$W_{1+\mathrm{\infty }}$ flows and multi-component hierarchy (KP case)

We show that abelian subalgebras of generalized $W_{1+\mathrm{\infty }}$ ($G{W}_{1+\mathrm{\infty }}$) algebra gives rise to the multicomponent KP flows. The matrix elements of the related group elements in the fermionic Fock space is expressed as a product of a certain factor (generalized content product) and of a number of the Schur functions and the skew Schur functions.