Editing Induced representation (section)

===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>