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I am trying to prove that the symmetric square of the fundamental representation $V$ of $SU(2)$ is irreducible.

Here is my approach:

Using the famous character formula of the symmetric representation and knowing that the fundamental representation is 2-dimensional, we can easily see that $\text{dim Sym}^2(V)=3$.

Using the theorem that states that each continuous representation is a direct sum of standard $n$-dimensional representations and comparing the values of the characters we can see that $\text{Sym}^2(V)$ is isomorphic to the standard 3-dimensional representation, which is irreducible.

Is that right? If yes, how can I prove the continuity of $\text{Sym}^2(V)$?

F.H.A
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  • "The symmetric" means some symmetric power $\mathrm{Sym}^k(V)$ here? For what $k$? -- That symmetric powers of continuous reps are continuous (and likewise for tensor products, duals, alternating powers) should be one of the first lemmata, if not entirely obvious, in the theory. – Torsten Schoeneberg May 05 '20 at 17:50
  • Yes, I am considering the symmetric square. – F.H.A May 05 '20 at 19:48
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    Then your approach seems reasonable to me, and as said, that $Sym^2(V)$ is a continuous rep if $V$ is one should be very easy and clear, long before comparing character values (how do you even compute these character values unless with some formula which at least implicitly already states that such symmetric powers are also continuous representations?). – Torsten Schoeneberg May 05 '20 at 21:31

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