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Some combinatorial results involving Young diagrams

  • G. D. James (a1)

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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(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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