NDLI: 2D Toda τ-Functions as Combinatorial Generating Functions

Content Provider	Springer Nature Link
Author	Guay Paquet, Mathieu Harnad, J.
Copyright Year	2015
Abstract	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 $${C_\lambda \in \mathbf{C}[S_n]}$$ in the group algebra by elements of an abelian group of central elements. Extending the characteristic map to the tensor product $${\mathbf{C}[S_n] \otimes \mathbf{C}[S_n]}$$ 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 specified ramification profiles at two branch points, and only simple branching at all the others, and various analogous combinatorial counting functions.
Starting Page	827
Ending Page	852
Page Count	26
File Format	PDF
ISSN	03779017
Journal	Letters in Mathematical Physics
Volume Number	105
Issue Number	6
e-ISSN	15730530
Language	English
Publisher	Springer Netherlands
Publisher Date	2015-05-12
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Hurwitz numbers Tau functions combinatorics branched coverings Cayley graph Theoretical, Mathematical and Computational Physics Statistical Physics, Dynamical Systems and Complexity Geometry Group Theory and Generalizations
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Mathematical Physics

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