aarjaneiro/planetmath huggingface dataset:
character

character

Let

ρ

:

G

⟶

GL

⁡

(

V

)

be a

finite dimensional

representation

of a group

G

(i.e.,

V

is a finite dimensional

vector space

over its scalar field

K

). The

character

of

ρ

is the function

χ

V

:

G

⟶

K

defined by

χ

V

⁢

(

g

)

:=

Tr

⁡

(

ρ

⁢

(

g

)

)

where

Tr

is the trace function.

Properties:

•

χ

V

⁢

(

g

)

=

χ

V

⁢

(

h

)

if

g

is

conjugate

to

h

in

G

. (Equivalently, a character is a

class function

on

G

.)

•

If

G

is finite, the characters of the

irreducible representations

of

G

over the

complex numbers

form a basis of the vector space of all class functions on

G

(with pointwise addition and

scalar multiplication

).

•

Over the complex numbers, the characters of the irreducible representations of

G

are orthonormal under the

inner product

(

χ

1

,

χ

2

)

:=

1

|

G

|

⁢

∑

g

∈

G

χ

1

⁢

(

g

)

¯

⁢

χ

2

⁢

(

g

)

Title

character

Canonical name

Character

Date of creation

2013-03-22 12:17:54

Last modified on

2013-03-22 12:17:54

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 20C99

Generated on Fri Feb 9 19:21:58 2018 by

LaTeXML
example of matrix representations

example of matrix representations

Sign

representation

of

S

n

Let

G

=

S

n

the

n

-th

symmetric group

, and consider

X

⁢

(

σ

)

=

sign

⁢

(

σ

)

where

σ

is any

permutation

in

S

n

.
That is,

sign

⁢

(

σ

)

=

1

when

σ

is an

even permutation

, and

sign

⁢

(

σ

)

=

-

1

when

σ

is an

odd permutation

.

The

function

X

is a

group homomorphism

between

S

n

and

G

⁢

L

⁢

(

ℂ

)

=

ℂ

∖

{

0

}

(that is

invertible matrices

of size

1

×

1

, which is the set of non-zero

complex numbers

). And thus we say that

ℂ

∖

{

0

}

carries a representation of the symmetric group.

Defining representation of

S

n

For each

σ

∈

S

n

, let

X

:

S

n

→

G

⁢

L

n

⁢

(

ℂ

)

the function given by

X

⁢

(

σ

)

=

(

a

i

⁢

j

)

n

×

n

where

(

a

i

⁢

j

)

is the

permutation matrix

given by

a

i

⁢

j

=

{

1

if

⁢

σ

⁢

(

i

)

=

j

0

if

⁢

σ

⁢

(

i

)

≠

j

Such matrices are called permutation matrices because they are obtained permuting the colums of the

identity matrix

. The function so defined is then a group homomorphism, and thus

G

⁢

L

n

⁢

(

ℂ

)

carries a representation of the symmetric group.

Title

example of

matrix representations

Canonical name

ExampleOfMatrixRepresentations

Date of creation

2013-03-22 14:53:31

Last modified on

2013-03-22 14:53:31

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Example

Classification

msc 20C99

Generated on Fri Feb 9 19:22:06 2018 by

LaTeXML
Frobenius reciprocity

Frobenius reciprocity

Let

V

be a finite-dimensional representation of a

finite group

G

, and let

W

be a

representation

of a

subgroup

H

⊂

G

. Then the

characters

of

V

and

W

satisfy the

inner product

relation

(

χ

Ind

⁡

(

W

)

,

χ

V

)

=

(

χ

W

,

χ

Res

⁡

(

V

)

)

where

Ind

and

Res

denote the

induced representation

Ind

H

G

and the

restriction representation

Res

H

G

.

