Part II Projective Representations II

Part II : Projective Representations II In Part I of Vol.1, we obtained a considerable amount of information concerning projective representations of finite groups. Here we treat the remaining topics, namely: splitting fields for twisted group algebras, projective Schur index and projective representations of abelian groups. Combined with the previous material on the subject, our coverage of projective representation theory and Schur multiplier spreads through 1300 pages. In many cases, the treatment is exhaustive and yields a complete record of the present status of the theory. The reader should gain a considerable amount of knowledge of the central ideas, the basic results, and the fundamental methods. In addition, a reservoir of folklore on the subject is easily accessible for the novice. In the spirit of our previous approach to projective representations, we emphasize modules over twisted group algebras rather than representations themselves. This approach allows us to bring into argument various deep results pertaining to the general theory of algebras over fields. In this way, projective representation theory takes a more perspicuous form and many unpleasant calculations involving cocycles can be entirely eliminated. A detailed account of the material presented below is given by the introductions to individual chapters, and therefore will not be repeated here. We only mention one fascinating topic: construction of irreducible projective representations of abelian groups over C. Although the body of knowledge concerning this topic has been slowly increasing, it may fairly be said that only its surface has been touched. In contrast to the ordinary case in which the corresponding theory is trivial, the projective case involves matrices of arbitrary size whose construction is the heart of the problem.

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We close on a philosophical note by quoting E.T. Bell : " When we begin unravelling a particular thread in the history of mathematics we soon get a discouraged feeling that mathematics itself is like a vast necropolis to which constant additions are being made for the eternal preservation of the newly dead"