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there are several Schur's lemma.

i am talking about the one saying this one:

Let E be a finite dimensional vector space over

**C**and U an irreducible subset of L(E). If an endomorphism ϕ of E commutes with any element of U, then ϕ is a dilation

I consider a vertex in a spinfoam with two incoming edges and one outgoing one.

we have SU2 irreps on D E F associated to these legs

if i have two matrices acting on D and E, an intertwiner will map them to a third one acting on F.

how to relate this to the Schur's lemma with only one vector space and a dilation.

i think that the vector space contains the tensor product of E and F and maybe F but i do not see how to get the map on this space.

thanks