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arXiv · 1309.0751

Laurent Phenomenon Sequences

Joshua Alman·Cesar Cuenca·Jiaoyang Huang
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Abstract

In this paper, we undertake a systematic study of recurrences x_{m+n}x_{m} = P(x_{m+1}, ..., x_{m+n-1}) which exhibit the Laurent phenomenon. Some of the most famous among these sequences come from the Somos and the Gale-Robinson recurrences. Our approach is based on finding period 1 seeds of Laurent phenomenon algebras of Lam-Pylyavskyy. We completely classify polynomials P that generate period 1 seeds in the cases of n=2,3 and of mutual binomial seeds. We also find several other interesting families of polynomials P whose generated sequences exhibit the Laurent phenomenon. Our classification for binomial seeds is a direct generalization of a result by Fordy and Marsh, that employs a new combinatorial gadget we call a double quiver.

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Joshua Alman, Cesar Cuenca, Jiaoyang Huang. 2013-10-04. Laurent Phenomenon Sequences. https://arxiv.org/abs/1309.0751

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