Induced representation

In the mathematical field of representation theory of groups, the induced representation can be used to construct a representation of the group containing it from a representation of a subgroup.

Problem

With the help of the restriction one can get a representation of a subgroup from a representation of a group . The question that now arises is that of the reverse process. Can one get a representation of the whole group from a given representation of a subgroup? It can be seen that the induced representation defined below delivers exactly what we are looking for. However, this construction is not inverse, but adjoint to the restriction. ${\displaystyle \phi }$ ${\displaystyle {\text{Res}}(\phi )}$
${\displaystyle \psi }$
${\displaystyle {\text{Ind}}(\psi )}$

definition

Let be a linear representation of Be a subgroup and the constraint. Let be a partial representation of Write for this representation. Let the vector space depend only on the left minor class of . Let be a representative system of then is a partial representation of ${\displaystyle \rho \colon G\to {\text{GL}}(V_{\rho })}$ ${\displaystyle G.}$ ${\displaystyle H}$ ${\displaystyle \rho |_{H}}$ ${\displaystyle W}$ ${\displaystyle \rho |_{H}.}$ ${\displaystyle \theta \colon H\to {\text{GL}}(W)}$ ${\displaystyle s\in G,}$ ${\displaystyle \rho (s)(W)}$ ${\displaystyle sH}$ ${\displaystyle s}$ ${\displaystyle R}$ ${\displaystyle G/H,}$ ${\displaystyle \textstyle \sum _{r\in R}\rho (r)(W)}$ ${\displaystyle V_{\rho }.}$

A representation of in is called induced by the representation of in if There is a system of representatives of as above and for each ${\displaystyle \rho }$ ${\displaystyle G}$ ${\displaystyle V_{\rho }}$ ${\displaystyle \theta }$ ${\displaystyle H}$ ${\displaystyle W,}$ ${\displaystyle \textstyle V_{\rho }=\bigoplus _{r\in R}W_{r}.}$ ${\displaystyle R}$ ${\displaystyle G/H}$ ${\displaystyle W_{r}=\rho (s)(W)}$ ${\displaystyle s\in rH.}$

In other words:
the representation is induced by if each can be clearly written as, where for each ${\displaystyle (\rho ,V_{\rho })}$ ${\displaystyle (\theta ,W),}$ ${\displaystyle v\in V_{\rho }}$ ${\displaystyle \textstyle \sum _{r\in R}w_{r}}$ ${\displaystyle w_{r}\in W_{r}}$ ${\displaystyle r\in R.}$

We write or short, if there is no likelihood of confusion for the representation of induced representation of Man also often used the representation of spaces rather than the representation of picture and writes or short if the representation of induced. ${\displaystyle {\text{Ind}}_{H}^{G}(\theta )}$ ${\displaystyle {\text{Ind}}(\theta )}$ ${\displaystyle \theta }$ ${\displaystyle H}$ ${\displaystyle G.}$ ${\displaystyle V={\text{Ind}}_{H}^{G}(W)}$ ${\displaystyle {\text{Ind}}(W),}$ ${\displaystyle V}$ ${\displaystyle W}$

Alternative description of the induced representation

Via the group algebra we get an alternative description of the induced representation:
Let a group, a module and a submodule of belong to the subgroup of Then is called by induced, if where operates on the first factor: for all ${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle \mathbb {C} [G]}$ ${\displaystyle W}$ ${\displaystyle \mathbb {C} [H]}$ ${\displaystyle V}$ ${\displaystyle H}$ ${\displaystyle G.}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle V=\mathbb {C} [G]\otimes _{\mathbb {C} [H]}W,}$ ${\displaystyle G}$ ${\displaystyle s\cdot (e_{t}\otimes w)=e_{st}\otimes w}$ ${\displaystyle s,t\in G,w\in W.}$

properties

The results presented in this section are presented without evidence. These can be found in.

