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Stasheff polytopes and the coordinate ring of the cluster \(\mathcal X \)-variety of type \(A_n\)

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Abstract

We define Stasheff polytopes in the spaces of tropical points of cluster \(\mathcal A \)-varieties. We study the supports of products of elements of canonical bases for cluster \(\mathcal X \)-varieties. We prove that, for the cluster \(\mathcal X \)-variety of type \(A_n\), such supports are Stasheff polytopes.

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Notes

  1. In the case of type \(A_n\), the Langlands dual \(\mathcal{A }_{{A_n}^\vee }\) coincides with \(\mathcal{A }_{A_n}.\)

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Acknowledgments

I wish to thank K. Peng, R. Raj for helpful conversations. I am especially grateful to my advisor A. Goncharov for suggesting the problem and for his enlightening suggestions and encouragement. In particular, the main definitions of this paper concerning convex polytopes in tropical positive spaces follow the idea in [6, Section 2.4]. Finally, I thank the referee for very careful reading of this paper and for many useful suggestions.

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  1. Department of Mathematics, Yale University, New Haven, CT, 06511, USA

    Linhui Shen

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  1. Linhui Shen
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Correspondence to Linhui Shen.

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Shen, L. Stasheff polytopes and the coordinate ring of the cluster \(\mathcal X \)-variety of type \(A_n\) . Sel. Math. New Ser. 20, 929–959 (2014). https://doi.org/10.1007/s00029-013-0124-8

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