The

Frobenius reciprocity

theorem is often given in the stronger form which states that

Res

and

Ind

are

adjoint functors

between the

category

of

G

–modules and the category of

H

–modules:

Hom

H

⁡

(

W

,

Res

⁡

(

V

)

)

=

Hom

G

⁡

(

Ind

⁡

(

W

)

,

V

)

,

or, equivalently

V

⊗

Ind

⁡

(

W

)

=

Ind

⁡

(

Res

⁡

(

V

)

⊗

W

)

.

Title

Frobenius reciprocity

Canonical name

FrobeniusReciprocity

Date of creation

2013-03-22 12:17:51

Last modified on

2013-03-22 12:17:51

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Theorem

Classification

msc 20C99

Generated on Fri Feb 9 19:22:08 2018 by

LaTeXML
G-module

G

-module

Let

V

a

vector space

over some field

K

(usually

K

=

ℚ

or

K

=

ℂ

). Let

G

be a group which acts on

V

. This means that there is an

operation

ψ

:

G

×

V

→

V

such that

1.

g

⁢

v

∈

V

.

2.

g

⁢

(

h

⁢

v

)

=

(

g

⁢

h

)

⁢

v

3.

e

⁢

v

=

v

where

g

⁢

v

stands for

ψ

⁢

(

g

,

v

)

and

e

is the

identity element

of

G

.

If in

addition

,

g

⁢

(

c

⁢

v

+

d

⁢

w

)

=

c

⁢

(

g

⁢

v

)

+

d

⁢

(

g

⁢

w

)

for any

g

∈

G

,

v

,

w

∈

V

,

c

,

d

∈

K

, we say that

V

is a

G

-module.
This is

equivalent

with the

existence

of a

group representation

from

G

to

G

⁢

L

⁢

(

V

)

.

Title

G

-module

Canonical name

Gmodule

Date of creation

2013-03-22 14:57:53

Last modified on

2013-03-22 14:57:53

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 20C99

Related topic

GroupRepresentation

Related topic

Group

Generated on Fri Feb 9 19:22:10 2018 by

LaTeXML
group representation

group representation

Let

G

be a group, and let

V

be a

vector space

. A

representation

of

G

in

V

is a

group homomorphism

ρ

:

G

→

GL

⁡

(

V

)

from

G

to the

general linear group

GL

⁡

(

V

)

of

invertible linear transformations

of

V

.

Equivalently, a representation of

G

is a vector space

V

which is a

G

-module, that is, a (left) module over the

group ring

ℤ

⁢

[

G

]

. The equivalence is achieved by assigning to each

homomorphism

ρ

:

G

→

GL

⁡

(

V

)

the module

structure

whose

scalar multiplication

is defined by

g

⋅

v

:=

(

ρ

⁢

(

g

)

)

⁢

(

v

)

, and extending linearly. Note that, although technically a

group representation

is a homomorphism such as

ρ

, most authors invariably denote a representation using the underlying vector space

V

, with the homomorphism being understood from

context

, in much the same way that vector spaces themselves are usually described as sets with the corresponding

binary operations

being understood from context.

Special kinds of representations

(preserving all notation from above)

A representation is

faithful

if either of the following

equivalent

conditions is satisfied:

•

ρ

:

G

→

GL

⁡

(

V

)

is

injective

,

•

V

is a faithful left

ℤ

⁢

[

G

]

–module.

A

subrepresentation

of

V

is a

subspace

W

of

V

which is a left

ℤ

⁢

[

G

]

–

submodule

of

V

;
such a subspace is sometimes called a

G

-

invariant subspace

of

V

. Equivalently, a subrepresentation of

V

is a subspace

W

of

V

with the property that

(

ρ

⁢

(

g

)

)

⁢

(

w

)

∈

W

⁢

for all

⁢

g

∈

G

⁢

and

⁢

w

∈

W

.

A representation

V

is called

irreducible

if it has no subrepresentations other than itself and the

zero module

.

