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CAF123

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## Homework Statement

Consider the following permutation representations of three elements in ##S_3##: $$\Gamma((1,2)) = \begin{pmatrix} 0&1&0\\1&0&0\\0&0&1 \end{pmatrix}\,\,\,\,;\Gamma((1,3)) = \begin{pmatrix} 0&0&1\\0&1&0\\1&0&0 \end{pmatrix}\,\,\,\,\,; \Gamma((1,3,2)) = \begin{pmatrix} 0&1&0\\0&0&1\\1&0&0\end{pmatrix}$$

The trivial representation is embedded in the subspace spanned by the vector ##\underline{r} = (1,1,1)##. Give ##x,y,\alpha, \beta## and then obtain the representation matrices ##\Gamma((1,2)), \Gamma((1,3,2))## and ##\Gamma((1,3))## in a basis where ##\underline{e}_1 = (x,y,y)## and ##\underline{e}_2 = (0,\alpha, \beta)##, with ##x,\beta > 0## and ##\underline{e}_i \cdot \underline{e}_j = \delta_{ij}\,\,\,, \underline{e}_i \cdot \underline{r} = 0##.

There is a hint with the question that states that the matrix ##\Gamma((1,2))## satisfies, in the standard representation, $$\Gamma((1,2)) = \frac{1}{2}\begin{pmatrix} ..&\sqrt{3}\\..&1\end{pmatrix}$$

## Homework Equations

$$\Gamma_{perm} = \Gamma_{triv} \oplus \Gamma_{stand} \Rightarrow \chi(\Gamma_{perm}) = \chi(\Gamma_{triv}) + \chi(\Gamma_{stand}) \Rightarrow \chi(\Gamma_{stand}) = 0\,\,\,\,\text{for}\,\,\,\Gamma((1,2))\,\,\,\,\text{and}\,\,\,\, \Gamma((1,3))$$

## The Attempt at a Solution

I think ##(1,1,1)## is a common eigenvector to all the permutation representations. Using the given condition involving the Kronecker tensor, I obtained the conditions ##\alpha = -\beta## and ##x = -2y##. I am looking for a hint on how to compute these matrices in the desired basis. Using the equation in the relevant equations, I get that the left hand entry for the matrix in the hint is ##-1## and by orthonormality of the rows the entry ##(21)## is ##\sqrt{3}##. But I am not sure how to actually obtain the matrices in the basis.

The RHS is a direct sum of the decomposition of two irreducible reps, which can be written like $$\begin{pmatrix}1&0&0\\0&a&b\\0&c&d \end{pmatrix},$$ but I don't see how to proceed.

Thanks.

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