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Shifted tableaux and the projective representations of symmetric groups. (English) Zbl 0677.20012
This paper explains connections between the theory of (complex) projective representations of the symmetric group \(S_ n\) and the theory of shifted tableaux. In this way the author obtains a proof of I. Schur’s description of the irreducible projective characters, a projective analogue of induction from parabolic subgroups and further representation theoretic and combinatorial results which generalize well known results established in the linear theory.
In more detail the content is as follows. At first the author reviews Schur’s theory of projective representations of finite groups and constructs the representation groups \(\tilde S_ n\) of \(S_ n\). In section 2 he describes the conjugacy classes of \(\tilde S_ n\), as well as those of \(\tilde A_ n\), the subgroup that doubly covers the alternating group \(A_ n\), and \(\tilde S^ J_ n\), the double covers of parabolic subgroups. In section 3 the basic spin representations of \(\tilde S_ n\) are constructed; these are obtained from an obvious action of \(S_ n\) on the standard Clifford algebra \(C_{n-1}\). Next he considers the problem of inducing representations from \(\tilde S^ J_ n\) to \(\tilde S_ n\). In this respect the author introduces in section 4 a projective version of the outer tensor product which is called the reduced Clifford product and shows that every irreducible \(\tilde S^ J_ n\)-module is a reduced Clifford product. Section 5 explains the connection between projective characters and symmetric functions. In section 6 Schur’s symmetric functions are defined in terms of shifted tableaux. Schur’s description of the irreducible characters is derived in section 7. In section 8 the theory of shifted tableaux is applied to prove a projective analogue of the Littlewood-Richardson rule; this rule can be used to decompose the induced characters into irreducibles. Section 9 considers the inner tensor product of \(\tilde S_ n\)-modules.
Reviewer: H.Opolka

20C30 Representations of finite symmetric groups
20C25 Projective representations and multipliers
05A17 Combinatorial aspects of partitions of integers
Full Text: DOI
[1] Andrews, G.E, The theory of partitions, (1976), Addison-Wesley Reading, MA · Zbl 0371.10001
[2] Curtis, C.W; Reiner, I, ()
[3] Garsia, A.M; Remmel, J, Shuffles of permutations and the Kronecker product, Graphs combin, 1, 217-263, (1985) · Zbl 0588.05005
[4] Humphreys, J.F, Blocks of projective representations of symmetric groups, J. London math. soc., 33, 441-452, (1986) · Zbl 0633.20007
[5] James, G.D; Kerber, A, The representation theory of the symmetric group, (1981), Addison-Wesley Reading, MA
[6] Knuth, D.E, Permutations, matrices and generalized Young tableaux, Pacific J. math., 34, 709-727, (1970) · Zbl 0199.31901
[7] Littlewood, D.E, On certain symmetric functions, Proc. London math. soc., 11, 3, 485-498, (1961) · Zbl 0099.25102
[8] Littlewood, D.E, The Kronecker product of symmetric group representations, J. London math. soc., 31, 89-93, (1956) · Zbl 0090.24803
[9] Macdonald, I.G, Symmetric functions and Hall polynomials, (1979), Oxford Univ. Press Oxford · Zbl 0487.20007
[10] Morris, A.O, The spin representation of the symmetric group, Proc. London math. soc., 12, 3, 55-76, (1962) · Zbl 0104.25202
[11] Morris, A.O, The spin representation of the symmetric group, Canad. J. math., 17, 543-549, (1965) · Zbl 0135.05602
[12] Morris, A.O, A survey on Hall-Littlewood functions and their applications to representation theory, () · Zbl 0365.20018
[13] Morris, A.O, The projective characters of the symmetric group—an alternative proof, J. London math. soc., 19, 2, 57-58, (1979) · Zbl 0393.20010
[14] Nazarov, M.L, An orthogonal basis of irreducible projective representations of the symmetric group, Functional anal. appl., 22, 66-68, (1988) · Zbl 0658.20010
[15] Read, E.W, On the projective characters of the symmetric group, J. London math. soc., 15, 2, 456-464, (1977) · Zbl 0379.20008
[16] Sagan, B.E, Shifted tableaux, Schur Q-functions and a conjecture of R. Stanley, J. combin. theory ser. A, 45, 62-103, (1987) · Zbl 0661.05010
[17] \scB. E. Sagan, private communication.
[18] Schur, I, Über die darstellung der symmetrischen und der alternierenden gruppe durch gebrochene lineare substitutionen, J. reine angew. math., 139, 155-250, (1911) · JFM 42.0154.02
[19] Stanley, R.P; Stanley, R.P, Theory and application of plane partitions I, II, Stud. appl. math., Stud. appl. math., 50, 259-279, (1971) · Zbl 0225.05012
[20] Weyl, H, The classical groups, (1946), Princeton Univ. Press Princeton, NJ · JFM 65.0058.02
[21] Worley, D.R, A theory of shifted Young tableaux, ()
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