# symmetry-d3-irrep-collapse.mp4

Chapter section: Representations

Caption (verbatim):

Reducing 3-dimensional representation to 2-dimensional

Image description (verbatim):

Subtracting the common component leaves the D3 representation in the zero-sum plane

## Before the animation

Chapter lines 132–138:

```math
\begin{pmatrix}1\\3\\5\end{pmatrix}
=
\begin{pmatrix}3\\3\\3\end{pmatrix}
+
\begin{pmatrix}-2\\0\\2\end{pmatrix}.
```

Chapter lines 140–140:

The common part lies on the rotation axis and is unchanged by every permutation. The zero-sum part lies in a plane through the origin. Although written with three components, it needs only two independent numbers, since the third must make their sum zero.

## After the animation

Chapter lines 148–148:

We also notice that vectors that lie on the axis of rotation itself are left unchanged by the transformation. A way to think of this is to allow the triangle's vertices to store some information, like a number or any numerical quantity. If the value they store is the same for all vertices, the symmetry actions have no effect, whereas if they are different, the actions permute those values in a way that can be represented in a 2-dimensional vector space. The 3-dimensional representation space we began with is thus decomposable into 2 subspaces, one 1-dimensional, the other 2-dimensional. These cannot be decomposed further. That is, there is no lower-dimensional space such that an allowable transformation of any state remains in that space. The 1-dimensional and 2-dimensional representations are called **irreducible representations** or **irreps** for short. This is admittedly heavy math. Why do we bother? In the story we have to tell of quantum physics, where constituents of matter must abide the dynamical symmetry of nature, each constituent, each type of **particle** such as electron or photon, corresponds to an irreducible representation of nature's symmetry. A particular state of the particle is encoded in an element in the corresponding irrep.