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Spin networks, Ehrhart quasipolynomials, and combinatorics of dormant indigenous bundles. (English) Zbl 1440.14147
Summary: It follows from work of S. Mochizuki [Foundations of \(p\)-adic Teichmüller theory. Providence, RI: American Mathematical Society (1999; Zbl 0969.14013)], F. Liu and B. Osserman [J. Algebr. Comb. 23, No. 2, 125–136 (2006; Zbl 1090.14009)] that there is a relationship between Ehrhart’s theory concerning rational polytopes and the geometry of the moduli stack classifying dormant indigenous bundles on a proper hyperbolic curve in positive characteristic. This relationship was established by considering the (finite) cardinality of the set consisting of certain colorings on a 3-regular graph called spin networks. In the present article, we recall the correspondences between spin networks, lattice points of rational polytopes, and dormant indigenous bundles and present some identities and explicit computations of invariants associated with the objects involved.

MSC:
14H10 Families, moduli of curves (algebraic)
05E05 Symmetric functions and generalizations
14G17 Positive characteristic ground fields in algebraic geometry
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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