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## Summary:

- What is the relation between tensors and group SO(3)?

## Main Question or Discussion Point

I am reading

I have big problems when learning chapter IV.1 Tensors and Representations of the Rotation Groups SO(N).

It reads

Also, I do not understand why the tensor came in. If I am correct, elements of groups should be some transformations that leave some objects invariant, but I can hardly imagin how tensors become transformations.

I became more frustrated when the following sections are full of tensors, and I got totally lost.

*Group Theory in a Nutshell for Physicists*by A. Zee.I have big problems when learning chapter IV.1 Tensors and Representations of the Rotation Groups SO(N).

It reads

I can understand why ##D\left ( R \right )## is a representation of SO(3), but I hardly can see why the tensor T can be a representation. I thought a representation should be scalar or square matrix, but why a column consisting of tensors can be a representation, as well.Mentally arrange the nine objects ##T^{ij}## in a column ##

\begin{pmatrix}

T^{11} \\

T^{12} \\

\vdots \\

T^{33}

\end{pmatrix}

##. The linear transformation on the nine objects can then be represetned by a 9-by-9 matrix ##D\left ( R \right )## acting on this column.

For every rotation, specified by a 3-by-3 matrix R, we can thus associate a 9-by-9 matrix ##D\left ( R \right )## transforming the nine objects ##T^{ij}## linearly among themselves.

...

The tensor T furnishes a 9-dimensional representation of the rotation group SO(3).

Also, I do not understand why the tensor came in. If I am correct, elements of groups should be some transformations that leave some objects invariant, but I can hardly imagin how tensors become transformations.

I became more frustrated when the following sections are full of tensors, and I got totally lost.