  We determine invariants like the Smith normal form and the determinant for
certain integral matrices which arise from the character tables of the
symmetric groups S_n and their double covers. In particular, we give a simple
computation, based on the theory of Hall-Littlewood symmetric functions, of the
determinant of the regular character table of S_n with respect to an integer
r>1. This result had earlier been proved by Olsson in a longer and more
indirect manner. As a consequence, we obtain a new proof of the Mathas'
Conjecture on the determinant of the Cartan matrix of the Iwahori-Hecke
algebra. When r is prime we determine the Smith normal form of the regular
character table. Taking r large yields the Smith normal form of the full
character table of S_n. Analogous results are then given for spin characters.
