Induced representation
In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the (whole) group G itself. Given a representation of H, the induced representation is, in a sense, the "most general" representation of G that extends the given one. Since it is often easier to find representations of the smaller group H than of G, the operation of forming induced representations is an important tool to construct new representations.
Induced representations were initially defined by Frobenius, for linear representations of finite groups. The idea is by no means limited to the case of finite groups, but the theory in that case is particularly well-behaved.
Let G be a finite group and H any subgroup of G. Furthermore let (π, V) be a representation of H. Let n = [G : H] be the index of H in G and let g1, ..., gn be a full set of representatives in G of the left cosets in G/H. The induced representation IndG
H π can be thought of as acting on the following space:
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