Editing Induced representation

{{Short description|Process of extending a representation of a subgroup to the parent group}}

In [[group theory]], the '''induced representation''' is a [[group representation|representation of a group]], {{mvar|G}}, which is constructed using a known representation of a [[subgroup]] {{mvar|H}}. Given a representation of {{mvar|H}}'','' the induced representation is, in a sense, the "most general" representation of {{mvar|G}} that extends the given one. Since it is often easier to find representations of the smaller group {{mvar|H}} than of {{mvar|G}}'','' the operation of forming induced representations is an important tool to construct new representations''.''

Induced representations were initially defined by [[Ferdinand Georg Frobenius|Frobenius]], for [[linear representation]]s of [[finite group]]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===
{{See also|Representation theory of finite groups#The induced representation}}
Let {{mvar|G}} be a finite group and {{mvar|H}} any subgroup of {{mvar|G}}.  Furthermore let {{math|(''π'', ''V'')}} be a representation of {{mvar|H}}.  Let {{math|''n'' {{=}} [''G'' : ''H'']}} be the [[Index of a subgroup|index]] of {{mvar|H}} in {{mvar|G}} and let {{math|''g''<sub>1</sub>, ..., ''g<sub>n</sub>''}} be a full set of representatives in {{mvar|G}} of the [[Coset|left cosets]] in {{math|''G''/''H''}}. The induced representation {{math|Ind{{su|b=''H''|p=''G''}} ''π''}} can be thought of as acting on the following space:

:<math>W=\bigoplus_{i=1}^n g_i V.</math>

Here each {{math|''g<sub>i</sub>&thinsp;V''}} is an [[isomorphic]] copy of the vector space ''V'' whose elements are written as {{math|''g<sub>i</sub>&thinsp;v''}} with {{math|''v'' &isin; ''V''}}. For each ''g'' in {{mvar|G}} and each ''g<sub>i</sub>'' there is an ''h<sub>i</sub>'' in {{mvar|H}} and ''j''(''i'') in {1, ..., ''n''} such that  {{math|1=''g'' ''g<sub>i</sub>'' = ''g''<sub>''j''(''i'')</sub> ''h<sub>i</sub>''}} . (This is just another way of saying that {{math|''g''<sub>1</sub>, ..., ''g<sub>n</sub>''}} is a full set of representatives.) Via the induced representation {{mvar|G}} acts on {{mvar|W}} as follows:

:<math> g\cdot\sum_{i=1}^n g_i v_i=\sum_{i=1}^n g_{j(i)} \pi(h_i) v_i</math>

where <math> v_i \in V</math> for each ''i''.

Alternatively, one can construct induced representations by [[Change of rings#Extension of scalars|extension of scalars]]: any ''K-''linear representation <math>\pi</math>  of the group ''H'' can be viewed as a [[Module (mathematics)|module]] ''V'' over the [[group ring]] ''K''[''H'']. We can then define

:<math>\operatorname{Ind}_H^G\pi= K[G]\otimes_{K[H]} V.</math>

This latter formula can also be used to define {{math|Ind{{su|b=''H''|p=''G''}} ''π''}} for any group {{mvar|G}} and subgroup {{mvar|H}}, without requiring any finiteness.<ref>Brown, Cohomology of Groups, III.5</ref>

====Examples====
For any group, the induced representation of the [[trivial representation]] of the [[trivial subgroup]] is the right [[regular representation]].  More generally the induced representation of the [[trivial representation]] 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 matrix|monomial matrices]].  Some groups have the property that all of their irreducible representations are monomial, the so-called [[monomial group]]s.

====Properties====
If {{mvar|H}} is a subgroup of the group {{mvar|G}}, then every {{mvar|K}}-linear representation {{mvar|ρ}} of {{mvar|G}} can be viewed as a {{mvar|K}}-linear representation of {{mvar|H}}; this is known as the [[restricted representation|restriction]] of {{mvar|ρ}} to {{mvar|H}} and denoted by {{math|Res(&rho;)}}. In the case of finite groups and finite-dimensional representations, the '''[[Frobenius reciprocity|Frobenius reciprocity theorem]]''' states that, given representations {{mvar|σ}} of {{mvar|H}} and {{mvar|ρ}} of {{mvar|G}}, the space of {{mvar|H}}-[[equivariant]] linear maps from {{mvar|σ}} to {{math|Res(''ρ'')}} has the same dimension over ''K'' as that of {{mvar|G}}-equivariant linear maps from {{math|Ind(''σ'')}} to {{mvar|ρ}}.<ref>{{Cite book|url=https://archive.org/details/linearrepresenta1977serr|title=Linear representations of finite groups|last=Serre|first=Jean-Pierre|date=1926–1977|publisher=Springer-Verlag|isbn=0387901906|location=New York|oclc=2202385|url-access=registration}}</ref>

