Exponents Associated with -Systems and their Relationship with -Series
arXiv:1812.05863 · doi:10.3842/SIGMA.2020.028
Abstract
Let be a finite type Dynkin diagram, and be a positive integer greater than or equal to two. The -system of type with level is a system of algebraic relations, whose solutions have been proved to have periodicity. For any pair , we define an integer sequence called exponents using formulation of the -system by cluster algebras. We give a conjectural formula expressing the exponents by the root system of type , and prove this conjecture for and cases. We point out that a specialization of this conjecture gives a relationship between the exponents and the asymptotic dimension of an integrable highest weight module of an affine Lie algebra. We also give a point of view from -series identities for this relationship.
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