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Decompose $\operatorname{Sym}^2 (V)$ into direct sum of irreducible sub representations. (Hint: Again consider the action on basis vectors.)"

Here, $V=\Bbb C^2$, with its standard basis, and the action $\pi$ we are considering is characterized by $$ \pi(12) = \pmatrix{0&1\\1&0} , \qquad \pi(123) = \pmatrix{\omega&0\\0&\omega^2} .$$

The hint says,

Consider the action on basic vectors.

(See the original problem statement here.)


So I think the basis vectors of $\operatorname{Sym}^2(V)$ are $e_1 \oplus e_1 , e_2 \oplus e_2$ $e_1 \oplus e_2 + e_2 \oplus e_1 $

To decompose it into irreducible representation I need to find it to be isomorphic to two vector spaces (?), I think.

Would really appreciate some advice, as I haven't quite 'clicked' with representation theory

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  • $\begingroup$ You should typeset the original question here---it will make the question more readable and make the question more likely to receive an answer besides. $\endgroup$ – Travis Nov 30 '15 at 14:18
  • $\begingroup$ ok thanks @Travis here it is: ********"Decompose $Sym^2(V)$ into a direct sum of irreducible subreps. (Hint: again consider the action on basis vectors.)" ******** $\endgroup$ – thinker Nov 30 '15 at 14:20
  • $\begingroup$ One really ought to make this post self-contained, which in particular means defining the representation $V$. At least this should contain the information in the problem statement, both the material before subquestion (i), as well as subquestion (iii). $\endgroup$ – Travis Nov 30 '15 at 14:28
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    $\begingroup$ You have the basis of vectors of ${\rm Sym}^2(V)$ correct, so what you should do next is to work out the $3 \times 3$ matrices for the actions of $(12)$ and $(123)$ on ${\rm Sym}^2(V)$ using these basis vectors. I think if you do that, then the decomposition of ${\rm Sym}^2(V)$ into irreducibles should be obvious. $\endgroup$ – Derek Holt Nov 30 '15 at 15:19
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    $\begingroup$ @thinker I've taken the liberty of formatting the question statement for readability. $\endgroup$ – Travis Nov 30 '15 at 15:21

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