If a subgroup of has a group representation , then there is a unique induced representation of on a vector space . The original space is contained in , and in fact,
(1)
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where is a copy of . The induced representation on is denoted .
Alternatively, the induced representation is the CG-module
(2)
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Also, it can be viewed as -valued functions on which commute with the action.
(3)
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The induced representation is also determined by its universal property:
(4)
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where is any representation of . Also, the induced representation satisfies the following formulas.
1. .
2. for any group representation .
3. when .
Some of the group characters of can be calculated from the group characters of , as induced representations, using Frobenius reciprocity. Artin's reciprocity theorem says that the induced representations of cyclic subgroups of a finite group generates a lattice of finite index in the lattice of virtual characters. Brauer's theorem says that the virtual characters are generated by the induced representations from P-elementary subgroups.