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Some combinatorial results involving Young diagrams

  • G. D. James (a1)
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In the first half of this paper we introduce a new method of examining the q-hook structure of a Young diagram, and use it to prove most of the standard results about q-cores and q-quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given q-weight and says how many of them are q-regular. In the case where q is prime, this tells us how many ordinary and q-modular irreducible representations of the symmetric group there are in a given q-block. None of the results of section 2 is original. In the next section we give a new definition, the p-power diagram, which is closely connected with the p-quotient. This concept is interesting because when p is prime a condition involving the p-power diagram appears to be a necessary and sufficient criterion for the diagram to be p-regular and the corresponding ordinary irreducible representation of to remain irreducible modulo p. In this paper we derive combinatorial results involving the p-power diagram, and in a later article we investigate the relevant representation theory.

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(1) Brauer, R. On a conjecture by Nakayama. Trans. Roy. Soc. Canada III 41 (1947), 1119.
(2) Chung, J. H. On the modular representations of the symmetric groups. Canad. J. Math. 3 (1951), 309327.
(3) Farahat, H. K. On p-quotients and star diagrams of the symmetric group. Proc. Cambridge Philos. Soc. 49 (1953), 157160.
(4) Frame, J. S. and Robinson, G. de B. On a theorem of Osima and Nagao. Canad. J. Math. 6 (1954), 125127.
(5) James, G. D. The irreducible representations of the symmetric groups. Bull. London Math. Soc. 8 (1976), 4244.
(6) Littlewood, D. E. Modular representations of symmetric groups. Proc. Roy. Soc. A 209 (1951), 333352.
(7) Meier, N. and Tappe, J. Ein neuer Beweis der Nakayama-Vermutung über die Block struktur Symmetrischer Gruppen. Bull. London Math. Soc. 8 (1976), 3437.
(8) Nagao, H. Note on the modular representations of the symmetric groups. Caned. J. Math. 5 (1953), 356363.
(9) Osima, M. Note on a paper by J. S. Frame and G. de B. Robinson. Okayama Math. J. 6 (1956), 7779.
(10) Robinson, G. De B. On a conjecture by Nakayama. Trans. Roy. Soc. Canada III 41 (1947), 2025.
(11) Robinson, G. De B. On a conjecture by J. H. Chung. Caned. J. Math. 4 (1952), 373380.
(12) Robinson, G. De B. Representation theory of the symmetric group (Toronto: University of Toronto Press, 1961).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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