Laurent phenomenon sequences
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Author(s)
Alman, Joshua
Cuenca, Cesar
Huang, Jiaoyang
Cuenca, Cesar A.
Date Issued
November 2015
Journal
Journal of Algebraic Combinatorics
Publisher
Springer US
Citation
Alman, Joshua, Cesar Cuenca, and Jiaoyang Huang. “Laurent Phenomenon Sequences.” Journal of Algebraic Combinatorics 43, no. 3 (November 5, 2015): 589–633.
Version
Author's final manuscript
Abstract
In this paper, we undertake a systematic study of sequences generated by recurrences x[subscript m+n]x[subscript m]=P(x[subscript m+1],…,x[subscript m+n−1])xm+nxm=P(xm+1,…,xm+n−1) which exhibit the Laurent phenomenon. Some of the most famous among these are the Somos and the Gale-Robinson sequences. 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s10801-015-0647-5
