Character Tables

Character tables classify irreducible representations by traces of representative matrices on conjugacy classes. This page gives small examples used for orientation; it is not a crystallographic point-group atlas.

Convention

Rows are irreducible representations. Columns are conjugacy classes. Characters are traces of representation matrices.

C2v

For the point group $C_{2v}$ with classes $E$, $C_2$, $\sigma_v(xz)$, and $\sigma_v'(yz)$:

Irrep	$E$	$C_2$	$\sigma_v(xz)$	$\sigma_v'(yz)$	Typical Functions
$A_1$	1	1	1	1	$z$, $x^2$, $y^2$, $z^2$
$A_2$	1	1	-1	-1	$R_z$, $xy$
$B_1$	1	-1	1	-1	$x$, $xz$, $R_y$
$B_2$	1	-1	-1	1	$y$, $yz$, $R_x$

C3v

For the point group $C_{3v}$ with classes $E$, $2C_3$, and $3\sigma_v$:

Irrep	$E$	$2C_3$	$3\sigma_v$	Typical Functions
$A_1$	1	1	1	$z$, $x^2+y^2$, $z^2$
$A_2$	1	1	-1	$R_z$
$E$	2	-1	0	$(x,y)$, $(R_x,R_y)$

How to Use Character Tables

Use characters to decompose a reducible representation into irreducibles:

$$n_\alpha =\frac{1}{\lvert G\rvert} \sum_{\text{classes }C} \lvert C\rvert\,\chi_\alpha(C)^*\,\chi(C).$$

The formula assumes finite groups and class characters in the same convention.

Common Mistakes

• Treating a character table as a multiplication table.
• Forgetting that columns represent conjugacy classes, not always individual elements.
• Comparing point-group labels across different axis conventions.
• Using molecular point-group labels for spinor representations without the double group.

Canonical Links

• Groups
• Molecular Symmetry
• Symmetry Constraints on Hamiltonians
• Projective Representations

References

• M. Tinkham, Group Theory and Quantum Mechanics, Dover, 2003.
• F. A. Cotton, Chemical Applications of Group Theory, 3rd ed., Wiley, 1990.
• H. Georgi, Lie Algebras in Particle Physics, 2nd ed., Westview Press, 1999.