 TOPICS  # Induced Representation

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)

where is a copy of . The induced representation on is denoted .

Alternatively, the induced representation is the CG-module (2)

Also, it can be viewed as -valued functions on which commute with the action. (3)

The induced representation is also determined by its universal property: (4)

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.

Artin's Reciprocity Theorem, Frobenius Reciprocity, Group, Group Representation, Group Representation Restriction, Irreducible Representation, Vector Space, Vector Space Tensor Product

This entry contributed by Todd Rowland

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Rowland, Todd. "Induced Representation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/InducedRepresentation.html