Character Table Exercises

These exercises pertain to the following clickable character tables.

1. Show that the number of operations in C_4v is equal to the sum of the squares of the dimensions of the irreducible representations.
2. Apply each of the operations of the C_4v point group to an atomic d_xy orbital to generate the characters of one of the irreducible representations. Which one is it?
3. Each of the irreducible representations of a point group is labeled with what is called a Mulliken symbol. What is the Mulliken symbol for the {1 -1 -1 1} irreducible representation of the C_2v point group?
4. Notice that g and u subscripts in Mulliken symbols occur only for those point groups where a center of inversion is present. Which signifies that the representation is symmetric with respect to inversion?
5. In the point group D_3h some irreducible representations have Mulliken symbols that carry a prime, and some a double prime. What is the meaning of the prime and double prime notation?