For any rep R of a Lie group G you have $R\otimes{R}=R_1\oplus{R_2}\oplus...\oplus{R_n}$ (eh, is this correct even if G is non-compact, non-simple, non-reductive or non-whatever?). Can you, for any n, finitely list all pairs {G,R} with such a Clebsch-Gordan expansion? "Finitely list" in the same sense as you can finitely list the decimal expansion of 1/3 by using a bar over the 3. I better elaborate for n=3: 100...^2=200...+010...+000... (all $BCD_n$ except $B_2$), 200...^2=400...+210...+020... (all $A_n$ except $A_1$), 3^2=1+3+5 ($A_1$), 010^2=020+101+000 ($A_3$), 10^2=20+02+00 and 01^2=02+10+00 ($B_2$) and 100000^2=200000+001000+100000 ($E_6$). (I hope my shorthand "^2" and $210=2\lambda_1+\lambda_2$ etc. is obvious.) Of course I used only the simple Lie groups, because I know nothing at all about the rest! (Do you have some non-whatever examples for n=3? If yes, can they also be sorted into series and a finite number of exceptions?)
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asked Oct 20, 2011 at 12:39
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$\begingroup$ Just to clarify, are you asking, for fixed $n$, what are all group $G$ and representations $R$ such that $R \otimes R$ splits into a direct sum of exactly $n$ irreducible / indecomposable representations? $\endgroup$– Steven SamCommented Oct 20, 2011 at 14:11
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6$\begingroup$ Please make an effort to write a readable question. $\endgroup$– S. Carnahan ♦Commented Oct 20, 2011 at 14:14
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$\begingroup$ @Steven - yes, exactly. @S. Carnahan - do you object to the notation or to the "flow"? $\endgroup$– Hauke ReddmannCommented Oct 21, 2011 at 10:43
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