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For $S_n,$ one can construct all the irreducible representations through the young diagrams. Is there any natural construction for the irreducible representation of $G\wr S_n$ (G is a finite group)?

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It depends what you mean by natural. In Theorem 4.3.34 of James and Kerber's book "The Representation Theory of the Symmetric Group" a complete list of irreducible representations is obtained via Clifford theory. However this is probably not what you're looking for. –  Jay Taylor Jul 29 '13 at 6:50
    
@Munees: Besides the older symmetric group literature, the more recent work on rational Cherednik algebras might be suggestive. But as Jay points out, it depends on what is meant by "natural". –  Jim Humphreys Jul 29 '13 at 14:01
    
@munees: you might also see this previous post: mathoverflow.net/questions/25625/… –  Ben Braun Jul 30 '13 at 15:40

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