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Tau functions and convolution symmetries: applications to random matrices November 23, 2010 (02:00PM PST - 03:00PM PST)
Location: MSRI: Simons Auditorium
Speaker(s) John Harnad
Description Abstract
Generalized convolution symmetries of integrable hierarchies of KP-Toda and 2KP-Toda
type have the e ect of multiplying the Fourier coecients of the elements of the Hilbert space
Grassmannian GrH+(H) de ned on H = L2(S1) by a speci ed sequence of constants. The induced
action on the associated fermionic Fock space is diagonal in the standard orthonormal
base determined by occupation sites and labeled by partitions. The coeffcients in the single
and double Schur function expansions of the  -functions associated to elements W 2 GrH+(H),
which are its Pllucker coordinates, are multiplied by the corresponding diagonal factors. Applying
such transformations to matrix integrals, we obtain new matrix models of externally coupled
type which are also KP-Toda or 2KP-Toda  -functions. More general multiple integral representations
of tau functions are similarly obtained, as well as nite determinantal expressions for
them.Flows consisting of generalized convolution symmetries are shown also to give rise to an
alternative fermionic operator representation of  -functions for the KP-chain and 2KP-Toda
hierarchies. The relation between such ows and the usual ones is explained, both in the rst
quantized and second quantized frameworks.

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