Lecture 1: Representations of Symmetric Groups, I
نویسنده
چکیده
In this lecture we start to study the representation theory of the symmetric groups Sn over C. Let us summarize a few things that we already know. 0) A representation of Sn is the same thing as a representation of the group algebra CSn. 1) As with all finite groups, any representation of Sn over C is completely reducible. 2) The number of irreducible representations of Sn coincides with the number of conjugacy classes. The conjugacy classes are in a natural bijection with partitions of n. Namely, we take an element σ ∈ Sn and decompose it into the product of disjoint cycles. The lengthes of cycles form a partition of n that is independent of the choice of σ in the conjugacy class. We assign this partition to the conjugacy class of σ. We would like to emphasize that 2) does not establish any preferred bijection between the irreducible representations of Sn and the partitions of n. To establish such a bijection is our goal in this part. We will follow a “new” approach to the representation theory of the groups Sn due to Okounkov and Vershik, [OV]. Our exposition follows [K, Section 2]. For a “traditional” approach based on Young symmetrizers, the reader is welcome to consult [E] or [F].
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