$$\underline{p}$$-reduced Multicomponent KP Hierarchy and Classical $$\mathcal {W}$$-algebras $$\mathcal {W}(\mathfrak {gl}_N,\underline{p})$$

Abstract

For each partition p of an integer N \geq 2, consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the p-reduced r-component KP hierarchy produces a solution of this integrable hierarchy. Along the way we provide an algorithm for the explicit construction of the generators of the corresponding classical W-algebra W(gl_N,p), and write down explicit formulas for evolution of these generators along the Hamiltonian flows.