2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/80163A family of polynomial τ-functions for the NLS-Toda hierarchy is constructed. The hierarchy is associated with the homogeneous vertex operator representation of the affine algebra \g of type A_1^{(1)}. These τ-functions are given explicitly in terms of Schur functions that correspond to rectangular Young diagrams. It is shown that an arbitrary polynomial τ-function which is an eigenvector of d, the degree operator of \g, is contained in the family. By the construction, any τ-function in the family becomes a Virasoro singular vector. This consideration gives rise to a simple proof of known results on the Fock representation of the Virasoro algebra with c=1.10 pages, no figureExactly Solvable and Integrable SystemsHigh Energy Physics - TheoryMathematical PhysicsRepresentation Theorytext