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On the Irreducible Representationsof the Symmetric Group

Published online by Cambridge University Press:  20 November 2018

Masaru Osima
Masaru Osima*
Affiliation:
Okayama University, Japan
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Let T be a Young diagram of n nodes:

(1)

ai being the length of its ith row. With respect to a prime p, we denote by T0 the p-core of T. If T0 consists of m nodes, then

(2) m=n-lp,

where l is the number of successive p-hooks [3] removable from T to yield its p-core T0. We have stated in [4] the following theorem :

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952 , pp. 381 - 384
Copyright
Copyright © Canadian Mathematical Society 1952

References

1. Brauer, R. and Robinson, G. de B., On a conjecture by Nakayama, Trans. Royal Soc. Canada, Sec III, vol. 40 (1947), 11–25.Google Scholar
2. Murnaghan, F. D., The theory of group representations (Baltimore, 1938).Google Scholar
3. Nakayama, T.,On some properties of irreducible representations of a symmetric group I, Jap. J. Math., vol. 17 (1941), 89–108.Google Scholar
4. Nakayama, T. and Osima, M., Note on blocks of symmetric groups, Nagoya (1951), 111–117.Google Scholar
5. Robinson, G. de B., On the representations of the symmetric group III, Amer. J. vol. 70 (1948), 277–294.Google Scholar