Title

group representation

Canonical name

GroupRepresentation

Date of creation

2013-03-22 12:13:30

Last modified on

2013-03-22 12:13:30

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 20C99

Synonym

representation

Related topic

GeneralLinearGroup

Defines

subrepresentation

Defines

irreducible

Defines

faithful

Generated on Fri Feb 9 19:22:12 2018 by

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induced representation

induced representation

Let

G

be a group,

H

⊂

G

a

subgroup

, and

V

a

representation

of

H

, considered as a

ℤ

⁢

[

H

]

–module. The

induced representation

of

ρ

on

G

, denoted

Ind

H

G

⁡

(

V

)

, is the

ℤ

⁢

[

G

]

–module whose underlying

vector space

is the

direct sum

⊕

σ

∈

G

/

H

σ

⁢

V

of formal

translates

of

V

by

left cosets

σ

in

G

/

H

, and whose

multiplication

operation

is defined by choosing a set

{

g

σ

}

σ

∈

G

/

H

of coset representatives and setting

g

⁢

(

σ

⁢

v

)

:=

τ

⁢

(

h

⁢

v

)

where

τ

is the unique left coset of

G

/

H

containing

g

⋅

g

σ

(i.e., such that

g

⋅

g

σ

=

g

τ

⋅

h

for some

h

∈

H

).

One easily verifies that the representation

Ind

H

G

⁡

(

V

)

is independent of the choice of coset representatives

{

g

σ

}

.

Title

induced representation

Canonical name

InducedRepresentation

Date of creation

2013-03-22 12:17:33

Last modified on

2013-03-22 12:17:33

Owner

djao (24)

Last modified by

djao (24)

Numerical id

4

Author

djao (24)

Entry type

Definition

Classification

msc 20C99

Generated on Fri Feb 9 19:22:15 2018 by

LaTeXML
matrix representation

matrix representation

A

matrix representation

of a group

G

is a

group homomorphism

between

G

and

G

⁢

L

n

⁢

(

ℂ

)

, that is, a

function

X

:

G

→

G

⁢

L

n

⁢

(

ℂ

)

such that

•

X

⁢

(

g

⁢

h

)

=

X

⁢

(

g

)

⁢

X

⁢

(

h

)

,

•

X

⁢

(

e

)

=

I

Notice that this definition is

equivalent

to the

group representation

definition when the

vector space

V

is finite dimensional over

ℂ

. The parameter

n

(or in the case of a group representation, the

dimension

of

V

) is called the

degree

of the representation.

References

1

Bruce E. Sagan.

The

Symmetric Group

: Representations, Combinatorial

Algorithms

and Symmetric Functions.

2a Ed. 2000. Graduate Texts in Mathematics. Springer.

Title

matrix representation

Canonical name

MatrixRepresentation

Date of creation

2013-03-22 14:53:56

Last modified on

2013-03-22 14:53:56

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Definition

Classification

msc 20C99

Related topic

PermutationRepresentation

Generated on Fri Feb 9 19:22:35 2018 by

LaTeXML
proof of Frobenius reciprocity

proof of Frobenius reciprocity

We prove the slightly more general result

Theorem 0.1

.

If

G

is a

finite group

with

subgroup

H

,

α

a

class function

on

H

and

β

a class function on

G

, then

⟨

α

↑

H

G

,

β

⟩

G

=

⟨

α

,

β

↓

H

G

⟩

H

Here we use

↑

H

G

to refer to the

induction

(

http://planetmath.org/InducedRepresentation

) to

G

of a class function on

H

, and

↓

H

G

to refer to the

restriction

(

http://planetmath.org/RestrictionRepresentation

) of a class function on

G

to one on

H

.

Proof.

