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On maximal green sequences for type \(\mathbb {A}\) quivers

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Abstract

Given a framed quiver, i.e., one with a frozen vertex associated with each mutable vertex, there is a concept of green mutation, as introduced by Keller. Maximal sequences of such mutations, known as maximal green sequences, are important in representation theory and physics as they have numerous applications, including the computations of spectrums of BPS states, Donaldson–Thomas invariants, tilting of hearts in derived categories, and quantum dilogarithm identities. In this paper, we study such sequences and construct a maximal green sequence for every quiver mutation equivalent to an orientation of a type \(\mathbb {A}\) Dynkin diagram.

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Notes

  1. In the sequel, we will identify an admissible sequence with the mutation sequence it defines.

  2. Note that since we identify an admissible sequence with the mutation sequence defined by it, we have that maximal green sequences are identified with maximal green mutation sequences, as they are referred to in [19].

  3. By identifying maximal green sequences with maximal green mutation sequences, we abuse notation and write an element of \(\text {green}(Q)\) either as an admissible sequence \(\mathbf i = (i_1,\ldots , i_d)\) or as its corresponding mutation sequence \({{\underline{\mu }} = \mu _{i_d} \circ \cdots \circ \mu _{i_1}}\).

  4. If \(d = 1\), then \(\widetilde{\text {tr}}(x(d)) = x^\prime _{m_1} = x_k\).

  5. If \(d = 1\), then \(\widetilde{\text {tr}}(x(d)) = x^\prime _{m_1} = x_k\). Furthermore, \(d=1\), in this case, if and only if \(k = 1\).

  6. Note that this can only happen if there exists \(j < k\) such that \(z_j=x_t\) and \(x(d,k)=y_j\) as in Definition 6.1.

  7. We define \(\widehat{\mathcal {Q}_{k,k}}|_{(\widehat{\mathcal {Q}_{k,k}})_0\setminus (\overline{R}_k)_0}\) (resp. ) to be the ice quiver that is a full subquiver of \(\widehat{\mathcal {Q}_{k,k}}\) (resp. ) on the vertices of \({(\widehat{\mathcal {Q}_{k,k}})_0\setminus (\overline{R}_k)_0}\).

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Acknowledgments

The authors would like to thank T. Brüstle, M. Del Zotto, B. Keller, S. Ladkani, R. Patrias, V. Reiner, and H. Thomas for useful discussions. We also thank the referees for their careful reading and numerous suggestions. The authors were supported by NSF Grants DMS-1067183, DMS-1148634, and DMS-1362980.

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  1. University of Minnesota, Minneapolis, MN, USA

    Alexander Garver & Gregg Musiker

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  1. Alexander Garver
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  2. Gregg Musiker
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Correspondence to Gregg Musiker.

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Garver, A., Musiker, G. On maximal green sequences for type \(\mathbb {A}\) quivers. J Algebr Comb 45, 553–599 (2017). https://doi.org/10.1007/s10801-016-0716-4

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