Editing Induced representation (section)

====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>