Some projective representations of Sn

Wales, David B.

doi:10.1016/0021-8693(79)90304-1

Cited by 50 publications

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“…However, no general construction of the irreducible representations has been produced so far (cf. [12]). In the present paper all irreducible representations T of the group T n with T(Z) = -id, are constructed.…”

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confidence: 99%

Young's Orthogonal Form of Irreducible Projective Representations of the Symmetric Group

Nazarov

1990

Journal of the London Mathematical Society

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The study of projective representations over C of the symmetric group S n leads to the investigation of linear representations of a central extension Namely, a group G can be so chosen that for each projective representation a: S n -* PGL (V) there is a linear representation T: G -> GL(V) making the following diagram commute. GLet n > 3. Then H 2 (S n , C*) = Z 2 . One can take for G the group T n generated by z,t x ,t 2 ,...,/ n _! subject to the relations z* = tl = (zt k f=\;The map % is defined for the generators: n(z) = 1, n(t k ) = s k being the transposition of k and k + 1 . This map expands to an epimorphism T n -*• S n since the group S n is generated by s v ..., s n _ x subject to the Coxeter relations = 1 ; (s k s k+l f = \; (s k s k ,? = \, \k-k'\>\.The linear operator T(Z) is scalar since the above diagram commutes. Thus T(Z) 2 = id implies that T(Z) = ±id. The projective representation a of S n corresponding to T is not equivalent to any linear one if x(z) = -id, otherwise it is.If n ^ 3 then H 2 (S n , C*) = 1 and all projective representations of S n are equivalent to linear ones. The extension T n of the group S n splits for n ^ 3: the monomorphism S n -> T n is defined by ^i-> t v s 2 t-*zt 2 . Nevertheless we shall consider the group T n for all n $s 1.The projective representations of S n were originally studied by Issai Schur [10]. It was Schur who had introduced the group T n and determined its irreducible characters. In recent years we have seen the increasing interest in the projective representations of S n (see, for example, [5] and the references therein). However, no general construction of the irreducible representations has been produced so far (cf. [12]). In the present paper all irreducible representations T of the group T n with T(Z) = -id, are constructed.

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confidence: 99%

Young's Orthogonal Form of Irreducible Projective Representations of the Symmetric Group

Nazarov

1990

Journal of the London Mathematical Society

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“…Furthermore, D n has dimension 2 n−1 2 . This has been shown by Wales in Wales (1979), and a proof can also be found in Benson (1988, Thm. 5.1).…”

Section: Simple F N -Modules In Characteristicmentioning

confidence: 57%

Vertices of Low-Dimensional Simple Modules for Symmetric Groups

Danz¹

2008

Communications in Algebra

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We determine the vertices of all simple modules for the symmetric groups over an algebraically closed field of prime characteristic whose dimensions are at most 1000 and give combinatorial descriptions of both the modules and their vertices. Moreover, we formulate some general conjectures on vertices of several series of simple modules for symmetric groups, and provide some evidence for these to hold true.

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“…It follows that χ(t) = −χ(1)/2. By the main result of [25], this implies that χ is a basic spin character. In the case n = 17 and χ ∈ Irr(G) is a spin character of degree less than 4d 1 = 512 , we can argue as before, embedding a double cover L of S 14 in G and using the character table of L as supplied in GAP.…”

Section: Double Covers Of Alternating Groupsmentioning

confidence: 83%