Induced Representation

If a subgroup H of G has a group representation phi:H×W->W, then there is a unique induced representation of G on a vector space V. The original space W is contained in V, and in fact,

 V= direct sum _(sigma in G/H)sigmaW,

where sigmaW is a copy of W. The induced representation on V is denoted Ind_H^G.

Alternatively, the induced representation is the CG-module

 Ind_H^G=CG tensor _(CH)W.

Also, it can be viewed as W-valued functions on G which commute with the H action.


The induced representation is also determined by its universal property:

 Hom_H(W,Res U)=Hom_G(IndW,U),

where U is any representation of G. Also, the induced representation satisfies the following formulas.

1. Ind direct sum W_i= direct sum IndW_i.

2. U tensor IndW=Ind(Res(U) tensor W) for any group representation U.

3. Ind_H^G(W)=Ind_K^G(Ind_H^KW) when H<=K<=G.

Some of the group characters of G can be calculated from the group characters of H, as induced representations, using Frobenius reciprocity. Artin's reciprocity theorem says that the induced representations of cyclic subgroups of a finite group G 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.

See also

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.

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