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Rigid and Schurian modules over cluster-tilted algebras of tame type

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Abstract

We give an example of a cluster-tilted algebra \(\Lambda \) with quiver Q, such that the associated cluster algebra \(\mathcal {A}(Q)\) has a denominator vector which is not the dimension vector of any indecomposable \(\Lambda \)-module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra \(\Lambda \), we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid \(\Lambda \)-modules. In order to do this it is necessary, and of independent interest, to first classify the indecomposable rigid \(\Lambda \)-modules in this case.

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Acknowledgments

Both authors would like to thank the referee for very helpful comments and would like to thank the MSRI, Berkeley for kind hospitality during a semester on cluster algebras in Autumn 2012. BRM was Guest Professor at the Department of Mathematical Sciences, NTNU, Trondheim, Norway, during the autumn semester of 2014 and would like to thank the Department for their kind hospitality.

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Authors and Affiliations

  1. School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Bethany R. Marsh

  2. Department of Mathematical Sciences, NTNU, 7491, Trondheim, Norway

    Bethany R. Marsh & Idun Reiten

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  1. Bethany R. Marsh
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  2. Idun Reiten
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Correspondence to Bethany R. Marsh.

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This work was supported by the Engineering and Physical Sciences Research Council (Grant Number EP/G007497/1), the Mathematical Sciences Research Institute, Berkeley and NFR FriNat (Grant Number 231000).

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Marsh, B.R., Reiten, I. Rigid and Schurian modules over cluster-tilted algebras of tame type. Math. Z. 284, 643–682 (2016). https://doi.org/10.1007/s00209-016-1668-z

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