Properties of some character tables related to the symmetric groups

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 Sn 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 XRC of Sn with respect to an integer r ≥ 2. 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 XRC . Taking r large yields the Smith normal form of the full character table of Sn. Analogous results are then given for spin characters. Christine Bessenrodt Institut für Mathematik Universität Hannover D-30167 Hannover, Germany [email protected] Jørn B. Olsson Matematisk Afdeling University of Copenhagen Copenhagen, Denmark [email protected] Richard P. Stanley Department of Mathematics 2-375 M.I.T. Cambridge, MA 02139, USA [email protected] Partially supported by The Danish National Research Council. Partially supported by NSF grant #DMS-9988459.