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arXiv · 1706.06866

A bijection between $m$-cluster-tilting objects and $(m+2)$-angulations in $m$-cluster categories

Lucie Jacquet-Malo
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Abstract

In this article, we study the geometric realizations of $m$-cluster categories of Dynkin types A, D, $\tilde{A}$ and $\tilde{D}$. We show, in those four cases, that there is a bijection between $(m+2)$-angulations and isoclasses of basic $m$-cluster tilting objects. Under these bijections, flips of $(m+2)$-angulations correspond to mutations of $m$-cluster tilting objects. Our strategy consists in showing that certain Iyama-Yoshino reductions of the $m$-cluster categories under consideration can be described in terms of cutting along an arc the corresponding geometric realizations. This allows to infer results from small cases to the general ones.

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BibTeXRIS

Lucie Jacquet-Malo. 2017-06-20. A bijection between $m$-cluster-tilting objects and $(m+2)$-angulations in $m$-cluster categories. https://arxiv.org/abs/1706.06866

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