Partition functions and a generalized coloring-flow duality for embedded graphs

Let G be a finite group and X : G → ℂ a class function. Let H = (V, E) be a directed graph with for each vertex a cyclic order of the edges incident to it. The cyclic orders give a collection F of faces of H. Define the partition function (Formula presented.), where K(S(v)) denotes the product of the κ-values of the edges incident with v (in cyclic order), where the inverse is taken for edges leaving v. Write X = ∑λ mλ Xλ, where the sum runs over irreducible representations λ of G with character Xλ and with mλ ∈ C for every λ. When H is connected, it is proved that (Formula presented.), where 1 is the identity element of G. Among the corollaries, a formula for the number of nowhere-identity G-flows on H is derived, generalizing a result of Tutte. We show that these flows correspond bijectively to certain proper G-colorings of a covering graph of the dual graph of H. This correspondence generalizes coloring-flow duality for planar graphs.