paper

Convolution symmetries of integrable hierarchies, matrix models and τ-functions

arXiv:0901.0323

Abstract

Generalized convolution symmetries of integrable hierarchies of KP and 2KP-Toda type multiply the Fourier coefficients of the elements of the Hilbert space $\HH= L^2(S^1)$ by a specified sequence of constants. This induces a corresponding transformation on the Hilbert space Grassmannian $\Gr_{\HH_+}(\HH)$ and hence on the Sato-Segal-Wilson τ-functions determining solutions to the KP and 2-Toda hierarchies. The corresponding action on the associated fermionic Fock space is also diagonal in the standard orthonormal base determined by occupation sites and labeled by partitions. The Plücker coordinates of the element element $W \in \Gr_{\HH_+}(\HH)$ defining the initial point of these commuting flows are the coefficients in the single and double Schur function of the associated τfunction, and are therefore multiplied by the corresponding diagonal factors under this action. Applying such transformations to matrix integrals, we obtain new matrix models of externally coupled type that are hence also KP or 2KP-Toda τ-functions. More general multiple integral representations of τfunctions are similarly obtained, as well as finite determinantal expressions for them.

29 pages. The paper has been revised to correct a number of typos, add details in the proofs of Propositions 4.1, 4.2 and 4.4, and clarify the introduction of convolution symmetries. This version will appear in a special MSRI volume based on the Fall 2011 semester devoted to Random Matrix Theory (eds. P. Forrester and P. Deift)

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