Abstract
Let χ be an irreducible character of a finite groupG. Letp=∞ or a prime. Letm p (χ) denote the Schur index of χ overQ p , the completion ofQ atp. It is shown that ifx is ap′-element ofG such that\(X_u \left( x \right) \in Q_p \left( X \right)\) for all irreducible charactersX u ofG thenm p (χ)/vbχ(x). This result provides an effective tool in computing Schur indices of characters ofG from a knowledge of the character table ofG. For instance, one can read off Benard’s Theorem which states that every irreducible character of the Weyl groupsW(E n), n=6,7,8 is afforded by a rational representation. Several other applications are given including a complete list of all local Schur indices of all irreducible characters of all sporadic simple groups and their covering groups (there is still an open question concerning one character of the double cover of Suz).
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This work was partly supported by NSF Grant MCS-8201333.
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Feit, W. The computations of some Schur indices. Israel J. Math. 46, 274–300 (1983). https://doi.org/10.1007/BF02762888
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DOI: https://doi.org/10.1007/BF02762888