Английская Википедия:Induced representation
In group theory, the induced representation is a representation of a group, Шаблон:Mvar, which is constructed using a known representation of a subgroup Шаблон:Mvar. Given a representation of Шаблон:Mvar, the induced representation is, in a sense, the "most general" representation of Шаблон:Mvar that extends the given one. Since it is often easier to find representations of the smaller group Шаблон:Mvar than of Шаблон:Mvar, the operation of forming induced representations is an important tool to construct new representations.
Induced representations were initially defined by Frobenius, for linear representations of finite groups. 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
Шаблон:See also Let Шаблон:Mvar be a finite group and Шаблон:Mvar any subgroup of Шаблон:Mvar. Furthermore let Шаблон:Math be a representation of Шаблон:Mvar. Let Шаблон:Math be the index of Шаблон:Mvar in Шаблон:Mvar and let Шаблон:Math be a full set of representatives in Шаблон:Mvar of the left cosets in Шаблон:Math. The induced representation Шаблон:Math can be thought of as acting on the following space:
- <math>W=\bigoplus_{i=1}^n g_i V.</math>
Here each Шаблон:Math is an isomorphic copy of the vector space V whose elements are written as Шаблон:Math with Шаблон:Math. For each g in Шаблон:Mvar and each gi there is an hi in Шаблон:Mvar and j(i) in {1, ..., n} such that Шаблон:Math . (This is just another way of saying that Шаблон:Math is a full set of representatives.) Via the induced representation Шаблон:Mvar acts on Шаблон:Mvar 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 extension of scalars: any K-linear representation <math>\pi</math> of the group H can be viewed as a 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 Шаблон:Math for any group Шаблон:Mvar and subgroup Шаблон:Mvar, without requiring any finiteness.[1]
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 matrices. Some groups have the property that all of their irreducible representations are monomial, the so-called monomial groups.
Properties
If Шаблон:Mvar is a subgroup of the group Шаблон:Mvar, then every Шаблон:Mvar-linear representation Шаблон:Mvar of Шаблон:Mvar can be viewed as a Шаблон:Mvar-linear representation of Шаблон:Mvar; this is known as the restriction of Шаблон:Mvar to Шаблон:Mvar and denoted by Шаблон:Math. In the case of finite groups and finite-dimensional representations, the Frobenius reciprocity theorem states that, given representations Шаблон:Mvar of Шаблон:Mvar and Шаблон:Mvar of Шаблон:Mvar, the space of Шаблон:Mvar-equivariant linear maps from Шаблон:Mvar to Шаблон:Math has the same dimension over K as that of Шаблон:Mvar-equivariant linear maps from Шаблон:Math to Шаблон:Mvar.[2]
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 Шаблон:Mvar-equivariant linear map <math>j:V\to\hat{V}</math> with the following property: given any representation Шаблон:Math of Шаблон:Mvar and Шаблон:Mvar-equivariant linear map <math>f:V\to W</math>, there is a unique Шаблон:Mvar-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 diagram commute:[3]
Файл:Universal property of the induced representation 2.svg
The Frobenius formula states that if Шаблон:Mvar is the character of the representation Шаблон:Mvar, given by Шаблон:Math, then the character Шаблон:Mvar 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 Шаблон:Mvar in Шаблон:Mvar and
- <math> \widehat{\chi} (k) = \begin{cases} \chi(k) & \text{if } k \in H \\ 0 & \text{otherwise}\end{cases}</math>
Analytic
If Шаблон:Mvar is a locally compact topological group (possibly infinite) and Шаблон:Mvar is a closed subgroup then there is a common analytic construction of the induced representation. Let Шаблон:Math be a continuous unitary representation of Шаблон:Mvar 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 Шаблон:Math means: the space G/H carries a suitable invariant measure, and since the norm of Шаблон:Math 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 Шаблон:Mvar acts on the induced representation space by translation, that is, Шаблон:Math for g,x∈G and Шаблон:Math.
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 Шаблон:Math are the modular functions of Шаблон:Mvar and Шаблон:Mvar respectively. With the addition of the normalizing factors this induction functor takes unitary representations 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 Шаблон:Math is compact then Ind and ind are the same functor.
Geometric
Suppose Шаблон:Mvar is a topological group and Шаблон:Mvar is a closed subgroup of Шаблон:Mvar. Also, suppose Шаблон:Mvar is a representation of Шаблон:Mvar over the vector space Шаблон:Math. Then Шаблон:Mvar acts on the product Шаблон:Math as follows:
- <math>g.(g',x)=(gg',x)</math>
where Шаблон:Math and Шаблон:Math are elements of Шаблон:Mvar and Шаблон:Math is an element of Шаблон:Math.
Define on Шаблон:Math 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 Шаблон:Mvar; consequently, Шаблон:Mvar acts on Шаблон:Math . The latter is a vector bundle over the quotient space Шаблон:Math with Шаблон:Math as the structure group and Шаблон:Math as the fiber. Let Шаблон:Math 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 Шаблон:Math. The group Шаблон:Mvar acts on a section <math>\phi : G/H \to \mathcal L_W</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 representations of locally compact groups, the induction construction can be formulated in terms of 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 parabolic subgroups. This leads, via the philosophy of cusp forms, to the Langlands program.
See also
Notes
References
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Cite book
- Шаблон:Cite book
- ↑ Brown, Cohomology of Groups, III.5
- ↑ Шаблон:Cite book
- ↑ Thm. 2.1 from Шаблон:Cite web
