Optimal bounds for the Schur index and the realizability of representations
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 χ of a finite group G of exponent n with Schur index m, which is either an odd number or has 2part no smaller than 4, over the field of rational numbers in a field L which is a subfield of Q({sup n}√1 ) and (L:Q(χ))=m. This condition generalizes the wellknown Fein condition obtained by him in the case of n=p{sup α}q{sup β}. The formulation of the GrunwaldWang problem on the realizability of representations is generalized, and some sufficient conditions are obtained. Bibliography: 10 titles. (paper)
 Authors:
 Faculty of Mechanics and Mathematics, Moscow State University (Russian Federation)
 Publication Date:
 OSTI Identifier:
 22365291
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Sbornik. Mathematics; Journal Volume: 205; Journal Issue: 4; Other Information: Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICAL METHODS AND COMPUTING; COMPLEXES; GROUP THEORY; INDEXES; IRREDUCIBLE REPRESENTATIONS; MATHEMATICAL SOLUTIONS; SYMMETRY GROUPS
Citation Formats
Kiselev, D. D., Email: denmexmath@yandex.ru. Optimal bounds for the Schur index and the realizability of representations. United States: N. p., 2014.
Web. doi:10.1070/SM2014V205N04ABEH004386.
Kiselev, D. D., Email: denmexmath@yandex.ru. Optimal bounds for the Schur index and the realizability of representations. United States. doi:10.1070/SM2014V205N04ABEH004386.
Kiselev, D. D., Email: denmexmath@yandex.ru. Wed .
"Optimal bounds for the Schur index and the realizability of representations". United States.
doi:10.1070/SM2014V205N04ABEH004386.
@article{osti_22365291,
title = {Optimal bounds for the Schur index and the realizability of representations},
author = {Kiselev, D. D., Email: denmexmath@yandex.ru},
abstractNote = {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 χ of a finite group G of exponent n with Schur index m, which is either an odd number or has 2part no smaller than 4, over the field of rational numbers in a field L which is a subfield of Q({sup n}√1 ) and (L:Q(χ))=m. This condition generalizes the wellknown Fein condition obtained by him in the case of n=p{sup α}q{sup β}. The formulation of the GrunwaldWang problem on the realizability of representations is generalized, and some sufficient conditions are obtained. Bibliography: 10 titles. (paper)},
doi = {10.1070/SM2014V205N04ABEH004386},
journal = {Sbornik. Mathematics},
number = 4,
volume = 205,
place = {United States},
year = {Wed Apr 30 00:00:00 EDT 2014},
month = {Wed Apr 30 00:00:00 EDT 2014}
}

We construct an optimal bound for the Schur index of irreducible complex representations of finite groups over the field of rational numbers, when only the prime divisors of the order of the group are known. We study relationships with compatible and universally compatible extensions of number fields. We give a simpler proof of the wellknown BermanYamada bound for the Schur index over the field Q{sub p}. Bibliography: 7 titles.

Communication Lower Bounds and Optimal Algorithms for Numerical Linear Algebra.
Abstract not provided. 
Optimal bounds for Reggepole parameters from low energy data
A method for determining highenergy parameters from lowenergy data is described and applied to the pi N forward chargeexchange amplitude A'. Unlike the usual FESR or CMSR which are based upon the Cauchy integral formula (dispersion relations) the present method uses the solution of an extremal problem in the space of analytic functions. This allows optimal bounds on Regge residues and correlations between Regge parameters and lowenergy data to be derived in a completely independent way. (auth)