Additional projective representations of the symmetric groups
Let \(G\) be a finite group and let \(F\) be an algebraically closed field of characteristic zero with multiplicative group \(F^*\). A mapping \(T\colon G\to\text{GL}_n(F)\) from \(G\) into the general linear group \(\text{GL}_n(F)\) is called a projective representation of \(G\) of degree \(n\) over \(F\) if there exists a function \(\alpha\colon G\times G\to F^*\) such that \(T(g)T(h)=\alpha(g,h)T(gh)\) for all \(g,h\in G\). Then, \(T\) is called irreducible if the vector space \(V=F^n\) has no nontrivial proper subspace invariant under all \(T(g)\), \(g\in G\). The projective representation theory of the symmetric groups from the point of view of Clifford algebras was considered for the first time in 1962 by \textit{A. O. Morris} [Proc. Lond. Math. Soc., III. Ser. 12, 55-76 (1962; Zbl 0104.25202)]. It was the reason to call them spin representations. In the paper under review, the authors attempt to follow \textit{V. F. Molchanov}'s approach [Vestn. Mosk. Univ., Ser. I 21, No. 1, 52-57 (1966; Zbl 0207.33404)], for ordinary representations of the symmetric group to construct the projective (spin) representations of the symmetric group. They investigate an alternative strategy which leads to the proof, more along the lines used by Molchanov. They determine all recurrence relations which should be useful in providing such a proof.
scientific article; zbMATH DE number 1568597