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On the minimal dimensions of irreducible representations of symmetric groups

  • G. D. James (a1)
Abstract

For each integer m, Rasala [6] has shown how to list all the ordinary irreducible representations of the symmetric group n which have degree less than nm, provided that n is large enough, and in this note we shall prove similar results for the irreducible representations of n over an arbitrary field K. Our estimates are very crude, so although we recover Rasala's results, we get nowhere near his precise information on how large n has to be.

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[1] James G. D.. The Representation Theory of the Symmetric Groups. Lecture Notes in Math. vol. 682 (Springer-Verlag, 1978).
[2] James G. D.. Representations of the symmetric groups over the field of order 2. J. Algebra 38 (1976), 280308.
(3)James G. D.. On the decomposition matrices of the symmetric groups II. J. Algebra 43 (1977), 4554.
[4] James G. D.. On the decomposition matrices of the symmetric groups III. J. Algebra 71 (1981), 115122.
[5] MacAogain E.. Decomposition Matrices of Symmetric and Alternating Groups. Trinity College, Dublin, Research Notes TCD 1976–10.
[6] Rasala R.. On the minimal degrees of characters of S n. J. Algebra 45 (1977), 132181.
[7] Wagner A.. The faithful linear representations of least degree of S n and A n over a field of characteristic 2. Math. Zeit. 151 (1976), 127137.
[8] Wagner A.. The faithful linear representations of least degree of S n and A n over a field of odd characteristic. Math. Zeit. 154 (1977), 103114.
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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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