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  • Print publication year: 2005
  • Online publication date: January 2010

13 - Sergeev superalgebras

from PART II - PROJECTIVE REPRESENTATIONS

Summary

By general theory, studying spin representations of the symmetric group Sn is equivalent to studying representations of its twisted group algebra Jn. It is convenient to consider Jn as a superalgebra with respect to the natural grading, where even (resp. odd) elements come from even (resp. odd) permutations. Even if we are only interested in the usual Jn-modules, Corollary 12.2.10 shows that, at least as far as irreducibles are concerned, we do not loose anything by working in the category of supermodules, providing we keep track of types of irreducible supermodules. Moreover, we even gain an additional insight into the usual irreducible modules, in view of Proposition 12.2.11. This additional information is exactly what we need in order to deal with spin representations of the alternating groups. It is interesting that the superalgebra approach is not useful for the linear representations of Sn, while spin representation theory of Sn has intrinsic features of a “supertheory”.

An important idea due to Sergeev is that instead of the superalgebra Jn it is more convenient to consider yn := JnCn, where Cn is the Clifford superalgebra, and ⊗ is the tensor product of superalgebras. On the one hand, nothing much is going to happen to representation theory when we tensor our superalgebra with a simple superalgebra (classically we get a Morita equivalence and in the “superworld” we get either a Morita equivalence or something almost as good as a Morita equivalence).

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