Uniqueness and existence of the induced representation

Let there be a linear representation of a subgroup of Then there is a linear representation of which is induced by and this is unique except for isomorphism. ${\displaystyle (\theta ,W_{\theta })}$ ${\displaystyle H}$ ${\displaystyle G.}$ ${\displaystyle (\rho ,V_{\rho })}$ ${\displaystyle G,}$ ${\displaystyle (\theta ,W_{\theta })}$

Transitivity of induction

Let be a representation of For an ascending chain of groups holds ${\displaystyle W}$ ${\displaystyle H.}$
${\displaystyle H\leq G\leq K}$

${\displaystyle {\text{Ind}}_{G}^{K}({\text{Ind}}_{H}^{G}(W))\cong {\text{Ind}}_{H}^{K}(W).}$

lemma

Be of induced and is a linear representation of and is a linear map with the property that for all Then there exists a uniquely determined linear map which continues and for all true. That is, if one understands as a module, the following applies: where denotes the vector space of all homomorphisms from to . Same goes for ${\displaystyle (\rho ,V_{\rho })}$ ${\displaystyle (\theta ,W_{\theta })}$ ${\displaystyle \rho '\colon G\to {\text{GL}}(V')}$ ${\displaystyle G}$ ${\displaystyle F\colon W_{\theta }\to V'}$ ${\displaystyle F\circ \theta (t)=\rho '(t)\circ F}$ ${\displaystyle t\in G.}$ ${\displaystyle F'\colon V_{\rho }\to V',}$ ${\displaystyle F}$ ${\displaystyle F'\circ \rho (s)=\rho '(s)\circ F'}$ ${\displaystyle s\in G}$
${\displaystyle V'}$ ${\displaystyle \mathbb {C} [G]}$ ${\displaystyle {\text{Hom}}^{H}(W_{\theta },V')\cong {\text{Hom}}^{G}(V_{\rho },V'),}$ ${\displaystyle {\text{Hom}}^{G}(V_{\rho },V')}$ ${\displaystyle \mathbb {C} [G]}$ ${\displaystyle V_{\rho }}$ ${\displaystyle V'}$ ${\displaystyle {\text{Hom}}^{H}(W_{\theta },V').}$

Induction on class functions

As with representations, we can also obtain a class function on the large group from class functions on a subgroup via so-called induction.
Be a class function , define the function on by ${\displaystyle \varphi }$ ${\displaystyle H.}$ ${\displaystyle \varphi '}$ ${\displaystyle G}$

${\displaystyle \varphi '(s)={\frac {1}{|H|}}\sum _{t\in G \atop t^{-1}st\in H}^{}\varphi (t^{-1}st).}$

We say is induced by and writing or ${\displaystyle \varphi '}$ ${\displaystyle \varphi }$ ${\displaystyle {\text{Ind}}_{H}^{G}(\varphi )=\varphi '}$ ${\displaystyle {\text{Ind}}(\varphi )=\varphi '.}$

Proposition

The function is a class function on If the character of a representation of is, then is the character of the induced representation of ${\displaystyle {\text{Ind}}(\varphi )}$ ${\displaystyle G.}$ ${\displaystyle \varphi }$ ${\displaystyle W}$ ${\displaystyle H}$ ${\displaystyle {\text{Ind}}(\varphi )}$ ${\displaystyle {\text{Ind}}(W)}$ ${\displaystyle G.}$

lemma

If a class function is open and a class function is open: ${\displaystyle \psi }$ ${\displaystyle H}$ ${\displaystyle \varphi }$ ${\displaystyle G,}$

${\displaystyle {\text{Ind}}(\psi \cdot {\text{Res}}\varphi )=({\text{Ind}}\psi )\cdot \varphi .}$

sentence

Let be the representation of and be the corresponding characters induced by the representation of the subgroup . Let be a representative system of For each : ${\displaystyle (\rho ,V_{\rho })}$ ${\displaystyle (\theta ,W_{\theta })}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle \chi _{\rho },\chi _{\theta }}$ ${\displaystyle R}$ ${\displaystyle G/H.}$ ${\displaystyle t\in G}$

