• Support PF! Buy your school textbooks, materials and every day products Here!

Decompose the permutation into the direct sum of irreducible reps.

  • Thread starter NasuSama
  • Start date
  • #1
326
3

Homework Statement



Note: I need help with part (c).

Consider the representation [itex]P: S_3 \rightarrow GL_3[/itex] where [itex]P_{\sigma}[/itex] is the permutation matrix associated to [itex]\sigma[/itex].

a) Determine the character [itex]\chi_P : S_3 \rightarrow \mathbb{C}[/itex]

b) Find all the irreducible representations of [itex]S_3[/itex].

c) Decompose [itex]P[/itex] into the direct sum of irreducible representations. That is, find a single matrix [itex]Q[/itex] so that [itex]Q^{-1}P_{\sigma}Q[/itex] is block diagonal where the blocks along the diagonal are either [itex]T_{\sigma}[/itex], [itex]\Sigma_{\sigma}[/itex] or [itex]A_{\sigma}[/itex]

2. The attempt at a solution

Here is my overall progress for the problem:

I let [itex]e[/itex] to be the identity permutation, [itex]x = (1 \ 2 \ 3)[/itex] and [itex]y = (1 \ 2)[/itex]

Then, I let [itex]P_x = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}[/itex], [itex]P_y = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}[/itex] and [itex]P_e = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}[/itex].

The conjugacy classes are as followed:

- [itex]\{e\}[/itex]

- [itex]\{y, xy, yx\}[/itex]

- [itex]\{x, x^2\}[/itex]

The character table shows that:

[itex]\chi (\{e\}) = 3[/itex]

[itex]\chi (\{x, x^2\}) = 0[/itex]

[itex]\chi (\{y, xy, yx\}) = 1[/itex]

By the theorem I have applied, I have found three representations, which is congruent to the number of conjugacy classes. They are:

- trivial representation [itex]T[/itex]
- sign representation [itex]\Sigma[/itex]
- two-dimensional representation [itex]A[/itex], presented as the symmetries of equilateral triangle

The sum of their dimensions corresponds to the theorem I applied:

[itex]d_1^2 + d_2^2 + d_3^2 = |S_3| = 6[/itex]

The only possibility for equality to hold is [itex]d_1 = d_2 = 1[/itex] and [itex]d_3 = 2[/itex].

Another character table shows that:

[itex]\chi_T (\{e\}) = 1[/itex]

[itex]\chi_T (\{x, x^2\}) = 1[/itex]

[itex]\chi_T (\{y, yx, xy\}) = 1[/itex]

[itex]\chi_A (\{e\}) = 2[/itex]

[itex]\chi_A (\{x, x^2\}) = -1[/itex]

[itex]\chi_A (\{y, xy, yx\}) = 0[/itex]

[itex]\chi_{\Sigma} (\{e\}) = 1[/itex]

[itex]\chi_{\Sigma} (\{x, x^2\}) = 1[/itex]

[itex]\chi_{\Sigma} (\{y, yx, xy\}) = -1[/itex]

Finally, I need to determine [itex]Q[/itex] for [itex]Q^{-1}P_xQ = \begin{bmatrix}1 & 0 & 0 \\ 0 & -1/2 & -\sqrt{3}/2 \\ 0 & \sqrt{3}/2 & -1/2 \end{bmatrix}[/itex] and [itex]Q^{-1}P_yQ = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}[/itex] We need to find the basis for [itex]\mathbb{R}^4[/itex].

I know that [itex]u = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}[/itex] is one of the vectors in the basis. That is because we want to find the vector [itex]u[/itex] such that [itex]P_xu = u[/itex] and [itex]P_yu = u[/itex].

Then, I need to find the remaining two vectors for the basis. Here is the hardest part. I attempt to find the vectors, say [itex]v[/itex] and [itex]w[/itex], but I can't find the nonzero basis. Here is what I have:

[itex]P_xv = w[/itex]

[itex]P_yv = v[/itex]

[itex]P_xw = -v[/itex]

[itex]P_yw = -w[/itex]

Here is what I have:

[itex]v = \begin{bmatrix} a \\ b \\ c\end{bmatrix} \rightarrow w = \begin{bmatrix} c \\ a \\ b\end{bmatrix}[/itex]

[itex]v = \begin{bmatrix} a \\ b \\ c\end{bmatrix} \rightarrow v = \begin{bmatrix} b \\ a \\ c\end{bmatrix}[/itex]

[itex]w = \begin{bmatrix} c \\ a \\ b\end{bmatrix} \rightarrow -v = \begin{bmatrix} b \\ c \\ a\end{bmatrix}[/itex]

[itex]w = \begin{bmatrix} c \\ a \\ b\end{bmatrix} \rightarrow -w = \begin{bmatrix} a \\ c \\ b\end{bmatrix}[/itex]

From here, I can't determine what are the nonzero vectors for [itex]Q[/itex].

Any advices or comments?
 
Last edited:

Answers and Replies

Related Threads on Decompose the permutation into the direct sum of irreducible reps.

Replies
4
Views
2K
Replies
4
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
6
Views
606
Replies
2
Views
2K
  • Last Post
Replies
10
Views
6K
Replies
5
Views
4K
Replies
6
Views
3K
Top