In this paper, we give an easy description of the elementary divisors of the Cartan matrices for symmetric groups in terms of the lengths of -regular partitions and their Glaisher correspondents. Moreover, for , it is done block wise. There we use certain kinds of cores and weights, which are similar to but different from the usual ones.
Katsuhiro UNO. Hiro-Fumi YAMADA. "Elementary divisors of Cartan matrices for symmetric groups." J. Math. Soc. Japan 58 (4) 1031 - 1036, October, 2006. https://doi.org/10.2969/jmsj/1179759536