An identity is derived expressing Schur functions as sums over products of pairs of Schur $Q$ -functions, generalizing previously known special cases. This is shown to follow from their representations as vacuum expectation values (VEV’s) of products of either charged or neutral fermionic creation and annihilation operators, Wick’s theorem and a factorization identity for VEV’s of products of two mutually anticommuting sets of neutral fermionic operators.

A multimatrix model can be built on two ingredients: the choice of an embedded graph and the choice of the integrand, that is, a tau function. We compare the simplest nontrivial tau function of the Toda lattice and the simplest nontrivial tau function of the \(\mathcal N\)-component Toda lattice in the context of their application to multimatrix integrals.

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.