Surveys in Mathematics and its Applications (Dec 2009)
On irreducible projective representations of finite groups
Abstract
The paper is a survey type article inwhich we present some results on irreducible projective representations offinite groups. Section 2 includes Curtis and Reiner's theorem inwhich is proved that a finite group has at most a finite number ofinequivalent irreducible projective representations in an algebraicallyclosed field K. Theorem 15 gives an alternative proofof the main theorem of Morris, where the structure of ageneralized Clifford algebra was determined. Similarly, Theorem 16 gives the structure theorem for a generalized Clifford algebrawhich arises in the study of the projective representations of thegeneralized symmetric group. Section 2 is also dedicated to the study ofdegrees of irreducible projective representations of a finite group G overan algebraically closed field K. In Theorem 20, H. N. NG proved ageneralization of Schur's result and showed that the degree of anirreducible 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 abeliannormal subgroups of G, which depends only on the 2-cohomology class c.In Theorem 27, Quinlan proved that the representationstheory of generic central extensions for a finite group G yieldsinformation on the irreducible projective representations of G overvarious fields. In Section 3 we give a necessary and sufficient condition fora nilpotent group G to have a class of faithful faithful irreducibleprojective representation. Section 4 includes NG's result in the case of a metacyclicgroup G with a faithful irreducible projective representation π overan algebraically closed field with arbitrary characteristic, which provedthat the degree of π is equal to the index of any cyclic normal subgroupN whose factor group G/ N is also cyclic and also a necessary andsufficient conditions for a metacyclic group to have a faithful irreducibleprojective representation. In Section 5 we remind Barannyk's results concerning the conditions for a finite p-group to have aclass of faithful irreducible projective representations. Section 6 contains the most important results of theadaptation to projective representations of Clifford's theory of inducingfrom normal subgroups.
