Properties of some character tables related to the symmetric groups

Christine Bessenrodt, Jorn B. Olsson, Richard P. Stanley

Published 2004-03-05Version 1

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.