Conformal Galilei algebras,symmetric polynomials and singular vectors

We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras ({mathfrak {cga}}_ell (d,mathbb {C})) with (d=1) for any integer value (ell in {mathbb {N}}). The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type and identified with specific polynomials arising as coefficients in the expansion of a parametric family of symmetric polynomials into power sum symmetric polynomials.