By Cohen A.
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Extra resources for Group theory for Maths, Physics and Chemistry students
Sample text
3. For which pairs (l, m) are the representations rl and rm equivalent? 13. 10. Show that ρa ∼ ρb if and only if ab = 0. 9, we saw that V ∗ is a invariant under the action of G on F (V ). This gives rise to the following definition. 52 CHAPTER 4. 14. Let r be a representation of a group G on V . A subspace W is invariant if r(g)W ⊂ W for all g ∈ G. In this case, r|W : G → GL(W ) given by r|W (g) = r(g)|W , is called a subrepresentation of r, and W is called a sub-G-module of V . If G is clear from the context, we leave it out and call W a submodule of V .
Set a := g(0). Then ta−1 g(0) = 0, so t−1 a g is an isometry fixing 0, and it is clear that a = g(0) is the only value for which this is the case. 4. Let r ∈ AO(n, R) be an isometry fixing 0. Then r is an orthogonal R-linear map. Proof. Let x, y ∈ Rn , and compute (rx, ry) = (−d(rx, ry)2 +d(rx, 0)2 +d(ry, 0)2 )/2 = (−d(x, y)2 +d(x, 0)2 +d(y, 0)2 )/2 = (x, y), where we use that r is an isometry fixing 0. Hence, r leaves the inner product invariant. This proves the second statement, if we take the first statement for granted.
For a finite group G, every finite-dimensional G-module is completely reducible. Proof. Proceed by induction. Suppose that the stament holds for all representations of dimension smaller than n, and let V be an n-dimensional representation. If V is irreducible, then we are done. Otherwise there exists a sub-G-module U with 0 U V . 19, U has a invariant complementary sub-Gmodule W . As both dim U and dim W are smaller than n, we may apply the induction hypothesis, and find that U is the direct sum of irreducible sub-Gmodules U1 , .
Group theory for Maths, Physics and Chemistry students by Cohen A.
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