⟨

α

↑

H

G

,

β

⟩

G

=

1

|

G

|

∑

g

∈

G

(

1

|

H

|

∑

t

∈

G

t

-

1

⁢

g

⁢

t

∈

H

α

(

t

-

1

g

t

)

)

β

⁢

(

g

)

¯

=

1

|

G

|

⁢

|

H

|

∑

t

∈

G

(

∑

g

∈

G

t

-

1

⁢

g

⁢

t

∈

H

α

(

t

-

1

g

t

)

)

β

⁢

(

g

)

¯

Since

β

is a class function, this is the same as

1

|

G

|

⁢

|

H

|

⁢

∑

t

∈

G

g

∈

G

t

-

1

⁢

g

⁢

t

∈

H

α

⁢

(

t

-

1

⁢

g

⁢

t

)

⁢

β

⁢

(

t

-

1

⁢

g

⁢

t

)

¯

=

1

|

G

|

⁢

|

H

|

⁢

∑

h

∈

H

∑

t

∈

G

g

∈

G

t

-

1

⁢

g

⁢

t

=

h

α

⁢

(

h

)

⁢

β

⁢

(

h

)

¯

Clearly for every

h

∈

H

,

t

∈

G

there is a unique

g

∈

G

with

t

-

1

⁢

g

⁢

t

=

h

, so every element of

H

is counted

|

G

|

times by the sum. Thus the sum is equal to

|

G

|

|

G

|

⁢

|

H

|

∑

h

∈

H

α

(

h

)

β

⁢

(

h

)

¯

=

1

|

H

|

∑

h

∈

H

α

(

h

)

β

⁢

(

h

)

¯

=

⟨

α

,

β

↓

H

G

⟩

H

∎

Title

proof of Frobenius reciprocity

Canonical name

ProofOfFrobeniusReciprocity

Date of creation

2013-03-22 18:36:23

Last modified on

2013-03-22 18:36:23

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

5

Author

rm50 (10146)

Entry type

Proof

Classification

msc 20C99

Generated on Fri Feb 9 19:22:38 2018 by

LaTeXML
proof that dimension of complex irreducible representation divides order of group

proof that dimension of complex irreducible representation divides order of group

Theorem

Let

G

be a

finite group

and

V

an

irreducible

complex representation of finite

dimension

d

. Then

d

divides

|

G

|

.

Proof: Given any

α

in the

group ring

of

G

(denoted

ℤ

⁢

G

) we may define a

sequence

of

submodules

of

ℤ

⁢

G

(regarded as a module over

ℤ

)
by

A

i

equals the

ℤ

linear span

of

{

1

,

α

,

α

2

,

⋯

,

α

i

}

.

ℤ

⁢

G

is Noetherian as a module over

ℤ

so we
must have

A

i

=

A

i

-

1

for some

i

. Hence

α

i

may be
expressed as a

ℤ

linear combination

of lower powers of

α

. In other

α

solves a

monic polynomial

of
degree

i

with

coefficients

in

ℤ

.

Given a

conjugacy class

C

in

G

, we may set

ϕ

C

=

∑

g

∈

C

g

. Then

ϕ

C

is

central

in

ℤ

⁢

G

, as given

h

∈

G

, we have:

ϕ

C

⁢

h

=

h

⁢

∑

g

∈

C

h

-

1

⁢

g

⁢

h

=

h

⁢

∑

g

∈

C

g

=

h

⁢

ϕ

C

Hence applying

ϕ

C

to

V

induces a

ℂ

⁢

G

linear map

V

→

V

. By Schur’s lemma this must be

multiplication

by some

complex number

λ

C

. Then

λ

C

is an

algebraic
integer

as it solves the same monic polynomial as

ϕ

C

.

Also any

g

∈

G

has finite order so the map it induces on

V

must have

eigenvalues

which are

roots of unity

and hence algebraic
integers. Hence the sum of the eigenvalues,

χ

V

⁢

(

g

)

, must
also be an algebraic integer.

