By Samiou E.

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Then (u, v) f-+ (u ® v) is a bilinear map of X x Y into X® Y. Moreover, the u ® v span X ® Y. j=l is a basis of X® Y so dim(X ® Y) = dim(X)dim(Y) (see Reed-Simon, Vol. 1 [14] for the proof of this and other facts about tensor products). X® Y can be given a unique inner product in which (u®v,B) = B(u,v). This is such that if {ei} and {/j} are orthonormal bases of X, Y respectively, then ei ® fJ is an orthonormal basis of X ® Y. This basis way of thinking about X® Y is useful; the abstract definition is useful in showing that the construction is not basis dependent.

5: REAL AND QUATERNIONIC REPRESENTATIONS 53 ~ ~) is a 2 x 2 unitary matrix since C preserves norms. But cW cW = li and C 2 = li implies -lo:l 2 + fJ'Y = 1. This is only possible if o: = 0 and then 8 = 0. We also get {3 = eiB' r = e+iB. If where ( then so the structure is unitarily equivalent to the one in the last proposition. 4. Let G be a finite group.

Let U(rp1, 11'2) = (V 11'1, cV crp2). Then U is a representation of G which obeys CUC = U. Moreover, there are no (U, C)-invariant subspaces if and only if V is an irrep, which is either complex or quaternionic. Proof That Cis a complex conjugate is easy, as is the calculation that CUC = U. If V is not irreducible and Yo = 1{ is an invariant subspace for V, then { (rp, crp) I rp E Yo} is a (U, C)-invariant subspace. Thus, no invariant (U, C)-subspaces implies that V must be an irrep. If V is a real representation, so cWV(g)W- 1 c = WV(g)W- 1 for some unitary Wand all g, then Y = {(rp,cW- 1 cWrp)} is a (U,C)-invariant subspace.