Direct Products of Irreducible Representations: a simple equation

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• Jon Goss
• AIMPRO

To reduce the direct product of two irredcible representations to its irreducible components, one may apply the following procedure. The procedure requires you to multiply characters together, along with the order of the group, Ω and the number of equivilent operations, n.

Let us label the irreducible representation in the product as 1 and 2. To test if the reduced form of the product contains irreducible representation 3, we sum of all the terms

n  ×  X₁  ×  X₂  ×  X₃,

where X_i are the characters of the relevant representations. Irreducible representation 3 is contained in the product if the sum is a multiple of the order, Ω.

We shall take as a worked example the product of T₂ and T₂ within the T_d point group.

	E	8C3	3C2	6σd	6S4
n	1	8	3	6	6
T2	3	0	-1	1	-1
n × T2 × T2	9	0	3	6	6

To test if this reducing this product yeilds an E irreducible representation, we mulitply further by the E characters.

× E

18

0

6

0

0

The sum of these is 24, which is equal to one times the order of the group. This means that the product contains E once. A similar analysis with the A₂ irreducible representation yields a sum of zero, indicating that the product does not reduce to contain A₂. You can check the result in the product table.