Lecture 21
نویسنده
چکیده
The representation given in Example 3 has an interesting property. The set of matrices {ρ(e), ρ(g)} have common invariant subspaces (other than the trivial ones, namely (0, 0) and C). Indeed, we can eas ily verify that the (orthogonal) onedimensional subspaces given by (t, t) and (t, −t) are invariant under the action of these matrices. Therefore, the restriction of ρ to those subspaces also gives representations of the group G. In this case, the one corresponding to the subspace (t, t) is “equivalent” (in a welldefined sense) to the trivial representation described in Example 2. The other subspace (t, −t) gives the one dimensional alternating representation of S2, namely ρA(e) = 1, ρA(g) = −1. Thus, the representation ρ decomposes as ρ = ρT ⊕ ρA, a direct sum of the trivial and the alternating representations. The same ideas extend to arbitrary finite groups.
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