Induced representation
From Groupprops
Contents
Definition
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.
Facts
Iteration
- 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