Optimal bounds for the Schur index and the realizability of representations
Sbornik. Mathematics, Tome 205 (2014) no. 4, pp. 522-531
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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.
@article{SM_2014_205_4_a3,
author = {D. D. Kiselev},
title = {Optimal bounds for the {Schur} index and the realizability of representations},
journal = {Sbornik. Mathematics},
pages = {522--531},
publisher = {mathdoc},
volume = {205},
number = {4},
year = {2014},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2014_205_4_a3/}
}
D. D. Kiselev. Optimal bounds for the Schur index and the realizability of representations. Sbornik. Mathematics, Tome 205 (2014) no. 4, pp. 522-531. http://geodesic.mathdoc.fr/item/SM_2014_205_4_a3/