Suppose is a representation of and is a subspace that is invariant under (so is a subrepresentation of ), but not necessarily under all of . For , the subspace depends only on the coset , and is a representation of the conjugate subgroup via , (conjugation by ).
The simplest scenario is when as a direct sum of vector spaces, where ranges over a set of coset representatives. In this case, , and the entire representation is completely determined by the -representation .
Definition 15.2.1 (Induced representation). A representation of , together with a subspace invariant under (i.e. is a subrepresentation of ), is said to be induced by if We write . Concretely: fix coset representatives of (where ). Then every can be written uniquely as with , and acts by permuting the summands: if for some , then .
Theorem 15.2.1 (Existence and uniqueness of induced representations). Given a representation of , the induced representation exists and is unique up to isomorphism of -representations.
Proof.Uniqueness. Suppose is a representation of and is -invariant with . For any , there is a unique such that , i.e. , and necessarily . This determines the -action on uniquely from the -action on .
Existence. Build where the are formal symbols (i.e. consists of copies of as a vector space). Define to act by: given , set . This is well-defined (independent of the choice of coset representatives) and defines a group homomorphism .∎
15.2.1 Examples
Example 15.2.1 (Permutation representation from induction). The permutation representation of on the coset space equals , where is the trivial representation of . Indeed, the permutation representation has basis , the element spans a 1-dimensional subspace fixed by , and , so .
Example 15.2.2 (Regular representation from induction). The regular representation of is , where is the regular representation of . Here .
15.2.2 Properties of Induction
Proposition 15.2.1 (Basic properties). Let , and let be representations of and a representation of .
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In particular, , where is the trivial representation of . The right side is tensored with the permutation representation on .
Proof of (ii). Write . Then where is invariant under , and as an -representation it is .∎
Note. In general, .
15.2.3 Character of an Induced Representation
Proposition 15.2.2 (Character formula for induction). Let be a representation of . Then
Proof. Choose coset representatives of . The element maps to where is determined by . If , this contributes nothing to the trace. If , then and acts on as composed with the identification , so the trace on is . Summing over all : Since each coset has elements, and the condition is the same for any representative of the coset, we can rewrite this as a sum over all with the normalization factor .∎