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I'm reading Lie Algebras in Particle Physics by Howard Georgi. When talking about an example of harmonic oscillator (on Page 27), it says

"The 2 dimensional space transforms by the representation $D_2$ below"

and then gives a unitary irreducible representation of $S_3$. I'm puzzled about what happened here. How does this representation come out? Is that a subgroup of $SO(2)$?

representation of D2

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    $\begingroup$ Is that a subgroup of $SO(2)$? Do you know what 2D rotation matrices look like? Do all of these have that form? $\endgroup$
    – Ghoster
    Nov 3, 2023 at 5:01
  • $\begingroup$ Sorry for my carelessness. What I'm thinking is whether these first 3 representations are connected with some rotation transformations. And it seems that these last 3 are something to do with rotation+reflection. $\endgroup$
    – Photon
    Nov 8, 2023 at 7:54

1 Answer 1

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Invoke Cayley's Theorem, which states that every group is isomorphic to a permutation group. Following that, if we consider the symmetry group of an equilateral triangle, which is isomorphic to $S_{3}$, the permutation group of order three. For a chemist $S_{3}$ is the symmetry group of the Ammonia.

The group elements are $\{E,R_{1},R_{2},R_{3},R_{4},R_{4}\}$, where $E$ is the identity element, $R_{1}$ and $R_{2}$ rotate the triangle by the angle $2\pi/3$ and $4\pi/3$ about the $z-$ axis, respectively. The remaining operations $R_3$, $R_4$, and $R_5$ denote reflections at the axes. Pictorially, enter image description here

Now start a coordinate system: enter image description here

One can see immediately that the action of $R_{1}$ is given by \begin{eqnarray} D(R_{1})\mathbf{e}_{x} & = & -\frac{1}{2} \mathbf{e}_{x} + \frac{\sqrt{3}}{2} \mathbf{e}_{y} ,\quad D(R_{1})\mathbf{e}_{y} & = & -\frac{\sqrt{3}}{2} \mathbf{e}_{x} - \frac{1}{2} \mathbf{e}_{y} ,\quad D(R_{1})\mathbf{e}_{z} & = & \mathbf{e}_{z} \end{eqnarray} Or \begin{eqnarray} D(R_{1}) = \begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix} \end{eqnarray}

This representation is reducible (look into the $2\times 2$ block). Do the same for the rest of the transformations, which will immediately give the matrices you are asking.

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