${\displaystyle \chi _{\rho }(t)=\sum _{r\in R, \atop r^{-1}tr\in H}^{}\chi _{\theta }(r^{-1}tr)={\frac {1}{|H|}}\sum _{s\in G, \atop s^{-1}ts\in H}^{}\chi _{\theta }(s^{-1}ts).}$

Frobenius reciprocity

→ Main article : Frobenius reciprocity

The Frobenius reciprocity says, on the one hand, that the images and are adjoint to one another. If, on the other hand, we consider an irreducible representation of and be an irreducible representation of, then with Frobenius reciprocity we also get that in is contained as often as in ${\displaystyle {\text{Res}}}$ ${\displaystyle {\text{Ind}}}$ ${\displaystyle W}$ ${\displaystyle H}$ ${\displaystyle V}$ ${\displaystyle G,}$ ${\displaystyle W}$ ${\displaystyle {\text{Res}}(V)}$ ${\displaystyle {\text{Ind}}(W)}$ ${\displaystyle V.}$

Then be and be is true ${\displaystyle \psi \in \mathbb {C} _{\text{class}}(H)}$ ${\displaystyle \varphi \in \mathbb {C} _{\text{class}}(G),}$

${\displaystyle \langle \psi ,{\text{Res}}\varphi \rangle _{H}=\langle {\text{Ind}}\psi ,\varphi \rangle _{G}}$

The statement also applies to the scalar product .

Mackey's criterion

→ Main article : Mackey's criterion

The induced representation is irreducible if and only if the following conditions are met: ${\displaystyle V={\text{Ind}}_{H}^{G}(W)}$

• ${\displaystyle W}$ is irreducible.
• For each , the two representations and of disjoint. ${\displaystyle s\in G\setminus H}$ ${\displaystyle \rho ^{s}}$ ${\displaystyle {\text{Res}}_{s}(\rho )}$ ${\displaystyle H_{s}}$

Applications to special groups

In this section some applications of the theory presented so far to normal subgroups and to a special group, the semi-direct product of a subgroup with an Abelian normal divisor, are presented.

Proposition

Let be a normal subgroup of the group and be an irreducible representation of Then: ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle \rho \colon G\to {\text{GL}}(V)}$ ${\displaystyle G.}$

• Either there is a real subgroup of which contains and an irreducible representation of which induces ${\displaystyle H}$ ${\displaystyle G,}$ ${\displaystyle A}$ ${\displaystyle \eta }$ ${\displaystyle H,}$ ${\displaystyle \rho }$
• or the restriction of to is isotypic. ${\displaystyle \rho }$ ${\displaystyle A}$

If Abelian, the second point of the above proposition is equivalent to that there is a homothety for everyone ${\displaystyle A}$ ${\displaystyle \rho (a)}$ ${\displaystyle a\in A.}$

We also get the following

Corollary

Be an Abelian normal subgroup of and any irreducible representation of whether the index of in true then: Is an Abelian subgroup of (not necessarily normal), so is no longer in general , however, still applies ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle \tau }$ ${\displaystyle G.}$ ${\displaystyle (G:A)}$ ${\displaystyle A}$ ${\displaystyle G.}$
${\displaystyle {\text{deg}}(\tau )|(G:A).}$
${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle {\text{deg}}(\tau )|(G:A),}$ ${\displaystyle {\text{deg}}(\tau )\leq (G:A).}$

In the following we show how all irreducible representations of a group, which are semi-direct products of an Abelian normal divisor and a subgroup, are classified . ${\displaystyle G,}$ ${\displaystyle A\vartriangleleft G}$ ${\displaystyle H\leq G}$

Be in the following and subsets of the group where is normal. In the following we assume that is abelian, and the semi-direct product of and thus . Now we classify the irreducible representations of such a group by showing that the irreducible representations of can be constructed from certain subgroups of . This is the "small group" method of Wigner and Mackey. ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle G,}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle A,}$ ${\displaystyle G=A\rtimes H}$
${\displaystyle G,}$ ${\displaystyle G}$ ${\displaystyle H}$

