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Mat. Sb., 2014, Volume 205, Number 4, Pages 69–78 (Mi msb8259)  

Optimal bounds for the Schur index and the realizability of representations

D. D. Kiselev

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: An optimal bound is given for the Schur index of an irreducible complex representation over the field of rational numbers on the class of finite groups of a chosen order or of a chosen exponent. We obtain a sufficient condition for the realizability of an irreducible complex character $\chi$ of a finite group $G$ of exponent $n$ with Schur index $m$, which is either an odd number or has $2$-part no smaller than $4$, over the field of rational numbers in a field $L$ which is a subfield of $\mathbb{Q}(\sqrt[n]{1} )$ and $(L:\mathbb{Q}(\chi))=m$. This condition generalizes the well-known Fein condition obtained by him in the case of $n=p^{\alpha}q^{\beta}$. The formulation of the Grunwald-Wang problem on the realizability of representations is generalized, and some sufficient conditions are obtained.
Bibliography: 10 titles.

Keywords: finite group, Schur index, realizability of a representation.

DOI: https://doi.org/10.4213/sm8259

Full text: PDF file (513 kB)
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English version:
Sbornik: Mathematics, 2014, 205:4, 522–531

Bibliographic databases:

UDC: 512.547.2+512.623.32
MSC: 20C15
Received: 12.06.2013

Citation: D. D. Kiselev, “Optimal bounds for the Schur index and the realizability of representations”, Mat. Sb., 205:4 (2014), 69–78; Sb. Math., 205:4 (2014), 522–531

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