# Direct Products of Irreducible Representations

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 non-degenerate (in most cases A or B). This process is embodied in an equation.

 Non-degenerate A × A = A A × E1 = E1 B × B = A A × E2 = E2 A × B = B B × E1 = E2 A × E = E B × E2 = E1 B × E = E A × T = T B × T = T
 Gerade/Ungerade Prime/double-prime Subscripts on A or B† g × g = g ’ × ’ = ’ 1 × 1 = 1 u × u = g ’’ × ’’ = ’ 2 × 2 = 1 u × g = u ’ × ’’ = ’’ 1 × 2 = 2
 2-fold Degenerate‡ 3-fold Degenerate* Infinite rotations E1 × E1 = A1+A2+E2 E × T1 = T1+T2 Σ+ × Σ+ = Σ+ Σ- × Σ- = Σ+ E2 × E2 = A1+A2+E1 E × T2 = T1+T2 Σ+ × Σ- = Σ- Σ × Π = Π E1 × E2 = B1+B2+E1 T1 × T1 = A1+E+T1+T2 Σ × Δ = Π × Π = Σ++ Σ-+ Δ E × E = A1+A2+E T2 × T2 = A1+E+T1+T2 Δ × Δ = Σ++ Σ-+ Γ Π × Δ = Π+ Φ T1 × T2 = A2+E+T1+T2
 Caveats † Groups where there are subscripts 1, 2 and 3 (eg D2 and D2h) - here 1x2=3, 2x3=1 and 1x3=2. ‡ For point groups where the principal axis is C2 or C4 (eg C4 and D2d), E × E=A1+A2+B1+B2. * Td and Oh
 Revised: © University of Newcastle upon Tyne, UK