Lecture 21

نویسنده

  • Pablo A. Parrilo
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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تاریخ انتشار 2006