Hegazi, A.; El Sayed, S. Additional projective representations of the symmetric groups. (English) Zbl 0998.20014 Adv. Appl. Clifford Algebr. 10, No. 1, 107-158 (2000). 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 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 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. Reviewer: Ali Iranmanesh (Tehran) MSC: 20C25 Projective representations and multipliers 20C30 Representations of finite symmetric groups 05E10 Combinatorial aspects of representation theory 15A66 Clifford algebras, spinors Keywords:projective representations; symmetric groups; spin representations; recurrence relations Citations:Zbl 0104.25202; Zbl 0207.33404 PDFBibTeX XMLCite \textit{A. Hegazi} and \textit{S. El Sayed}, Adv. Appl. Clifford Algebr. 10, No. 1, 107--158 (2000; Zbl 0998.20014) Full Text: DOI References: [1] Schur I., Uber die Darstellung der endlichen Gruppen durch gebrochene lineare substitutionen,J. Reine Angew. Math.,127 20–50, (1904). · JFM 35.0155.01 · doi:10.1515/crll.1904.127.20 [2] Schur I., Chtersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare substitutionen,Journal für Reine und angewandte Mathematik,132 85–137, (1907). · JFM 38.0174.02 · doi:10.1515/crll.1907.132.85 [3] Schur I., Uber die Darstellung der symmetrischen und der alternievender Gruppe durch gebrochene lineare substitutionen,Journal für reine and angewandte Methematik,139 155–250, (1911). · JFM 42.0154.02 · doi:10.1515/crll.1911.139.155 [4] Morris A. O., The spin representations of the symmetric group,Proc. London Math. Soc.,12 (3), 55–76 (1962). · Zbl 0104.25202 · doi:10.1112/plms/s3-12.1.55 [5] Morris A. O., A note on the multiplication of Hall functions,Journal London Math. Soc.,39, 481–488 (1964). · Zbl 0125.01702 · doi:10.1112/jlms/s1-39.1.481 [6] Nazarov M. L., An orthogonal basis of irreducible projective representations of the symmetric group,Functional Analysis and its Applications,22 66–68, (1988). · Zbl 0658.20010 · doi:10.1007/BF01077731 [7] Nazarov M. L., Young’s orthogonal form of irreducible projective representations of the symmetric group,J. London Math. Soc.,42 (2), 437–451 (1990). · Zbl 0677.20011 · doi:10.1112/jlms/s2-42.3.437 [8] Nazarov M. L., ”Young’s symmetrizers for projective representations of the symmetric group”, RIMS 900-Koyoto Univ. Japan (1992). · Zbl 0930.20011 [9] James G. D., The representation theory of symmetric groups,Lecture Notes in Mathematics,682, Springer-Verlag, Berlin, (1978). · Zbl 0393.20009 [10] Macdonald I. G., Symmetric Functions and Hall Polynomials, ”Oxford Mathematical Monographs” Carendon Press, Oxford, (1979). · Zbl 0487.20007 [11] Molchanov V. F., ”On matrix elements of irreducible representations of symmetric groups”, Vestnik, Moscow, Univ. 11, 52–57, (1986). [12] Stembridge J., Shifted tableaux and the projective representation of symmetric groups,Advances in Mathematics 74 87–134, (1989). · Zbl 0677.20012 · doi:10.1016/0001-8708(89)90005-4 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.