Soliton Theory, Symmetric Functions and Matrix Integrals

We consider a certain scalar product of symmetric functions which is parameterized by a function r and an integer n. On the one hand we have a fermionic representation of this scalar product. On the other hand we get a representation of this product with the help of multi-integrals. This gives links between a theory of symmetric functions, soliton theory and models of random matrices (such as a model of normal matrices).