# Assignment 3

Problem 1.  Prove that the conjugacy classes for the group $S_3$ are given by

$\displaystyle [e] = \{e\}, \left[(1,2)\right] = \left\{(1,2), (1,3), (2,3)\right\}, \text{ and } \left[(1,2,3)\right] = \left\{(1,2,3), (1,3,2)\right\}.$

Problem 2.  More generally, let $G$ be a group, and suppose $g_1, g_2 \in G$ are conjugate to each other (i.e. that there exists $h \in G$ so that $h^{-1}g_1h = g_2$).

Prove that $\text{ord}(g_1) = n$ if and only if $\text{ord}(g_2) = n$.

Note: This is the key step in understanding the conjugacy classes of any $S_n$ group.  The conjugacy class of a transposition must contain only elements of order 2 which therefore must be other transpositions.  Similarly, the conjugacy class of a 3-cycle must contain only 3-cycles.

Problem 3.  We developed a “natural” representation of $S_3$ wherein each element is associated with a $3\times 3$permutation matrix.”  In particular, this matrix, $\rho(\sigma) = M_{\sigma}$ is defined by

$\displaystyle M_{\sigma}\left(c_1\vec{e}_1 + c_2\vec{e}_2 + c_3\vec{e}_3\right) = c_1\vec{e}_{\sigma(1)} + c_2\vec{e}_{\sigma(2)} + c_3\vec{e}_{\sigma(3)}.$

Unfortunately, this representation is reducible.  Indeed, every such matrix $M_{\sigma}$ leaves the following subspace $W$ invariant:

$\displaystyle W = \left\{ c(1, 1, 1) : c \in \mathbb{R} \right\}$.

As a result, the “complementary subpsace” $U$, defined by

$\displaystyle U = \left\{ (x,y,z) \in \mathbb{R}^3 : x + y + z = 0 \right\}$,

is also left invariant by every matrix $M_{\sigma}$.  We argued that every matrix $M_{\sigma}$ acts on the subspace $W$ as the identity function, i.e. that as a $1\times 1$ matrix acting on the 1-dimensional space $W$ we have

$\displaystyle M_{\sigma} \cong [1]$ as a transformation $M_{\sigma} : W \to W$.

The piece of our 3-dimensional representation we obtain by restricting to $W$ gives us the trivial irreducible representation, one that we already had.  Hence we focus on representing the permutations $\sigma \in S_3$ as $2\times 2$ matrices $\tilde{M}_{\sigma} : U \to U$.

Using the basis $\left\{\vec{u}_1, \vec{u}_2\right\} = \left\{ (-1, 1, 0), (-1, 0, 1) \right\}$ we found that

$\displaystyle \tilde{M}_{\sigma_1}\left(\vec{u}_1\right) = -\vec{u}_1$

$\displaystyle \tilde{M}_{\sigma_1}\left(\vec{u}_2\right) = -\vec{u}_1 + \vec{u}_2$.

This means that as a linear transformation on $U$, the element $\sigma_1 = (1, 2)$ can be represented by the matrix

$\tilde{M}_{\sigma_1} = \left[ \begin{array}{cc} -1 & -1 \\ 0 & 1 \end{array}\right]$

Compute the other five $2\times 2$ matrices, $\tilde{M}_{\sigma_i}$ for every $\sigma_i \in S_3$.

Note: We should expect that if $\sigma_i$ and $\sigma_j$ are in the same conjugacy class, then their matrix representatives $\tilde{M}_{\sigma_i}$ and $\tilde{M}_{\sigma_j}$ will look different but will have the same trace.

Problem 4.  Complete the character table for $S_3$.  (This should be easy after finishing problem 3.)

Problem 5.  Suppose $G$ is a finite Abelian group of size $|G| = n$.  Prove that $G$ must have $n$ conjugacy classes.  What does this result tell us about how simplified a character table is for an Abelian group?

Problem 6.  Complete the character table for the group $G = \mathbb{Z}_3$.  Do this by following this step-by-step procedure.

Step (1) : Figure out how large the table is by counting the number of conjugacy classes (Problem 5. should be helpful.)

Step (2) : Include as many “obvious” or “easy” irreducible representations as you can.  For instance, the trivial irreducible representation

$\displaystyle \rho_1 (g) = \left[1\right] \, \text{ for all } g \in \mathbb{Z}_3$

should be used to fill out the first row.  For remaining rows (assuming there are any), see if you can think of any “natural” transformations that the elements in $\mathbb{Z}_3$ remind you of.

Step (3) Cry?  Try and invent new representations??  Look things up?  Consult a Chemistry textbook with an example worked out where $\mathbb{Z}_3$ is the symmetry group???