Since Abelian is to have the irreducible representations of degree and its characters form a group , the group operated on by for Be a representative system of the railway from in Each was This is a subset of Be the corresponding subgroup of Then we extend the function on from in which we, for putting. This means that there is a class function on Da for all one can show that there is also a group homomorphism from to . It is therefore a representation of the degree that corresponds to its own character. Let now be an irreducible representation of Then one gets an irreducible representation of in which one connects with the canonical projection . Finally we build the tensor product of and and get an irreducible representation of. To show the classification now we consider the representation of which is induced by. This gives us the following result: ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle 1}$ ${\displaystyle \mathrm {X} ={\text{Hom}}(A,\mathbb {C} ^{\times }).}$ ${\displaystyle G}$ ${\displaystyle \mathrm {X} }$ ${\displaystyle (s\chi )(a)=\chi (s^{-1}as)}$ ${\displaystyle s\in G,\chi \in \mathrm {X} ,a\in A.}$
${\displaystyle (\chi _{j})_{j\in \mathrm {X} /H}}$ ${\displaystyle H}$ ${\displaystyle \mathrm {X} .}$ ${\displaystyle j\in \mathrm {X} /H}$ ${\displaystyle H_{j}=\{t\in H\mid t\chi _{j}=\chi _{j}\}.}$ ${\displaystyle H.}$ ${\displaystyle G_{j}=A\cdot H_{j}}$ ${\displaystyle G.}$ ${\displaystyle \chi _{j}}$ ${\displaystyle G_{j}}$ ${\displaystyle \chi _{j}(at)=\chi _{j}(a)}$ ${\displaystyle a\in A,t\in H_{j}}$
${\displaystyle \chi _{j}}$ ${\displaystyle G_{j}.}$ ${\displaystyle t\chi _{j}=\chi _{j}}$ ${\displaystyle t\in H_{j},}$ ${\displaystyle \chi _{j}}$ ${\displaystyle G_{j}}$ ${\displaystyle \mathbb {C} ^{\times }}$ ${\displaystyle G_{j}}$ ${\displaystyle 1,}$
${\displaystyle \rho }$ ${\displaystyle H_{j}.}$ ${\displaystyle {\tilde {\rho }}}$ ${\displaystyle G_{j},}$ ${\displaystyle \rho }$ ${\displaystyle G_{j}\to H_{j}}$ ${\displaystyle \chi _{j}}$ ${\displaystyle {\tilde {\rho }}}$ ${\displaystyle \chi _{j}\otimes {\tilde {\rho }}}$ ${\displaystyle G_{j}.}$
${\displaystyle \theta _{j,\rho }}$ ${\displaystyle G,}$ ${\displaystyle \chi _{j}\otimes {\tilde {\rho }}}$

Proposition

• ${\displaystyle \theta _{j,\rho }}$ is irreducible.
• If and are isomorphic, then is and is isomorphic to ${\displaystyle \theta _{j,\rho }}$ ${\displaystyle \theta _{j',\rho '}}$ ${\displaystyle j=j'}$ ${\displaystyle \rho }$ ${\displaystyle \rho '.}$
• Every irreducible representation of is isomorphic to one of the ${\displaystyle G}$ ${\displaystyle \theta _{j,\rho }.}$

For the proof of the proposition, among other things, Mackey's criterion and a conclusion from Frobenius reciprocity are required. More details can be found in.
That is, we have classified all irreducible representations in the group . ${\displaystyle G=A\rtimes H}$

Artin's theorem

sentence

Let be a family of subsets of a finite group Let be the homomorphism defined by the family of Then the following properties are equivalent: ${\displaystyle X}$ ${\displaystyle G.}$ ${\displaystyle {\text{Ind}}\colon \bigoplus _{H\in X}{\mathcal {R}}(H)\to {\mathcal {R}}(G)}$ ${\displaystyle {\text{Ind}}_{H}^{G},\,\,H\in X.}$

