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Let $G$ be a compact semi-simple Lie group, (or to be even more concrete let $G = SL(N)$), and let $V$ and $W$ be finite dimensional irreducible representations of $G$. Surely it is very well-known how to classify the $G$-equivariant linear maps between $V$ and $W$. Can anyone enlighten me as to how this works? I am most interested in the case where $\text{dim}(V) = {\text {dim}}(W)$ and the maps are isomorphisms.

Also, in the quantum setting, ie for $SL_q(N)$, does this classification pass over to a classification of comodule maps between the modules $V_q$ and $W_q$?

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$SL(N)$ is not compact. You need to be more precise here. Do you mean a complex or real semisimple group? Do you mean compact semisimple? In that case you probably want $SU(N)$ rather than $SL(N)$. In any case this is probably too elementary for MO. The short answer is that representations are classified by highest weights, and you have a nonzero morphism between two irreducible representations if and only if they have the same highest weight. The space of intertwiners in that case is 1-dimensional. The same holds true in the quantum case as well. –  MTS Jul 12 '13 at 16:17
Where could I get this, in say Humphrey's book? –  Milan Bernolak Jul 12 '13 at 16:31
In Humphreys' book it is in Chapter VI, Representation Theory. Alternatively you can find it in Chapter V of Knapp's book Lie Groups Beyond an Introduction, 2nd edition. –  MTS Jul 12 '13 at 18:40
Great, thanks a lot. –  Milan Bernolak Jul 13 '13 at 17:25

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