# Convolution symmetries of integrable hierarchies, matrix models and \tau-functions

J. Harnad and A. Yu. Orlov

Arxiv ID: 0901.0323•Last updated: 11/30/2021

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 \tau-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
Pl\"ucker 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 \tau function, 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 \tau-functions. More
general multiple integral representations of \tau functions are similarly
obtained, as well as finite determinantal expressions for them.

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