Now

V

is irreducible so,

|

G

|

=

∑

g

∈

G

χ

V

⁢

(

g

)

⁢

χ

V

⁢

(

g

)

*

=

∑

C

⊂

G

tr

⁢

(

ϕ

C

)

⁢

χ

V

⁢

(

C

)

*

=

d

⁢

∑

C

⊂

G

λ

C

⁢

χ

V

⁢

(

C

)

*

Therefore

|

G

|

/

d

is both

rational

and an algebraic integer. Hence
it is an

integer

and

d

divides

|

G

|

.

Title

proof that dimension of complex irreducible representation divides order of group

Canonical name

ProofThatDimensionOfComplexIrreducibleRepresentationDividesOrderOfGroup

Date of creation

2013-03-22 17:09:04

Last modified on

2013-03-22 17:09:04

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

8

Author

whm22 (2009)

Entry type

Proof

Classification

msc 20C99

Generated on Fri Feb 9 19:22:41 2018 by

LaTeXML
quotient representations

quotient representations

We assume that all

representations

(

G

-modules) are

finite-dimensional

.

Definition 1

If

N

1

and

N

2

are

G

-modules over a field

k

(i.e. representations of

G

in

N

1

and

N

2

), then a map

φ

:

N

1

→

N

2

is a

G

-map

if

φ

is

k

-linear and preserves the

G

-action, i.e. if

φ

⁢

(

σ

⋅

x

)

=

σ

⋅

φ

⁢

(

x

)

G

-maps have subrepresentations, also called

G

-

submodules

, as their kernel and image. To see this, let

φ

:

N

1

→

N

2

be a

G

-map; let

M

1

⊂

N

1

and

M

2

⊂

N

2

be the kernel and image respectively of

φ

.

M

1

is a submodule of

N

1

if it is stable under the action of

G

, but

x

∈

M

1

⇒

φ

⁢

(

σ

⋅

x

)

=

σ

⋅

φ

⁢

(

x

)

=

0

⇒

σ

⋅

x

∈

M

1

M

2

is a submodule of

N

2

if it is stable under the action of

G

, but

y

=

φ

⁢

(

x

)

∈

M

2

⇒

σ

⋅

y

=

σ

⋅

φ

⁢

(

x

)

=

φ

⁢

(

σ

⋅

x

)

⇒

σ

⋅

y

∈

M

2

Finally, we define the intuitive concept of a

quotient

G

-module. Suppose

N

′

⊂

N

is a

G

-submodule. Then

N

/

N

′

is a finite-dimensional

vector space

. We can define an action of

G

on

N

/

N

′

via

σ

⁢

(

n

+

N

′

)

=

σ

⁢

(

n

)

+

σ

⁢

(

N

′

)

=

σ

⁢

(

n

)

+

N

′

, so that

n

+

N

′

is well-defined under the action and

N

/

N

′

is a

G

-module.

Title

quotient representations

Canonical name

QuotientRepresentations

Date of creation

2013-03-22 16:37:59

Last modified on

2013-03-22 16:37:59

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Definition

Classification

msc 20C99

Generated on Fri Feb 9 19:22:44 2018 by

LaTeXML
regular representation

regular representation

Given a group

G

, the

regular representation

of

G

over a field

K

is the

representation

ρ

:

G

⟶

GL

⁡

(

K

G

)

whose underlying

vector space

K

G

is the

K

–vector space of formal

linear combinations

of elements of

G

, defined by

ρ

⁢

(

g

)

⁢

(

∑

i

=

1

n

k

i

⁢

g

i

)

:=

∑

i

=

1

n

k

i

⁢

(

g

⁢

g

i

)

for

k

i

∈

K

,

g

,

g

i

∈

G

.

Equivalently, the regular representation is the

induced representation

on

G

of the

trivial representation

on the

subgroup

{

1

}

of

G

.

