Geodesic
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. 
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/
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[1] D. D. Kiselev, “A bound for the Schur index of irreducible representations of finite groups”, Sb. Math., 204:8 (2013), 1152–1160 | DOI | DOI | MR | Zbl

[2] P. Schmid, “Schur indices and Schur groups”, J. Algebra, 169:1 (1994), 226–247 | DOI | MR | Zbl

[3] B. I. Fein, “Minimal splitting fields for group representations”, Pacific J. Math., 51:2 (1974), 427–431 | DOI | MR | Zbl

[4] R. A. Mollin, “Splitting fields and group characters”, J. Reine Angew. Math., 315 (1980), 107–114 | DOI | MR | Zbl

[5] E. Spiegel, A. Trojan, “Minimal splitting fields in cyclotomic extensions”, Proc. Amer. Math. Soc., 87:1 (1983), 33–37 | DOI | MR | Zbl

[6] I. Reiner, Maximal orders, London Math. Soc. Monogr. (N.S.), 28, Oxford Univ. Press, Oxford, 2003, xiv+395 pp. | MR | Zbl

[7] T. Yamada, The Schur subgroup of the Brauer group, Lecture Notes in Math., 397, Springer-Verlag, Berlin–New York, 1974, v+159 pp. | MR | Zbl

[8] M. Benard, M. M. Schacher, “The Schur subgroup. II”, J. Algebra, 22:2 (1972), 378–385 | DOI | MR | Zbl

[9] V. V. Ishkhanov, B. B. Lur'e, D. K. Faddeev, The embedding problem in Galois theory, Transl. Math. Monogr., 165, Amer. Math. Soc., Providence, RI, 1997, xii+182 pp. | MR | MR | Zbl | Zbl

[10] V. V. Ishkhanov, “The embedding problem with given localizations”, Math. USSR-Izv., 9:3 (1975), 481–492 | DOI | MR | Zbl