  Let S be the direct sum of algebra of symmetric groups C S_n for a
non-negative integer n. We show that the Grothendieck group K_0(S) of the
category of finite dimensional modules of S is isomorphic to the differential
algebra of polynomials Z[D^n x]. Moreover, for a non-negative integer m, we
define m-th products on K_0(S) which make the algebra K_0(S) isomorphic to an
integral form of the Virasoro-Magri Poisson vertex algebra. Also, we
investigate relations between K_0(S) and K_0(N) where K_0(N) is the direct sum
of Grothendieck groups K_0(N_n) of finitely generated projective N_n-modules.
Here N_n is the nil-Coxeter algebra generated by n-1 elements.
