Invariant Functions On Cluster Ensembles
Dani Kaufman
TL;DR
This paper introduces invariant functions on cluster ensembles under modular group actions, linking them to geometric and number theoretic concepts, and classifies these invariants for affine Dynkin diagram cases.
Contribution
It defines invariant functions on cluster ensembles, unifies previous examples, and classifies invariants for affine Dynkin diagram related ensembles.
Findings
Many known functions are realized as invariants.
Invariants have geometric and number theoretic interpretations.
Complete classification for affine Dynkin diagram ensembles.
Abstract
We define the notion of an invariant function on a cluster ensemble with respect to an action of the cluster modular group on its associated function fields. We realize many examples of previously studied functions as elements of this type of invariant ring and give many new examples. We show that these invariants have geometric and number theoretic interpretations, and classify them for ensembles associated to affine Dynkin diagrams.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic structures and combinatorial models · Advanced Combinatorial Mathematics