Title

regular representation

Canonical name

RegularRepresentation

Date of creation

2013-03-22 12:17:40

Last modified on

2013-03-22 12:17:40

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Definition

Classification

msc 20C99

Generated on Fri Feb 9 19:22:46 2018 by

LaTeXML
representation ring

representation ring

Let

G

be a group and

k

a field. Consider the class

ℛ

=

{

X

|

X

⁢

is a representation of

⁢

G

⁢

over

⁢

k

}

and its

subclass

ℛ

f

consisting of those

representations

which are

finite-dimensional

as

vector spaces

. We consider a special representation

ℱ

=

(

V

,

⋅

)

where

V

is a fixed vector space with a basis

ℬ

which is in

bijective

correspondence with

G

. If

f

:

ℬ

→

G

is a required bijection, then we define ,,

⋅

” on basis

ℬ

by

g

⋅

b

=

g

⁢

f

⁢

(

b

)

where on the right side we have a

multiplication

in

G

. It can be shown that this gives us a well-defined representation and further more, if

X

∈

ℛ

f

, then there exists an

epimorphism

of representations

e

:

ℱ

n

→

X

for some

n

∈

ℕ

(

ℱ

is a ,,free” representation). In particular every finite-dimensional representation is a

quotient

of a

direct sum

of copies of

ℱ

. This fact shows that a

maximal

subclass

𝒳

⊂

ℛ

f

consisting of pairwise nonisomorphic representations is actually a set (note that

𝒳

is never unique). Fix such a set.

Definition.

The representation

semiring

R

k

⁢

(

G

)

¯

of

G

is defined as a triple

(

𝒳

,

+

,

⋅

)

, where

𝒳

is a maximal set of pairwise nonisomorphic representations taken from

ℛ

f

. Addition and multiplication are given by

X

+

Y

=

Z

where

Z

is a representation in

𝒳

isomorphic

to the direct sum

X

⊕

Y

and

X

⋅

Y

=

Z

′

where

Z

′

is a representation in

𝒳

isomorphic to the

tensor product

X

⊗

Y

. Note that

R

k

⁢

(

G

)

¯

is not a ring, because there are no

additive inverses

.

The

representation ring

R

k

⁢

(

G

)

is defined as the Grothendieck ring (

http://planetmath.org/GrothendieckGroup

) induced from

R

k

⁢

(

G

)

¯

. It can be shown that the definition does not depend on the choice of

𝒳

(in the sense that it always gives us naturally isomorphic rings).

It is convenient to forget about formal definition which includes the choice of

𝒳

and simply write elements of

R

k

⁢

(

G

)

¯

as isomorphism classes of representations

[

X

]

. Thus every

element

in

R

k

⁢

(

G

)

can be written as a formal

difference

[

X

]

-

[

Y

]

. And we can write

[

X

]

+

[

Y

]

=

[

X

⊕

Y

]

;

[

X

]

⁢

[

Y

]

=

[

X

⊗

Y

]

.

Title

representation ring

Canonical name

RepresentationRing

Date of creation

2013-03-22 19:19:02

Last modified on

2013-03-22 19:19:02

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

7

Author

joking (16130)

Entry type

Definition

Classification

msc 20C99

Generated on Fri Feb 9 19:22:48 2018 by

LaTeXML
representation ring vs burnside ring

representation ring vs burnside ring

Let

G

be a

finite group

and let

k

be any field. If

X

is a

G

-set, then we may consider the

vector space

V

k

⁢

(

X

)

over

k

which has

X

as a basis. In this manner

V

k

⁢

(

X

)

becomes a

representation

of

G

via action induced from

X

and linearly extended to

V

k

⁢

(

X

)

. It can be shown that

V

k

⁢

(

X

)

only depends on the isomorphism class of

X

, so we have a

well-defined

mapping

:

[

X

]

↦

[

V

k

⁢

(

X

)

]

which can be easily extended to the

function

β

:

Ω

⁢

(

G

)

→

R

k

⁢

(

G

)

;

β

⁢

(

[

X

]

)

=

[

V

k

⁢

(

X

)

]

where on the left side we have the

Burnside ring

and on the right side the

representation ring

. It can be shown, that

β

is actually a

ring homomorphism

, but in most cases it neither

injective

nor

surjective

. But the following

theorem

due to Segal gives us some

properties

of

β

:

Theorem (Segal).

