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<p>We prove divisibility results concerning the number of inequivalent irreducible complex representations of certain finite groups whose degrees are not divisible by a prime <i>p</i>. In particular, we study the symmetric group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(S_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, wreath product groups of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(G \wr S_n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>≀</mo> <msub> <mi>S</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for a finite group <i>G</i>, and index 2 subgroups of these groups. Our proofs use the combinatorics of rim hook tableaux, <i>p</i>-cores and <i>p</i>-quotients of integer partitions.</p>

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Irreducible Representations with Nonzero Degree mod p

  • 期刊论文

  • 发表日期：

    2026年3月30日

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  • Vanessa Beeler,
  • Jeffrey Liese,
  • Anthony Mendes

摘要

We prove divisibility results concerning the number of inequivalent irreducible complex representations of certain finite groups whose degrees are not divisible by a prime p. In particular, we study the symmetric group \(S_n\) S n , wreath product groups of the form \(G \wr S_n\) G ≀ S n for a finite group G, and index 2 subgroups of these groups. Our proofs use the combinatorics of rim hook tableaux, p-cores and p-quotients of integer partitions.