Hypergeometric functions of complex matrices were introduced by James in the 1960s, building on earlier work of Bochner and Herz. They are defined using Schur functions, so connections with algebraic combinatorics are inevitable. A longstanding problem, which has for decades been without an organizing principle, is to find a general approach to approximating such functions as the number of variables goes to infinity. I will explain how to do this using ideas from Hurwitz theory.
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