Induction Functor

In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" representation of that extends the given one. Since it is often easier to find representations of the smaller group than of '','' 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

Let be a finite group and any subgroup of . Furthermore let be a representation of . Let be the index of in and let be a full set of representatives in of the left cosets in . The induced representation can be thought of as acting on the following space:

:$W=\bigoplus_^n g_i V.$

Here each is an isomorphic copy of the vector space ''V'' whose elements are written as with . For each ''g'' in and each ''g_i'' there is an ''h_i'' in and ''j''(''i'') in such that . (This is just another way of saying that is a full set of representatives.) Via the induced representation acts on as follows:

:$g\cdot\sum_^n g_i v_i=\sum_^n g_ \pi(h_i) v_i$

where $v_i \in V$ for each ''i''.

Alternatively, one can construct induced representations using the tensor product: any ''K-''linear representation $\pi$ of the group ''H'' can be viewed as a module ''V'' over the group ring ''K'' 'H'' We can then define

:\operatorname_H^G\pi= K otimes_ V.

This latter formula can also be used to define for any group and subgroup , without requiring any finiteness.

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 is a subgroup of the group , then every -linear representation of can be viewed as a -linear representation of ; this is known as the restriction of to and denoted by . In the case of finite groups and finite-dimensional representations, the Frobenius reciprocity theorem states that, given representations of and of , the space of -equivariant linear maps from to has the same dimension over ''K'' as that of -equivariant linear maps from to .

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 $(\sigma,V)$ is a representation of ''H'' and $(\operatorname(\sigma),\hat)$ is the representation of ''G'' induced by $\sigma$, then there exists a -equivariant linear map $j:V\to\hat$ with the following property: given any representation of and -equivariant linear map $f:V\to W$, there is a unique -equivariant linear map $\hat: \hat\to W$ with $\hatj=f$. In other words, $\hat$ is the unique map making the following diagram commute:Thm. 2.1 from

The Frobenius formula states that if is the character of the representation , given by , then the character of the induced representation is given by

: $\psi(g) = \sum_ \widehat\left(x^gx \right),$

where the sum is taken over a system of representatives of the left cosets of in and

:$\widehat (k) = \begin \chi(k) & \text k \in H \\ 0 & \text\end$

Analytic

If is a locally compact topological group (possibly infinite) and is a closed subgroup then there is a common analytic construction of the induced representation. Let be a continuous unitary representation of into a Hilbert space ''V''. We can then let:

:$\operatorname_H^G\pi= \left\.$

Here means: the space ''G''/''H'' carries a suitable invariant measure, and since the norm of 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 acts on the induced representation space by translation, that is, for ''g,x''∈''G'' and .

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:

:$\operatorname_H^G\pi= \left \.$

Here are the modular functions of and 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:

:$\operatorname_H^G\pi= \left\.$

Note that if is compact then Ind and ind are the same functor.

Geometric

Suppose is a topological group and is a closed subgroup of . Also, suppose is a representation of over the vector space . Then acts on the product as follows:
:$g.(g',x)=(gg',x)$
where and are elements of and is an element of .

Define on the equivalence relation

:$(g,x) \sim (gh,\pi(h^)(x)) \texth\in H.$

Denote the equivalence class of $(g,x)$ by ,x/math>. Note that this equivalence relation is invariant under the action of ; consequently, acts on . The latter is a vector bundle over the quotient space with as the structure group and as the fiber. Let be the space of sections $\phi : G/H \to (G \times V)/ \! \sim$ of this vector bundle. This is the vector space underlying the induced representation . The group acts on a section $\phi : G/H \to \mathcal L_W$ given by gH \mapsto ,\phi_g/math> as follows:
:(g\cdot \phi)(g'H)= ',\phi_\ \text g,g'\in G.

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

*Restricted representation
* Nonlinear realization
*Frobenius character formula

Notes

References

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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

Group theory