In
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the induced representation is a
representation of a group
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...
, , which is constructed using a known representation of a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
. Given a representation of '','' the induced representation is, in a sense, the "most general" representation of that extends the given one. Since it is often easier to find representations of the smaller group than of '','' the operation of forming induced representations is an important tool to construct new representations''.''
Induced representations were initially defined by
Frobenius, for
linear representation
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
s of
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
s. The idea is by no means limited to the case of finite groups, but the theory in that case is particularly well-behaved.
Constructions
Algebraic
Let be a finite group and any subgroup of . Furthermore let be a representation of . Let be the
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
of in and let be a full set of representatives in of the
left cosets in . The induced representation can be thought of as acting on the following space:
:
Here each is an
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
copy of the vector space ''V'' whose elements are written as with . For each ''g'' in and each ''g
i'' there is an ''h
i'' in and ''j''(''i'') in such that . (This is just another way of saying that is a full set of representatives.) Via the induced representation acts on as follows:
:
where
for each ''i''.
Alternatively, one can construct induced representations using the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
: any ''K-''linear representation
of the group ''H'' can be viewed as a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
''V'' over the
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
''K''
'H'' We can then define
:
This latter formula can also be used to define for any group and subgroup , without requiring any finiteness.
Examples
For any group, the induced representation of the
trivial representation In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ...
of the
trivial subgroup
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
is the right
regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation.
One distinguishes the left regular rep ...
. More generally the induced representation of the
trivial representation In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ...
of any subgroup is the permutation representation on the cosets of that subgroup.
An induced representation of a one dimensional representation is called a monomial representation, because it can be represented as
monomial matrices
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
. Some groups have the property that all of their irreducible representations are monomial, the so-called
monomial group In mathematics, in the area of algebra studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters are all monomial, that is, induced from characters of degree 1 .
In this s ...
s.
Properties
If is a subgroup of the group , then every -linear representation of can be viewed as a -linear representation of ; this is known as the
restriction
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
of to and denoted by . In the case of finite groups and finite-dimensional representations, the
Frobenius reciprocity theorem states that, given representations of and of , the space of -
equivariant
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
linear maps from to has the same dimension over ''K'' as that of -equivariant linear maps from to .
The
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
of the induced representation, which is also valid for infinite groups, is equivalent to the adjunction asserted in the reciprocity theorem. If
is a representation of ''H'' and
is the representation of ''G'' induced by
, then there exists a -equivariant linear map
with the following property: given any representation of and -equivariant linear map
, there is a unique -equivariant linear map
with
. In other words,
is the unique map making the following
diagram commute:
[Thm. 2.1 from ]
The Frobenius formula states that if is the
character
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of the representation , given by , then the character of the induced representation is given by
:
where the sum is taken over a system of representatives of the left cosets of in and
:
Analytic
If is a
locally compact topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
(possibly infinite) and is a
closed subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
then there is a common analytic construction of the induced representation. Let be a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
unitary representation of into a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''V''. We can then let:
:
Here means: the space ''G''/''H'' carries a suitable invariant measure, and since the norm of is constant on each left coset of ''H'', we can integrate the square of these norms over ''G''/''H'' and obtain a finite result. The group acts on the induced representation space by translation, that is, for ''g,x''∈''G'' and .
This construction is often modified in various ways to fit the applications needed. A common version is called normalized induction and usually uses the same notation. The definition of the representation space is as follows:
:
Here are the
modular functions of and respectively. With the addition of the ''normalizing'' factors this induction
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
takes
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
s to unitary representations.
One other variation on induction is called compact induction. This is just standard induction restricted to functions with
compact support. Formally it is denoted by ind and defined as:
:
Note that if is compact then Ind and ind are the same functor.
Geometric
Suppose is a
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
and is a
closed subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of . Also, suppose is a representation of over the vector space . Then
acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
on the product as follows:
:
where and are elements of and is an element of .
Define on the
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
:
Denote the equivalence class of
by