
Here are listed some helpful general rules for the product of two irreducible representations. For specific combinations not listed here, one can work out the product by multiplying the characters of each irreducible representation and solving the linear combination of the irreducible representations from the point group that generates that product. Often ths process is simple, especially when one or both of the irreducible representations are nondegenerate (in most cases A or B). This process is embodied in an equation.
Nondegenerate 
A × A 
= 
A 
A × E_{1} 
= 
E_{1} 
B × B 
= 
A 
A × E_{2} 
= 
E_{2} 
A × B 
= 
B 
B × E_{1} 
= 
E_{2} 
A × E 
= 
E 
B × E_{2} 
= 
E_{1} 
B × E 
= 
E 
A × T 
= 
T 
B × T 
= 
T 
Gerade/Ungerade 
Prime/doubleprime 
Subscripts on A or B^{†} 
g × g 
= 
g 
’ × ’ 
= 
’ 
1 × 1 
= 
1 
u × u 
= 
g 
’’ × ’’ 
= 
’ 
2 × 2 
= 
1 
u × g 
= 
u 
’ × ’’ 
= 
’’ 
1 × 2 
= 
2 
2fold Degenerate^{‡} 
3fold Degenerate^{*} 
Infinite rotations 
E_{1} × E_{1} 
= 
A_{1}+A_{2}+E_{2} 
E_{} × T_{1} 
= 
T_{1}+T_{2} 
Σ^{+} × Σ^{+} 
= 
Σ^{+} 
Σ^{} × Σ^{} 
= 
Σ^{+} 
E_{2} × E_{2} 
= 
A_{1}+A_{2}+E_{1} 
E_{} × T_{2} 
= 
T_{1}+T_{2} 
Σ^{+} × Σ^{} 
= 
Σ^{} 
Σ × Π 
= 
Π 
E_{1} × E_{2} 
= 
B_{1}+B_{2}+E_{1} 
T_{1} × T_{1} 
= 
A_{1}+E_{}+T_{1}+T_{2} 
Σ × Δ 
= 
 Π × Π 
= 
Σ^{+}+
Σ^{}+
Δ

E_{} × E_{} 
= 
A_{1}+A_{2}+E_{} 
T_{2} × T_{2} 
= 
A_{1}+E_{}+T_{1}+T_{2} 
Δ × Δ 
= 
Σ^{+}+
Σ^{}+
Γ

Π × Δ 
= 
Π+
Φ




T_{1} × T_{2} 
= 
A_{2}+E_{}+T_{1}+T_{2} 
Caveats 
† 
Groups where there are subscripts 1, 2 and 3 (eg D_{2} and D_{2h})  here 1x2=3, 2x3=1 and 1x3=2. 
‡ 
For point groups where the principal axis is C_{2} or C_{4} (eg C_{4} and D_{2d}), E × E=A_{1}+A_{2}+B_{1}+B_{2}. 
* 
T_{d} and O_{h} 