The [[universal property]] of the induced representation, which is also valid for infinite groups, is equivalent to the adjunction asserted in the reciprocity theorem. If <math>(\sigma,V)</math> is a representation of ''H'' and <math>(\operatorname{Ind}(\sigma),\hat{V})</math> is the representation of ''G'' induced by <math>\sigma</math>, then there exists a {{mvar|H}}-equivariant linear map <math>j:V\to\hat{V}</math> with the following property: given any representation {{math|(ρ,''W'')}} of {{mvar|G}} and {{mvar|H}}-equivariant linear map <math>f:V\to W</math>, there is a unique {{mvar|G}}-equivariant linear map <math>\hat{f}: \hat{V}\to W</math> with <math>\hat{f}j=f</math>. In other words, <math>\hat{f}</math> is the unique map making the following [[Commutative diagram|diagram commute]]:<ref>Thm. 2.1 from {{cite web|url=https://canvas.harvard.edu/files/1502130/download?download_frd=1&verifier=Ms6OjK8y2wqN6WKri4v4vrjnxsgOMYOzEb5KoyRt|title=Math 221 : Algebra notes Nov. 20|last=Miller|first=Alison|archive-url=https://archive.today/20180801043646/https://canvas.harvard.edu/files/1502130/download?download_frd=1&verifier=Ms6OjK8y2wqN6WKri4v4vrjnxsgOMYOzEb5KoyRt|archive-date=2018-08-01|url-status=live|access-date=2018-08-01}}</ref>

[[Image:Universal property of the induced representation 2.svg|200px|class=skin-invert]]

The '''Frobenius formula''' states that if {{mvar|χ}} is the [[character theory|character]] of the representation {{mvar|σ}}, given by {{math|''χ''(''h'') {{=}} Tr ''σ''(''h'')}}, then the character {{mvar|ψ}} of the induced representation is given by

: <math>\psi(g) = \sum_{x\in G / H} \widehat{\chi}\left(x^{-1}gx \right),</math>

where the sum is taken over a system of representatives of the left cosets of {{mvar|H}} in {{mvar|G}} and

:<math> \widehat{\chi} (k) = \begin{cases} \chi(k) & \text{if } k \in H \\ 0 & \text{otherwise}\end{cases}</math>

===Analytic===
If {{mvar|G}} is a [[locally compact]] [[topological group]] (possibly infinite) and {{mvar|H}} is a [[Closed set|closed]] [[subgroup]] then there is a common analytic construction of the induced representation.  Let {{math|(''π'', ''V'')}} be a [[continuous function|continuous]] unitary representation of {{mvar|H}} into a [[Hilbert space]] ''V''.  We can then let:

:<math>\operatorname{Ind}_H^G\pi= \left\{\phi\colon G \to V \ : \ \phi(gh^{-1})=\pi(h)\phi(g)\text{ for all }h\in H,\; g\in G \text{ and } \ \phi \in L^2(G/H)\right\}.</math>

Here {{math|&phi;&isin;''L''<sup>2</sup>(''G''/''H'')}} means: the space ''G''/''H'' carries a suitable invariant measure, and since the norm of  {{math|&phi;(''g'')}}  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 {{mvar|G}} acts on the induced representation space by translation, that is, {{math|1=(''g''.&phi;)(''x'')=&phi;(''g''<sup>−1</sup>''x'')}} for ''g,x''&isin;''G'' and {{math|&phi;&isin;Ind{{su|b=''H''|p=''G''}} ''π''}}.

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:

:<math>\operatorname{Ind}_H^G\pi= \left \{\phi \colon G \to V \ : \  \phi(gh^{-1})=\Delta_G^{-\frac{1}{2}}(h)\Delta_H^{\frac{1}{2}}(h)\pi(h)\phi(g)  \text{ and }  \phi\in L^2(G/H) \right \}.</math>

Here {{math|Δ<sub>''G''</sub>, Δ<sub>''H''</sub>}} are the [[Haar measure#The modular function|modular functions]] of {{mvar|G}} and {{mvar|H}} respectively.  With the addition of the ''normalizing'' factors this induction [[functor]] takes [[unitary representation]]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:

:<math>\operatorname{ind}_H^G\pi= \left\{\phi\colon G \to V \ : \ \phi(gh^{-1})=\pi(h)\phi(g) \text{ and } \phi \text{ has compact support mod } H \right\}.</math>

Note that if {{math|''G''/''H''}} is compact then Ind and ind are the same functor.

===Geometric===
Suppose {{mvar|G}} is a [[topological group]] and {{mvar|H}} is a [[Closed set|closed]] [[subgroup]] of {{mvar|G}}. Also, suppose {{mvar|&pi;}} is a representation of {{mvar|H}} over the vector space {{math|''V''}}. Then {{mvar|G}}  [[Group action (mathematics)|acts]] on the product {{math|''G'' × ''V''}}  as follows:
:<math>g.(g',x)=(gg',x)</math>
where {{math|''g''}} and {{math|''g''′}} are elements of {{mvar|G}} and {{math|''x''}} is an element of {{math|''V''}}.

