# Irreducible representation of S3

1. Jan 20, 2010

### sineontheline

okay so i was reading a book on representations and found this discussion and was confused:
it starts at the bottom of pg 53 and ends at the top of pg 54

so I understood the beginning of the discussion:
there are matrix representations of S3 and they permute the vector components
but (1,1,1) constitutes an invariant subspace cause what ever you permutation occurs on that, will bring you back to (1,1,1)

but then it goes on to say:
"To find another invariant subspace, we note that every 3 X 3 matric in the representation belongs to O(3) and hence preserves the ordinary Euclidean scalar product. Therefore, the subspace W' orthogonal to (1,1,1) is also invariant."
It then goes on to list the invariant subspace.

I got lost. Can anyone help? Why did they come up with those numbers? (and how too)

The example is particularly important cause he uses it later:

2. Jan 21, 2010

### George Jones

Staff Emeritus
The matrix

$$\left[ \begin{array}{ccc} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{array} \right]$$

is with respect to the standard basis

$$\left\{ \left( \begin{array}{c} 1\\ 0\\ 0 \end{array} \right), \left( \begin{array}{c} 0\\ 1\\ 0 \end{array} \right), \left( \begin{array}{c} 0\\ 0\\ 1 \end{array} \right) \right\}.$$

How is this expressed with respect to the basis $\left\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 \right\}$?

3. Jan 22, 2010

### George Jones

Staff Emeritus
How to do this typically is covered at the end of a first course in linear algebra or at the beginning of a second course. Perform a similarity transform using the change of basis matrix.

4. Jan 22, 2010

### wofsy

If a matrix preserves angles then the orthogonal complement of an invariant subspace must also be invariant because it remains orthogonal to the first invariant subspace under the action of the matrix.