An instance of umbral methods in representation theory: the parking function module.

We test the umbral methods introduced by Rota and Taylor within the theory of representation of the symmetric group. We prove that the volume polynomial of Pitman and Stanley represents the Frobenius characteristic of the Haiman parking function module, when the set of its variables consists of suitable umbrae. We also show that the volume polynomial in any set of similar and uncorrelated umbrae is umbrally equivalent, up to a constant term, to an Abel-like umbral polynomial. An analogous treatment of the parking function module of type B is given. Keywords: parking functions, noncrossing partitions, volume polynomial, umbral calculus, Abel polynomials. AMS subject