• The coke of is finite. ${\displaystyle {\text{Ind}}:\bigoplus _{H\in X}{\mathcal {R}}(H)\to {\mathcal {R}}(G)}$
• ${\displaystyle G}$is the union of the conjugates of the subgroups belonging to it, ie ${\displaystyle X}$ ${\displaystyle G=\bigcup _{H\in X \atop s\in G}sHs^{-1}.}$

Since the group is finite, the first point can be reformulated as follows: ${\displaystyle {\mathcal {R}}(G)}$

• For every character from there exist virtual characters and an integer so ${\displaystyle \chi }$ ${\displaystyle G}$ ${\displaystyle \chi _{H}\in {\mathcal {R}}(H),\,H\in X}$ ${\displaystyle d\geq 1,}$ ${\displaystyle d\cdot \chi =\sum _{H\in X}{\text{Ind}}_{H}^{G}(\chi _{H}).}$

The same applies to the rings and there ${\displaystyle R(H)}$ ${\displaystyle R(G),}$ ${\displaystyle R(G)\cong {\mathcal {R}}(G).}$

This theorem is proven in

Corollary

Each character of is a rational linear combination of characters induced by characters of cyclic subsets of . ${\displaystyle G}$ ${\displaystyle G}$

This follows immediately from Artin's theorem, since it is the union of all conjugates of its cyclic subsets. ${\displaystyle G}$

Induced representations for compact groups

If there is a closed subgroup of finite index in the compact group , the definition of the induced representation can be adopted as for finite groups. However, the induced representation can also be defined more generally, so that the definition is also valid if the index of in is not finite. Be to a unitary representation of the completed sub-group The steadily induced representation is defined as follows: With we denote the Hilbert space of all measurable, square-integrable functions with the property that for all is the standard and the presentation is given by law Translation: The induced representation then again a unitary representation. Since is compact, the induced representation is broken down into the direct sum of irreducible representations of. It is true that all irreducible representations belonging to the same isotype occur with the multiplicity that corresponds. If there is a representation of then there is a canonical isomorphism ${\displaystyle H}$ ${\displaystyle G}$
${\displaystyle H}$ ${\displaystyle G}$
${\displaystyle (\eta ,V_{\eta })}$ ${\displaystyle H.}$ ${\displaystyle {\text{Ind}}_{H}^{G}(\eta )=(I,V_{I})}$
${\displaystyle V_{I}}$ ${\displaystyle \Phi \colon G\to V_{\eta }}$ ${\displaystyle \Phi (ls)=\eta (l)\Phi (s)}$ ${\displaystyle l\in H,s\in G.}$ ${\displaystyle ||\Phi ||_{G}={\text{sup}}_{s\in G}||\Phi (s)||}$ ${\displaystyle I}$ ${\displaystyle I(s)\Phi (k)=\Phi (ks).}$

${\displaystyle G}$ ${\displaystyle G.}$ ${\displaystyle {\text{dim}}({\text{Hom}}_{G}(V_{\eta },V_{I}))}$
${\displaystyle (\rho ,V_{\rho })}$ ${\displaystyle G,}$

${\displaystyle T\colon {\text{Hom}}_{G}(V_{\rho },I_{H}^{G}(\eta ))\to {\text{Hom}}_{H}(V_{\rho }|_{H},V_{\eta })=\langle V_{\eta },V_{I}\rangle _{G}.}$

The Frobenius reciprocity is carried over to compact groups with the modified definition of the scalar product and the bilinear form, whereby the theorem applies here to quadratically integrable functions instead of to class functions and the subgroup must be closed. ${\displaystyle G}$ ${\displaystyle H}$

literature

1. ^ Jean-Pierre Serre: Linear Representations of Finite Groups. Springer Verlag, New York 1977, ISBN 0-387-90190-6
2. ^ William Fulton, Joe Harris: Representation Theory A First Course. Springer-Verlag, New York 1991, ISBN 0-387-97527-6
3. ↑ Serre, op. Cit.
4. ↑ Serre, op. Cit.