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We explore the Fock spaces of the parafermionic algebra introduced by H.S.Green. Each parafermionic Fock space allows for a free minimal resolution bygraded modules of the graded 2-step nilpotent subalgebra of the parafermioniccreation operators. Such a free resolution is constructed with the help of aclassical Kostant's theorem computing Lie algebra cohomologies of the nilpotentsubalgebra with values in the parafermionic Fock space. The Euler-Poincar\'echaracteristics of the parafermionic Fock space free resolution yields someinteresting identities between Schur polynomials. Finally we briefly comment onparabosonic and general parastatistics Fock spaces.

Parafermionic algebras, their modules and cohomologies