By Makhnev A. A.

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**Extra info for 3-Characterizations of finite groups**

**Example text**

If (·, ·) is another G-invariant inner product on V , deﬁne a second bijective linear map T : V → V ∗ by T v = (·, v) . 12) shows that dim HomG (V , V ∗ ) = 1. Since T, T ∈ HomG (V , V ∗ ), there exists c ∈ C, so T = cT . Thus (·, v) = c(·, v) for all v ∈ V . It is clear that c must be in R and positive. 21. Let V be a ﬁnite-dimensional representation of a compact Lie group G with a G-invariant inner product (·, ·). , (V1 , V2 ) = 0. Proof. Consider W = {v1 ∈ V1 | (v1 , V2 ) = 0}. Since (·, ·) is G-invariant, W is a submodule of V1 .

8) G acts on V by the same action as it does on V . It needs to be veriﬁed that each of these actions deﬁne a representation. All are simple. 14. For number (3), smoothness and invertibility are clear. It remains to verify the homomorphism property so we calculate [g1 (g2 T )] (v) = g1 (g2 T ) g1−1 v = g1 g2 T (g2−1 g1−1 v) = [(g1 g2 )T ] (v) for gi ∈ G, T ∈ Hom(V, W ), and v ∈ V . For number (5), recall that k V is simply k V modulo Ik , where Ik is k V intersect the ideal generated by {v ⊗ v | v ∈ V }.

Suppose V is a representation of a compact Lie group G that is reducible. Let (·, ·) be a G-invariant inner product. If W ⊆ V is a proper G-invariant subspace, then V = W ⊕ W ⊥ . Moreover, W ⊥ is also a proper G-invariant subspace since (gw , w) = (w , g −1 w) = 0 for w ∈ W ⊥ and w ∈ W . By the ﬁnite dimensionality of V and induction, the proof is ﬁnished. 18) V ∼ = N i=1 n i Vi , 38 2 Representations where {Vi | 1 ≤ i ≤ N } is a collection of inequivalent irreducible representations of G and n i Vi = Vi ⊕· · ·⊕ Vi (n i copies).