In abstract terms
Suppose is a group, is a subgroup of , and is a linear representation of on a vector space over a field . The induced representation of is a linear representation of on a new (bigger) vector space over the same field , defined as follows.
- is defined as the -vector space of functions (under pointwise addition and scalar multiplication) satisfying for all and . Note here that while is a linear transformation of , so it makes sense to apply to . The dimension of is the product of the dimension of and the index of in .
- The action of on is defined by the following map . For , is the linear transformation that sends to the following function : for . Note that the gets multiplied on the right in order to make this a left action, because the multiplication is happening on the inside rather than the outside.
In matrix terms
Using the same notation as the previous definition, this more concrete description works when the index as well as the degree of are both finite. Suppose and has degree , with identified with via a basis, so that is now a map from to . Then, we do the following:
- Choose a left transversal of in , i.e., a set that intersects every left coset of in at exactly one point. Choose a bijection between this left transversal and , and label the coset representatives .
- If is the induced representation, we define , for any , as the following matrix. We first begin by viewing it as a block matrix with each block a matrix. Define . For , the block is defined as if and otherwise. Note that the matrix is a block monomial matrix in the sense that, as a block matrix, every row has exactly one nonzero block and every column has exactly one nonzero block.
- Induction of representations is transitive: If are groups then .
Relation with induced class functions
Relation with restriction of representations
- Frobenius reciprocity relates the representation-theoretic operations of induction and restriction
Particular cases of induction of representations
- Induced representation from regular representation of subgroup is regular representation of group
- Induced representation from trivial representation on normal subgroup factors through regular representation of quotient group
- Induced representation from trivial representation of subgroup is permutation representation for action on coset space