Positivity in coefficient-free rank two cluster algebras

2009/03/15 by Grégoire Dupont, G. Dupont, Dupont, G.
Mathematics · #05E99 #16G20 #16S99 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:05E99 #msc:16G20 #msc:16S99

paper · pdf · doi:10.48550/arxiv.0903.2677

9 pages

arxiv created 2009/03/15 · openalex publication_date 2009/03/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let b,c be positive integers, x₁,x₂ be indeterminates over \Z and x_m, m ∈ \mathbb Z be rational functions defined by x_m-1x_m+1=x_m^b+1 if m is odd and x_m-1x_m+1=x_m^c+1 if m is even. In this short note, we prove that for any m,k ∈ \Z, x_k can be expressed as a substraction-free Laurent polynomial in \Z[x_m^{± 1},x_m+1^{± 1}]. This proves Fomin-Zelevinsky's positivity conjecture for coefficient-free rank two cluster algebras.

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