40590-015-0063-9

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