Let

β

:

Ω

⁢

(

G

)

→

R

ℚ

⁢

(

G

)

be defined as above with

rationals

as the underlying field. If

G

is a

p

-group for some prime number

p

, then

β

is surjective. Furthermore

β

is an

isomorphism

if and only if

G

is cyclic.

Title

representation ring vs burnside ring

Canonical

name

RepresentationRingVsBurnsideRing

Date of creation

2013-03-22 19:19:05

Last modified on

2013-03-22 19:19:05

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 20C99

Generated on Fri Feb 9 19:22:50 2018 by

LaTeXML
representations vs modules

representations vs modules

Let

G

be a group and

k

a field. Recall that a pair

(

V

,

⋅

)

is a

representation

of

G

over

k

, if

V

is a

vector space

over

k

and

⋅

:

G

×

V

→

V

is a linear group action (compare with parent object). On the other hand we have a

group algebra

k

⁢

G

, which is a vector space over

k

with

G

as a basis and the

multiplication

is induced from the multiplication in

G

. Thus we can consider modules over

k

⁢

G

. These two

concepts

are related.

If

𝕍

=

(

V

,

⋅

)

is a representation of

G

over

k

, then define a

k

⁢

G

-module

𝕍

¯

by putting

𝕍

¯

=

V

as a vector space over

k

and the action of

k

⁢

G

on

𝕍

¯

is given by

(

∑

λ

i

⁢

g

i

)

∘

v

=

∑

λ

i

⁢

(

g

i

⋅

v

)

.

It can be easily checked that

𝕍

¯

is indeed a

k

⁢

G

-module.

Analogously if

M

is a

k

⁢

G

-module (with action denoted by ,,

∘

”), then the pair

M

¯

=

(

M

,

⋅

)

is a representation of

G

over

k

, where ,,

⋅

” is given by

g

⋅

v

=

g

∘

v

.

As a simple exercise we leave the following

proposition

to the reader:

Proposition.

Let

𝕍

be a representation of

G

over

k

and let

M

be a

k

⁢

G

-module. Then

𝕍

¯

¯

=

𝕍

;

M

¯

¯

=

M

.

This means that modules and representations are the same concept. One can generalize this even further by showing that

⋅

¯

and

⋅

¯

are both

functors

, which are (mutualy invert)

isomorphisms

of appropriate

categories

.

Therefore we can easily define such concepts as ,,

direct sum of representations

” or ,,

tensor product

of representations”, etc.

Title

representations vs modules

Canonical name

RepresentationsVsModules

Date of creation

2013-03-22 19:18:59

Last modified on

2013-03-22 19:18:59

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Definition

Classification

msc 20C99

Generated on Fri Feb 9 19:22:53 2018 by

LaTeXML
restriction representation

restriction representation

Let

ρ

:

G

⟶

GL

⁡

(

V

)

be a

representation

on a group

G

. The

restriction representation

of

ρ

to a

subgroup

H

of

G

, denoted

Res

H

G

⁡

(

V

)

, is the representation

ρ

|

H

:

H

⟶

GL

⁡

(

V

)

obtained by restricting the function

ρ

to the subset

H

⊂

G

.

Title

restriction representation

Canonical name

RestrictionRepresentation

Date of creation

2013-03-22 12:17:30

Last modified on

2013-03-22 12:17:30

Owner

djao (24)

Last modified by

djao (24)

Numerical id

4

Author

djao (24)

Entry type

Definition

Classification

msc 20C99

Generated on Fri Feb 9 19:22:55 2018 by

LaTeXML