Friezes |
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Authors: | Ibrahim Assem David Smith |
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Institution: | a Département de Mathématiques, Université de Sherbrooke, 2500 boulevard de l'Université, Sherbrooke, Canada J1K 2R1 b Département de Mathématiques, Université du Québec à Montréal, CP 8888 succ. Centre-Ville, Montréal, Canada H3C 3P8 |
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Abstract: | The construction of friezes is motivated by the theory of cluster algebras. It gives, for each acyclic quiver, a family of integer sequences, one for each vertex. We conjecture that these sequences satisfy linear recursions if and only if the underlying graph is a Dynkin or an Euclidean (affine) graph. We prove the “only if” part, and show that the “if” part holds true for all non-exceptional Euclidean graphs, leaving a finite number of finite number of cases to be checked. Coming back to cluster algebras, the methods involved allow us to give formulas for the cluster variables in case Am and ; the novelty is that these formulas use 2 by 2 matrices over the semiring of Laurent polynomials generated by the initial variables (which explains simultaneously positivity and the Laurent phenomenon). One tool involved consists of the SL2-tilings of the plane, which are particular cases of T-systems of Mathematical Physics. |
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Keywords: | Rationality Periodicity Dynkin diagrams Euclidean diagrams Cluster algebras Positivity Laurent phenomenon _method=retrieve& _eid=1-s2 0-S0001870810002136& _mathId=si4 gif& _pii=S0001870810002136& _issn=00018708& _acct=C000069490& _version=1& _userid=6211566& md5=ece649786b32416ea804bc8849bb3c57')" style="cursor:pointer SL2 tilings" target="_blank">" alt="Click to view the MathML source" title="Click to view the MathML source">SL2 tilings Frieze patterns |
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