# Representation Theory Block Diagonalizing

I am currently examining the symmetric group $$S_4$$, and I was tasked with finding a $$2$$-D, a $$3$$-D, and a $$4$$-D representation of the group. The $$4$$-D representation is reducible, so I first found it and then found that it was the direct sum of a $$3$$-D irreducible representation plus the fully symmetric representation. Consider one element:

This is not in block diagonal form. In python, there is a function available for a matrix m, m.jordan_form(), which applies a symmetry operation to the input matrix and returns that matrix in Jordan form. The result for this matrix was

I assume that $$I$$ is $$i$$, the imaginary number. Please correct me if I am wrong on this.

The issue with this result is we know that the above matrix is the direct sum of the fully symmetric representation and a $$3$$-D representation; but the corner elements are $$-1$$ and $$I$$, neither of which are $$1$$. The fully symmetric representation is always just $$1$$, so I expect that the block form result should either have a $$1$$ in the top left corner; and then the bottom right block is the $$3$$-D representation; or I expect the bottom right corner to be $$1$$ and then the top left block is the $$3$$-D representation. To clarify, if the fully symmetric representation is represented by $$\Gamma^{(1)}$$, then

$$\Gamma^{(1)}[(1342)]=1$$

and for the general group element $$R$$

$$\Gamma^{(1)}[R]=1$$

My question is then, what am I doing wrong? Is there a different block form other than Jordan form which I need to use in this situation? Thanks for your help!

• In the 3D representation, $S_4$ acts by symmetries of the cube, which includes $90^{\circ}$ rotations around axes through faces. These are order four elements corresponding to $4$-cycles in $S_4$. This clearly acts as $1$ in the fixed axis and the usual $90^{\circ}$ 2D rotation matrix for the other block. However, when complexified that 2D block diagonalizes to ${\rm diag}(i,-i)$, since these are conjugate the fourth roots of unity. – arctic tern Jun 2 at 23:00

The problem is you need to decompose all of the matrices corresponding to the representation of $$G$$ at the same time, that is, find a basis $$v_1, \ldots, v_4$$ such that every matrix of the representation becomes block-diagonal in that basis. In the language of matrices, you need to find a single conjugating matrix $$P$$ such that $$P \Gamma[\sigma] P^{-1}$$ is block diagonal for all $$\sigma \in S_4$$. Once you have done this, the 1-dimensional and 3-dimensional subrepresentations will be clear.
• do you know of any procedure (other than simply generating a set of equations) that would help me to find that $P$ matrix? – Kraig Jun 3 at 12:15
• @Kraig Well, it's the same problem as breaking a representation into a sum of irreducibles. In this simple case, you have the projector $P(v) = \frac{1}{4!} \sum_{\sigma \in S_4} \Gamma[\sigma] v$ onto the one-dimensional representation, and so $1 - P$ will project onto the leftover three-dimensional representation. – Joppy Jun 3 at 15:05