Character Computation for Representations

1. From Matrices to Invariant Numbers: The Character

A significant challenge in working with representations is their dependence on the chosen basis. A representation $\rho$ in one basis consists of matrices $\rho (g)$, while in another basis related by a similarity transformation $S$, the matrices become ${\rho }^{\prime }(g)=S\rho (g)S^{-1}$. We need a "fingerprint" of the representation that remains unchanged regardless of the basis.

The solution lies in a familiar property of matrices: the trace. Recall that the trace of a matrix is invariant under cyclic permutations of a product: $\text{Tr}(ABC)=\text{Tr}(BCA)$. Applying this to our similarity transformation gives:

The trace is basis-independent. This motivates our central definition.

Definition: The Character
The character of a representation $\rho$ for a group element $g$, denoted ${\chi }^{(\rho )}(g)$, is the trace of the corresponding representation matrix:

2. The Link to Conjugacy Classes

The utility of characters is immensely enhanced by another key property. Recall from our first module that two group elements $g_1$ and $g_2$ are in the same conjugacy class if there exists an element $h\in G$ such that $g_2=h{g}_1{h}^{-1}$. Let's see how the character behaves for two such elements.

Since a representation $\rho$ is a homomorphism, we have $\rho (g_2)=\rho (h{g}_1{h}^{-1})=\rho (h)\rho (g_1)\rho (h{)}^{-1}$. Now, we take the trace:

Using the cyclic property of the trace once more, we find:

This proves a fundamental theorem: all elements in the same conjugacy class have the same character. Functions with this property are called class functions.

This is a massive simplification. We don't need to compute the character for every element in the group; we only need to compute it once for each conjugacy class. This naturally leads to the idea of a character table: a table where rows correspond to the different irreducible representations (irreps) and columns correspond to the conjugacy classes.

3. Why Trace? The Case of Higher-Dimensional Irreps

For one-dimensional representations, the matrix is just a $1\times 1$ matrix, a single number, so its trace is the number itself. But for the multi-dimensional irreps that arise in non-abelian groups, we need a way to extract a single representative number. Why is the trace the right choice, and not, for example, the determinant? The determinant is also a single number and is also basis-independent.

4. Constructing a Character Table: The $S_3$ Group

Let's make this concrete by constructing the character table for the smallest non-abelian group, $S_3$ (the permutation group of three objects), which is isomorphic to the point group $C_{3v}$ (the symmetry group of an equilateral triangle or an ammonia molecule).

Step 1: Identify Conjugacy Classes.
As we found in Module 1, $S_3$ has three conjugacy classes:

1. $C_1=\{e\}$: The identity (1 element).
2. $C_2=\{(12),(13),(23)\}$: The three transpositions (3 elements).
3. $C_3=\{(123),(132)\}$: The two 3-cycles (2 elements).
   The columns of our character table will be labeled by these classes.

Step 2: Identify Irreducible Representations.
It can be proven (as we'll see how in the next lesson) that the number of irreps is equal to the number of conjugacy classes. So, $S_3$ has three irreps.

1. The trivial representation ($A_1$): A 1D rep where every group element is mapped to the number 1.
2. The sign representation ($A_2$): A 1D rep where elements are mapped to their sign (parity): +1 for even permutations ($e$, cycles), -1 for odd permutations (transpositions).
3. A 2D representation ($E$): This corresponds to the symmetries of an equilateral triangle in a plane.

Step 3: Compute the Characters for each Irrep and Class.

Let's summarize the calculation based on the video and our knowledge:

• For the trivial rep, ${\chi }^{(A_1)}$:

  • All matrices are $[1]$. The trace is always 1.
  • Character row: (1, 1, 1)

• For the sign rep, ${\chi }^{(A_2)}$:

  • The matrix for $e$ and the 3-cycles is $[1]$ (trace = 1).
  • The matrix for the transpositions is $[-1]$ (trace = -1).
  • Character row: (1, -1, 1)

• For the 2D rep, ${\chi }^{(E)}$:

  • Class $C_1$ (Identity): $\rho (e)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$. Thus, ${\chi }^{(E)}(e)=\text{Tr}(\rho (e))=1+1=2$.
  • Class $C_2$ (Transpositions/Reflections): A representative matrix is a reflection, e.g., $\rho ({\sigma }_v)=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$. Thus, ${\chi }^{(E)}(C_2)=\text{Tr}(\rho ({\sigma }_v))=-1+1=0$.
  • Class $C_3$ (3-cycles/Rotations): A representative matrix is a rotation by $120^{\circ }$, $\rho (C_3)=\left(\begin{array}{cc}\cos (2\pi /3)& \sin (2\pi /3)\\ -\sin (2\pi /3)& \cos (2\pi /3)\end{array}\right)=\left(\begin{array}{cc}-1/2& \sqrt{3}/2\\ -\sqrt{3}/2& -1/2\end{array}\right)$. Thus, ${\chi }^{(E)}(C_3)=\text{Tr}(\rho (C_3))=-1/2-1/2=-1$.
  • Character row: (2, 0, -1)

Step 4: Assemble the Character Table.
We conventionally list the number of elements in each class at the top.

This table is a complete and incredibly dense summary of the representation theory of $S_3$. For comparison, here is the character table for the larger group $S_4$, which has five irreps and five conjugacy classes.

5. Properties and a Look Ahead

Take a close look at the first column of our $S_3$ table: (1, 1, 2). These are the characters of the identity element, $\chi (e)$. Since $\rho (e)$ is always the identity matrix, its trace is simply the sum of 1s down the diagonal, which is equal to the dimension of the matrix.

Property: The character of the identity element, $\chi (e)$, is the dimension of the representation.

Notice another "coincidence" for $S_3$. The sum of the squares of the dimensions of the irreps is $1^2+1^2+2^2=1+1+4=6$, which is exactly the order of the group, $|S_3|$! This is not a coincidence but a general theorem.

These are just two of several "miraculous" properties that govern the structure of character tables. These properties are not just mathematical curiosities; they are powerful computational rules. The most important of these is the great orthogonality theorem for characters.

Conclusion

In this lesson, we have moved from complex, basis-dependent matrices to simple, invariant numbers called characters. This simplification is the key that unlocks the computational power of representation theory.

Here are the essential takeaways:

• The character $\chi (g)$ is the trace of the representation matrix $\rho (g)$. It is independent of the basis chosen for the representation.
• Characters are class functions: all elements within the same conjugacy class share the same character.
• This allows us to summarize the information for all irreps of a group in a compact character table, with rows for irreps and columns for conjugacy classes.
• The character of the identity element $\chi (e)$ gives the dimension of the representation.

We have learned what a character is and how to compute it from known representation matrices. However, we often don't know the matrices beforehand. In our next lesson, we will introduce the great orthogonality theorem. This theorem exposes the deep geometric structure of the character table itself and provides a powerful algorithm for computing the characters and decomposing any reducible representation without ever needing to see the matrices.