Induced representation

Induction extends a representation of a subgroup to the whole group; Frobenius reciprocity computes its multiplicities.

In the compact picture of a principal-series group representation of the Lorentz spinor double cover, functions on SU(2) have a fixed right-torus character

e^{- in ϑ}

. Expanding into spin matrix coefficients leaves one right weight in each allowed spin, giving the displayed multiplicity-one rotation decomposition. The boost action couples these spaces. The central element acts by

(- 1)^{n}

; even

n

yields integer rotation spins and a group representation of the connected Lorentz group.

Conjugate subgroup representation ( $^{g} V$ )

 0  0

For an

H

-representation

V

and

g \in G

, the conjugate representation

^{g} V

g H g^{- 1}

is defined by

(g h g^{- 1}) v = h v

. This transports the acting subgroup; it is distinct from conjugating complex matrix entries. It occurs in the Mackey restriction formula.

Character formula for an induced representation

 0  0

For

H \leq G

and an

H

-character

χ

(Ind_{H}^{G} χ) (g) = \frac{1}{∣ H ∣} \sum_{x \in G x^{- 1} gx \in H} χ (x^{- 1} gx) .

(1)

Induction from an abelian normal subgroup with a free character orbit

 0  0

Let

N ◃ G

be abelian and let a linear character

θ

N

have stabilizer exactly

N

under conjugation by

G

. Then

Ind_{N}^{G} θ

is irreducible of degree

[G : N]

. Its value is zero outside

N

, while on

n \in N

it is the sum of the characters in the

G

-orbit of

θ

Irreducible characters of the affine semidirect product of orders eleven and five

 0  0

For

G = C_{11} ⋊ C_{5}

with faithful action, the abelianization

G / C_{11} ≅ C_{5}

supplies five linear characters. The two orbits of size five on the nontrivial characters of

C_{11}

induce two irreducible characters of degree five. Their squared degrees satisfy

5 \cdot 1^{2} + 2 \cdot 5^{2} = 55,

(1)

so these seven characters are all the irreducible characters of

G

Tensor identity for induced representations

 0  0

For a

G

-representation

W

and an

H

-representation

V

Ind_{H}^{G} (Res_{H}^{G} W \otimes V) ≅ W \otimes Ind_{H}^{G} V .

(1)

On group-algebra tensors the isomorphism sends

x \otimes (w \otimes v)

x w \otimes (x \otimes v)

Frobenius reciprocity

 0  0

For

H \leq G

, an

H

-representation

V

, and a

G

-representation

W

Hom_{G} (Ind_{H}^{G} V, W) ≅ Hom_{H} (V, Res_{H}^{G} W) .

(1)

Equivalently, induction and restriction are adjoint for character inner products.

Mackey theory

 0  0

Mackey theory studies how induced representations behave under restriction and conjugation of subgroups.

Mackey restriction formula

 0  0

For

H, K \leq G

and a

K

-representation

V

Res_{H}^{G} Ind_{K}^{G} V ≅ ⨁_{x \in H \ G / K} Ind_{H \cap x K x^{- 1}}^{H} Res_{H \cap x K x^{- 1}}^{x K x^{- 1}} (^{x} V),

(1)

where

^{x} V (x k x^{- 1}) = V (k)

To prove the formula, write induction as

C [G] \otimes_{C [K]} V

and decompose the

(H, K)

-bimodule

C [G] = ⨁_{x \in H \ G / K} C [H x K] .

(2)

For

L_{x} = H \cap x K x^{- 1}

, the map

C [H] \otimes_{C [L_{x}]}^{x} V ⟶ C [H x K] \otimes_{C [K]} V, h \otimes v ⟼ h x \otimes v,

(3)

is a well-defined

H

-equivariant isomorphism. Taking the direct sum over the double cosets proves the result.

Mackey irreducibility criterion

 0  0

For an irreducible

H

-representation

V

, the induced representation

Ind_{H}^{G} V

is irreducible exactly when, for every nonidentity double coset representative

x \in H \ G / H

, the restrictions of

V

and its

x

-conjugate to

H \cap x H x^{- 1}

have inner product zero.

Bruhat decomposition of SL2 over a finite field

 0  0

For the upper-triangular subgroup

B

S L_{2} (F_{p})

and

w = (01 - 1 0),

(1)

the two double cosets are

B

and

BwB

, and

B \cap wB w^{- 1} = T

, the diagonal subgroup.

Mackey inner product for the finite SL2 principal series

 0  0

For one-dimensional representations

θ, φ

of the upper-triangular subgroup

B \leq S L_{2} (F_{p})

⟨ Ind_{B}^{G} θ, Ind_{B}^{G} φ ⟩_{G} = 1_{θ = φ} + 1_{θ^{w} ∣_{T} = φ ∣_{T}},

(1)

where

θ^{w} (t) = θ (w^{- 1} tw)

Irreducible principal series of finite SL2

 0  0

Every linear character of the upper-triangular subgroup

B \leq S L_{2} (F_{q})

with

q \geq 4

is obtained from a multiplicative character

θ

F_{q}^{\times}

. Its induction to

S L_{2} (F_{q})

is irreducible exactly when

θ^{2} \neq = 1

Square orbits of additive characters of a finite field

 0  0

The diagonal subgroup of

S L_{2} (F_{p})

conjugates

u (x)

u (a^{2} x)

. It therefore has two orbits, each of size

(p - 1) /2

, on the nontrivial characters

χ (v \cdot)

of the unipotent subgroup: one indexed by squares and one by nonsquares.

Induction from the diagonal subgroup of upper-triangular SL2

 0  0

B = U ⋊ T

is the upper-triangular subgroup of

S L_{2} (F_{p})

and

θ

is one-dimensional on

T

, then

Ind_{T}^{B} θ

(1)

is the direct sum of three pairwise nonisomorphic irreducibles. Their restrictions to

U

have character supports given by the trivial character, the square orbit, and the nonsquare orbit, so their dimensions are

1, (p - 1) /2, (p - 1) /2

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Induced representation is a concept from representation theory in mathematics, particularly in the study of group theory. It allows one to construct a representation of a larger group from a representation of a subgroup. To understand induced representations, consider the following key ideas: 1. **Groups and Representations**: A group is a mathematical structure consisting of a set of elements equipped with an operation that satisfies certain axioms (closure, associativity, identity, and invertibility).

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