Shifted tableaux and the projective representations of symmetric groups.

*(English)*Zbl 0677.20012This 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.

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

##### MSC:

20C30 | Representations of finite symmetric groups |

20C25 | Projective representations and multipliers |

05A17 | Combinatorial aspects of partitions of integers |

##### Keywords:

projective representations; symmetric group; shifted tableaux; irreducible projective characters; induction; representation groups; double covers; basic spin representations; symmetric functions; Littlewood-Richardson rule; induced characters; inner tensor product
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