2013/09/03 by Joshua Alman, Alman, Joshua, Cesar Cuenca +3
Mathematics · #05E10 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT #msc:05E10
paper · pdf · doi:10.48550/arxiv.1309.0751
38 pages
arxiv created 2013/10/04 · arxiv updated 2013/10/08
In this paper, we undertake a systematic study of recurrences xm+nxm = P(xm+1, ..., xm+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.