Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 4 (2009), 191 -- 214

ON IRREDUCIBLE PROJECTIVE REPRESENTATIONS OF FINITE GROUPS

Tania-Luminiţa Costache

Abstract. The paper is a survey type article in which we present some results on irreducible projective representations of finite groups.
Section 2 includes Curtis and Reiner's theorem in which is proved that a finite group has at most a finite number of inequivalent irreducible projective representations in an algebraically closed field K. Theorem 15 gives an alternative proof of the main theorem of Morris, where the structure of a generalized Clifford algebra was determined. Similarly, Theorem 16 gives the structure theorem for a generalized Clifford algebra which arises in the study of the projective representations of the generalized symmetric group. Section 2 is also dedicated to the study of degrees of irreducible projective representations of a finite group G over an algebraically closed field K. In Theorem 20, H. N. NG proved a generalization of Schur's result and showed that the degree of an irreducible projective representation of a finite group G belonging to c∈ H2(G;K*), where K is an algebraically closed field such that char{K} does not divide |G|, divides the index of a class of abelian normal subgroups of G, which depends only on the 2-cohomology class c. In Theorem 27, Quinlan proved that the representations theory of generic central extensions for a finite group G yields information on the irreducible projective representations of G over various fields.
In Section 3 we give a necessary and sufficient condition for a nilpotent group G to have a class of faithful faithful irreducible projective representation.
Section 4 includes NG's result in the case of a metacyclic group G with a faithful irreducible projective representation π over an algebraically closed field with arbitrary characteristic, which proved that the degree of π is equal to the index of any cyclic normal subgroup N whose factor group G/ N is also cyclic and also a necessary and sufficient conditions for a metacyclic group to have a faithful irreducible projective representation.
In Section 5 we remind Barannyk's results concerning the conditions for a finite p-group to have a class of faithful irreducible projective representations.
Section 6 contains the most important results of the adaptation to projective representations of Clifford's theory of inducing from normal subgroups.

2000 Mathematics Subject Classification: 20C25; 19C09; 20B05; 20C30; 20D15; 20F18; 22E27; 20F16; 20D10; 20K01; 20K10.
Keywords: Multiplier; Cohomology class; Schur multiplier; Faithful irreducible projective representation; Degree of a projective representation; Finite group; Nilpotent group; Metacyclic group; p-group.

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Tania-Luminiţ a Costache
Faculty of Applied Sciences, University "Politehnica" of Bucharest
Splaiul Independentei 313, Bucharest, Romania.
e-mail: lumycos1@yahoo.com

http://www.utgjiu.ro/math/sma