Define on  {{math|''G'' × ''V''}}  the [[equivalence relation]]

:<math>(g,x) \sim (gh,\pi(h^{-1})(x)) \text{ for all }h\in H.</math>

Denote the equivalence class of <math>(g,x)</math> by <math>[g,x]</math>. Note that this equivalence relation is invariant under the action of {{mvar|G}}; consequently, {{mvar|G}}  acts on {{math|(''G'' × ''V'')/~}} . The latter is a [[vector bundle]] over the [[Quotient space (topology)|quotient space]] {{math|''G''/''H''}} with {{math|''H''}} as the [[structure group]] and {{math|''V''}} as the fiber. Let {{math|''W''}} be the space of sections <math>\phi : G/H \to (G \times V)/ \! \sim</math> of this vector bundle. This is the vector space underlying the induced representation <math>\operatorname{Ind}_H^G\pi : W \to \mathcal L_W</math>. The group {{mvar|G}} acts on a section <math>\phi : G/H \to (G \times V)/ \! \sim</math> given by <math>gH \mapsto [g,\phi_g]</math> as follows:
:<math>(g\cdot \phi)(g'H)=[g',\phi_{g^{-1}g'}] \ \text{ for } g,g'\in G.</math>

=== Systems of imprimitivity ===
In the case of [[unitary representation]]s of locally compact groups, the induction construction can be formulated in terms of [[system of imprimitivity|systems of imprimitivity]].

== Lie theory ==
In [[Lie theory]], an extremely important example is [[parabolic induction]]: inducing representations of a [[reductive group]] from representations of its [[Borel subgroup|parabolic subgroups]]. This leads, via the [[philosophy of cusp forms]], to the [[Langlands program]].

== See also ==
*[[Restricted representation]]
*[[Nonlinear realization]]
*[[Frobenius character formula]]
*[[Frobenius reciprocity]], an important result that relates induced representations to [[restricted representation|restricted representations]]

== Notes ==
{{reflist}}

== References ==
*{{cite book|last=Alperin|first=J. L.|author-link=J. L. Alperin|author2=Rowen B. Bell|title=Groups and Representations|url=https://archive.org/details/groupsrepresenta00jlal|url-access=limited|publisher=[[Springer-Verlag]]|year=1995|isbn=0-387-94526-1|pages=[https://archive.org/details/groupsrepresenta00jlal/page/n174 164]–177 }}
*{{cite book |last = Folland | first = G. B. |author-link=Gerald Folland| title= A Course in Abstract Harmonic Analysis |url=https://archive.org/details/courseabstractha00foll_844|url-access=limited| publisher = [[CRC Press]] | year = 1995 | isbn = 0-8493-8490-7 | pages = [https://archive.org/details/courseabstractha00foll_844/page/n158 151]–200 }}
*{{cite book|last1 = Kaniuth | first1 = E. | last2 = Taylor | first2 = K. | title= Induced Representations of Locally Compact Groups | publisher = [[Cambridge University Press]] | year = 2013 |isbn=9780521762267}}
* {{Citation | last = Mackey | first = G. W. |author-link=George Mackey| title = On induced representations of groups | journal = American Journal of Mathematics | volume = 73 | issue = 3 | pages = 576–592 | year = 1951 | doi=10.2307/2372309|jstor=2372309}}
* {{Citation | last = Mackey | first = G. W. | title = Induced representations of locally compact groups I  | journal = Annals of Mathematics | volume = 55 | issue = 1 | pages = 101–139 | year = 1952 | doi=10.2307/1969423|jstor=1969423}}
* {{Citation | last = Mackey | first = G. W. | title = Induced representations of locally compact groups II : the Frobenius reciprocity theorem  | journal = Annals of Mathematics | volume = 58 | issue = 2 | pages = 193–220 | year = 1953 | doi=10.2307/1969786|jstor=1969786}}
*{{Cite book|url=https://www.springer.com/gp/book/9781461412304|title= Representing Finite Groups, A Semimsimple Introduction |last=Sengupta |first=Ambar N.|author-link=Ambar Sengupta |publisher=Springer | year= 2012|chapter = Chapter 8: Induced Representations|isbn=978-1-4614-1232-8|oclc=875741967}} 
[[Category:Representation theory of groups]]
*{{Cite book|url=https://www.springer.com/gp/book/9780387493855|title=Geometry of Quantum Theory|last= Varadarajan |first=V. S. |author-link=Veeravalli S. Varadarajan |publisher=Springer | year= 2007|chapter = Chapter VI: Systems of Impritivity|isbn=978-0-387-49385-5}} 

[[Category:Group theory]]