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Worksheet: Reducible and Irreducible Representations

Q1:

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

D 3 h E 2 C 3 3 C 2 𝜎 h 2 S 3 3 𝜎 v    
A β€² 1 1 1 1 1 1 1   π‘₯ + 𝑦 , 𝑧 2 2 2
A β€² 2 1 1 βˆ’ 1 1 1 βˆ’ 1 𝑅 𝑧  
E β€² 2 βˆ’ 1 0 2 βˆ’ 1 0 π‘₯ , 𝑦 π‘₯ βˆ’ 𝑦 , π‘₯ 𝑦 2 2
A β€³ 1 1 1 1 βˆ’ 1 βˆ’ 1 βˆ’ 1    
A β€³ 2 1 1 βˆ’ 1 βˆ’ 1 βˆ’ 1 1 𝑧  
E β€³ 2 βˆ’ 1 0 βˆ’ 2 1 0 𝑅 , 𝑅 π‘₯ 𝑦 π‘₯ 𝑧 , 𝑦 𝑧