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<p>The study of modular representation theory of the double covering groups of the symmetric and alternating groups reveals rich and subtle combinatorial and algebraic phenomena involving their irreducible characters and the structure of their <i>p</i>-blocks, where <i>p</i> is an odd prime number. In this paper, we investigate the action of certain Galois automorphisms, those that act on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p'\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>-roots of unity by a power of <i>p</i>, on spin characters, with an emphasis on their interaction with perfect isometries and block theory. In particular, we prove that perfect isometries constructed by the first author and J.&#xa0;B. Gramain in [<CitationRef CitationID="CR2">2</CitationRef>], which were used to establish a weaker form of the Kessar–Schaps conjecture, remain preserved under this Galois action whenever certain natural compatibility conditions occur.</p>

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Galois automorphisms and littlewood decompositions

  • 期刊论文

  • 发表日期：

    2026年7月10日

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  • Olivier Brunat,
  • Rishi Nath

摘要

The study of modular representation theory of the double covering groups of the symmetric and alternating groups reveals rich and subtle combinatorial and algebraic phenomena involving their irreducible characters and the structure of their p-blocks, where p is an odd prime number. In this paper, we investigate the action of certain Galois automorphisms, those that act on \(p'\) p ′ -roots of unity by a power of p, on spin characters, with an emphasis on their interaction with perfect isometries and block theory. In particular, we prove that perfect isometries constructed by the first author and J. B. Gramain in [2], which were used to establish a weaker form of the Kessar–Schaps conjecture, remain preserved under this Galois action whenever certain natural compatibility conditions occur.