decomposition of a module using orthogonal idempotents

decomposition of a module using orthogonal idempotents

Let

K

be a field and let

G

be a finite abelian group. For simplicity, we will assume that the

characteristic

of

K

does not divide the order of

G

. Let

φ

1

,

…

,

φ

n

be a complete set (up to equivalence) of distinct

irreducible

(

http://planetmath.org/GroupRepresentation

) (linear) representations of

G

over

K

, so that

φ

i

is a

homomorphism

:

φ

i

:

G

⟶

GL

⁡

(

n

i

,

K

)

where

n

i

is the degree of the representation

φ

i

and

∑

i

n

i

=

|

G

|

. Let

χ

1

,

…

,

χ

n

be the irreducible

characters

attached to the

φ

i

, i.e. the function

χ

i

:

G

→

K

is defined by

χ

i

⁢

(

g

)

=

Trace

⁢

(

φ

i

⁢

(

g

)

)

.

Notice, however, that in general the map

χ

i

is not a homomorphism from the group into either the additive or

multiplicative group

of

K

. We define a system of

primitive

orthogonal idempotents

of the

group ring

K

⁢

[

G

]

, one for each

χ

i

, by:

𝟏

χ

i

=

1

|

G

|

⁢

∑

g

∈

G

χ

i

⁢

(

g

-

1

)

⁢

g

∈

K

⁢

[

G

]

so that

∑

i

𝟏

χ

i

=

1

∈

K

and

𝟏

χ

i

⋅

𝟏

χ

⁢

j

=

δ

i

⁢

j

where

δ

i

⁢

j

is the

Kronecker delta

function. We define the

χ

i

component

of

K

⁢

[

G

]

to be the ideal

K

⁢

[

G

]

χ

i

=

𝟏

χ

i

⋅

K

⁢

[

G

]

. Notice that

V

i

=

K

⁢

[

G

]

χ

i

is a finite dimensional

K

-

vector space

, on which

G

acts. Thus, the representation of

G

afforded by the

K

⁢

[

G

]

-module

V

i

, call it

φ

, must be one of the representations

φ

j

defined above. Comparing the trace, one concludes that

φ

=

φ

i

and

V

i

=

K

⁢

[

G

]

χ

i

is a vector space of

dimension

n

i

. In particular, there is a

decomposition

:

K

⁢

[

G

]

=

⊕

χ

K

⁢

[

G

]

χ

.

If

k

∈

K

⁢

[

G

]

then by the previous decomposition, we can write:

k

=

∑

χ

k

χ

where

k

χ

∈

K

⁢

[

G

]

χ

. Notice that the representations

φ

i

can be retrieved as:

φ

i

:

G

⟶

GL

⁢

(

K

⁢

[

G

]

χ

i

)

.

Lemma

.

Let

M

be a

K

⁢

[

G

]

-module and define

submodules

M

χ

=

1

χ

⋅

M

, for each irreducible character

χ

. Then:

1.

There is a decomposition

M

=

⊕

χ

M

χ

.

2.

The group

K

⁢

[

G

]

acts on

M

χ

via

K

⁢

[

G

]

χ

. In other words, if

k

∈

K

⁢

[

G

]

, with

k

=

∑

χ

k

χ

then:

k

⋅

m

=

k

χ

⋅

m

,

for all

⁢

m

∈

M

χ

.

3.

The representation

φ

of

G

afforded by the

K

-vector space

M

χ

i

is, up to equivalence, a number of copies of

φ

i

, i.e.

φ

=

φ

i

⊕

…

⊕

φ

i

=

φ

i

⊕

r

for some integer

r

≥

0

. In other words,

M

χ

i

is the submodule consisting of the sum of all

K

⁢

[

G

]

-submodules of

M

isomorphic

to

K

⁢

[

G

]

χ

i

.

4.

Suppose that

M

,

N

and

R

are

K

⁢

[

G

]

-modules which fit in the

short exact sequence

:

0

⟶

R

⟶

M

⟶

N

⟶

0

where every map above is a

K

⁢

[

G

]

-

module homomorphism

, i.e. each map is a

K

-homomorphism which is

compatible

with the action of

G

. Then, the

exact sequence

above yields an exact sequence of

χ

components:

0

⟶

R

χ

⟶

M

χ

⟶

N

χ

⟶

0

for every irreducible character

χ

.

Title

decomposition of a module using orthogonal idempotents

Canonical name

DecompositionOfAModuleUsingOrthogonalIdempotents

Date of creation

2013-03-22 15:12:22

Last modified on

2013-03-22 15:12:22

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

9

Author

alozano (2414)

Entry type

Application

Classification

msc 13C05

Classification

msc 16S34

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