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Projective representations of the symmetric groups. $$Q$$-functions and shifted tableaux. (English) Zbl 0777.20005
Oxford Mathematical Monographs. Oxford: Clarendon Press. xiii, 304 p. (1992).
More than eighty years ago I. Schur discovered non-trivial central $$\mathbb{Z}_ 2$$-extensions of the symmetric group $$S_ n$$, $$n>3$$ [J. Reine Angew. Math. 139, 155-250 (1911; JFM 42.0154.02)]. One of these extensions is the group with generators $$z,t_ 1,\dots,t_{n-1}$$ and relations $$z^ 2=1$$; $$zt_ i=t_ iz$$; $$t_ i^ 2=z$$; $$(t_ it_{i+1})^ 3=z$$; $$(t_ i t_ j)^ 2=z$$, $$i-j>1$$. That is, the group $$S_ n$$ possesses projective representations which cannot be reduced to linear ones.
The group $$S_ n$$ being an object of unceasing interest to many scholars, surprisingly not so much attention was paid to its projective representations until the recent time. However, last years the interest in these representations increased and the monograph under review is the most complete treatise on the subject now available.
The above cited paper by I. Schur is one of the basic works in the classical representation theory. The authors give a modern exposition of Schur’s results and systematize the recent developments in projective representations of the group $$S_ n$$.
Reviewer: M.Nazarov (Moskva)

##### MSC:
 20C30 Representations of finite symmetric groups 20C35 Applications of group representations to physics and other areas of science 20-02 Research exposition (monographs, survey articles) pertaining to group theory