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 3permutation 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???

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