I am studying representation theory and got stuck on the following problem.

Let $V$ be the vector space of degree 3, real homogeneous polynomials of three variables, then $S_3$ acts on $V$ by permuting the variables. Then, describe the decomposition of $V$ into irreducible representations of $S_3$.

Where do I even get started with this?

  • $\begingroup$ Have you done characters already? Because it is not difficult to compute the character of this representation, and then check which irreducible reps of $S_{3}$ appear in it with which multiplicity. $\endgroup$ – Andreas Caranti Oct 28 '16 at 10:19
  • $\begingroup$ Yes, but I am having trouble describing the representation to begin with. If $\rho$ is the representation, then how do I obtain the matrix $\rho(\pi)$, for given $\pi\in S_3$. I know that $\pi.f = f(x_{\pi(1)}, x_{\pi(2)}, x_{\pi(3)})$ and $\dim V = 10,$ but cannot see where to get from here. $\endgroup$ – dezdichado Oct 28 '16 at 11:07
  • $\begingroup$ OK, I'll describe this in an answer. $\endgroup$ – Andreas Caranti Oct 28 '16 at 11:25

Start by checking how the identity, the 3-cycle $(1 2 3)$ and the 2-cycle $(1 2)$ operate on the basis of $V$ given by $$ x_{1}^{3}, x_{2}^{3}, x_{3}^{3},\\ x_{1}^{2} x_{2}, x_{2}^{2} x_{1},\\ x_{1}^{2} x_{3}, x_{3}^{2} x_{1},\\ x_{2}^{2} x_{3}, x_{3}^{2} x_{2},\\ x_{1} x_{2} x_{3}. $$ For instance, $(1 23)$ cyclically permutes the first three elements.

This will yield matrices for the actions of the identity, the 3-cycle $(1 2 3)$ and the 2-cycle $(1 2)$, and thus the character $\chi$ of the representation.

Then it is not difficult to use the usual inner product to compute the multiplicities of the irreducible characters of $S_{3}$ in $\chi$.

So you get the character $\chi$ $$ \begin{matrix} 1 & (123) & (12)\\\hline 10 & 1 & 2 \end{matrix} $$ meaning that it has value $10$ on the identity, value $1$ on the elements of the conjugacy class of $(123)$, and $2$ on the elements of the conjugacy class of $(12)$.

Now the character table of $S_{3}$ is $$ \begin{matrix} & 1 & (123) & (12)\\\hline \chi_{1} & 1 & 1 & 1\\ \chi_{2} &1 & 1 & -1\\ \chi_{3} &2 & -1 & 0\\ \end{matrix} $$ Now compute the inner product of $\chi$ with the $\chi_{i}$ to get the multiplicities of each $\chi_{i}$ in $\chi$.

  • $\begingroup$ I see. We get the whole character because the 3-cycle and the 2-cycle generates S_3. But how do I determine the decomposition of $V$ corresponding irreducible representations of S_3? $\endgroup$ – dezdichado Oct 28 '16 at 12:14
  • $\begingroup$ I'm answering in an edit to the answer. $\endgroup$ – Andreas Caranti Oct 28 '16 at 12:17
  • $\begingroup$ I think I got it now after writing down the matrices and the product formula in detail. $\endgroup$ – dezdichado Oct 28 '16 at 13:09
  • $\begingroup$ Great, thanks!! $\endgroup$ – Andreas Caranti Oct 28 '16 at 13:10

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