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2D Toda τ-functions as combinatorial generating functions

Two methods of constructing 2D Toda $τ$-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group $S_n$ by multiplication of the conjugacy class sums $C_λ\in C[S_n]$ in the group algebra by elements of an abelian group of central elements. Extending the characteristic map to the tensor product $C[S_n]\otimes C[S_n]$ leads to double expansions in terms of power sum symmetric functions, in which the coefficients count the number of such paths. Applying the same map to sums over the orthogonal idempotents leads to diagonal double Schur function expansions that are identified as $τ$-functions of hypergeometric type. The second method is the standard construction of $τ$-functions as vacuum state matrix elements of products of vertex operators in a fermionic Fock space with elements of the abelian group of convolution symmetries. A homomorphism between these two group actions is derived and shown to be intertwined by the characteristic map composed with fermionization. Applications include Okounkov's generating function for double Hurwitz numbers, which count branched coverings of the Riemann sphere with nonminimal branching at two points, and various analogous combinatorial counting functions.