Worksheet: Reducible and Irreducible Representations

In this worksheet, we will practice defining reducible and irreducible representations and relating their characters to the order of the group.

Q1:

The character table for the D h group is given below. How many 3D irreducible representations are there in this group?

D h E 2 C 3 C 𝜎 h 2 S 3 𝜎 v
A 1 1 1 1 1 1 𝑥 + 𝑦 , 𝑧
A 1 1 1 1 1 1 𝑅
E 2 1 0 2 1 0 𝑥 , 𝑦 𝑥 𝑦 , 𝑥 𝑦
A 1 1 1 1 1 1
A 1 1 1 1 1 1 𝑧
E 2 1 0 2 1 0 𝑅 , 𝑅 𝑥 𝑧 , 𝑦 𝑧

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