orthogonality relations

orthogonality relations

First

orthogonality relations

:
Let

ρ

α

:

G

→

V

α

and

ρ

β

:

G

→

V

β

be

irreducible representations

of a

finite group

G

over the field

ℂ

. Then

1

|

G

|

⁢

∑

g

∈

G

ρ

i

⁢

j

(

α

)

⁢

(

g

)

¯

⁢

ρ

k

⁢

l

(

β

)

⁢

(

g

)

=

δ

α

⁢

β

⁢

δ

i

⁢

k

⁢

δ

j

⁢

l

dim

⁡

V

α

.

We have the following useful corollary.
Let

χ

1

,

χ

2

be

characters

of

representations

V

1

,

V

2

of a finite group

G

over a field

k

of

characteristic

0

. Then

(

χ

1

,

χ

2

)

=

1

|

G

|

⁢

∑

g

∈

G

χ

1

⁢

(

g

)

¯

⁢

χ

2

⁢

(

g

)

=

dim

⁡

(

Hom

⁢

(

V

1

,

V

2

)

)

.

Proof.

First of all, consider the special case where

V

=

k

with the trivial action
of the group. Then

Hom

G

⁢

(

k

,

V

2

)

≅

V

2

G

, the

fixed points

.
On the other hand, consider the map

ϕ

=

1

|

G

|

⁢

∑

g

∈

G

g

:

V

2

→

V

2

(with the sum in

End

⁢

(

V

2

)

). Clearly,
the image of this map is contained in

V

2

G

, and it is the

identity

restricted
to

V

2

G

. Thus, it is a

projection

with image

V

2

G

. Now, the rank of
a projection (over a field of characteristic 0) is its trace. Thus,

dim

k

⁡

Hom

G

⁢

(

k

,

V

2

)

=

dim

⁡

V

2

G

=

tr

⁢

(

ϕ

)

=

1

|

G

|

⁢

∑

χ

2

⁢

(

g

)

which is exactly the orthogonality formula
for

V

1

=

k

.

Now, in general,

Hom

⁢

(

V

1

,

V

2

)

≅

V

1

*

⊗

V

2

is
a representation, and

Hom

G

⁢

(

V

1

,

v

2

)

=

(

Hom

⁢

(

V

1

,

V

2

)

)

G

. Since

χ

V

1

*

⊗

V

2

=

χ

1

¯

⁢

χ

2

,

dim

k

⁡

Hom

G

⁢

(

V

1

,

V

2

)

=

dim

k

⁡

(

Hom

⁢

(

V

1

,

V

2

)

)

G

=

∑

g

∈

G

χ

1

¯

⁢

χ

2

which is exactly the

relation

we desired.
∎

In particular, if

V

1

,

V

2

irreducible

, by Schur’s Lemma

Hom

⁢

(

V

1

,

V

2

)

=

{

D

V

1

≅

V

2

0

V

1

≇

V

2

where

D

is a

division algebra

. In particular, non-isomorphic irreducible
representations have

orthogonal

characters. Thus, for any representation

V

,
the

multiplicities

n

i

in the unique

decomposition

of

V

into the

direct sum

(

http://planetmath.org/DirectSum

)
of irreducibles

V

≅

V

1

⊕

n

1

⊕

⋯

⊕

V

m

⊕

n

m

where

V

i

ranges over irreducible representations of

G

over

k

, can be
determined in terms of the character

inner product

:

n

i

=

(

ψ

,

χ

i

)

(

χ

i

,

χ

i

)

where

ψ

is the character of

V

and

χ

i

the character of

V

i

.
In particular, representations over a field of characteristic zero are determined by their character. Note: This is not true over fields of positive
characteristic.

If the field

k

is

algebraically closed

,
the only finite division algebra over

k

is

k

itself, so
the characters of irreducible representations form an

orthonormal basis

for
the

vector space

of

class functions

with respect to this inner product.
Since

(

χ

i

,

χ

i

)

=

1

for all irreducibles, the multiplicity formula
above reduces to

n

i

=

(

ψ

,

χ

i

)

.

Second orthogonality relations

:
We assume now that

k

is algebraically closed.
Let

g

,

g

′

be elements of a finite group

G

.
Then

∑

χ

χ

⁢

(

g

)

⁢

χ

⁢

(

g

′

)

¯

=

{

|

C

G

⁢

(

g

1

)

|

g

∼

g

′

0

g

≁

g

′

where the sum is over the characters of irreducible representations, and

C

G

⁢

(

g

)

is the

centralizer

of

g

.

Proof.

Let

χ

1

,

…

,

χ

n

be the characters of the irreducible representations,
and let

g

1

,

…

,

g

n

be representatives of the

conjugacy classes

.

Let

A

be the matrix whose

i

⁢

j

th entry is

|

G

:

C

G

(

g

j

)

|

⁢

(

χ

i

⁢

(

g

j

)

¯

)

.
By first orthogonality,

A

⁢

A

*

=

|

G

|

⁢

I

(here

*

denotes

conjugate transpose

),
where

I

is the

identity matrix

. Since

left inverses

(

http://planetmath.org/MatrixInverse

) are right ,

A

*

⁢

A

=

|

G

|

⁢

I

. Thus,

|

G

:

C

G

(

g

i

)

|

|

G

:

C

G

(

g

k

)

|

⁢

∑

j

=

1

n

χ

j

⁢

(

g

i

)

⁢

χ

j

⁢

(

g

k

)

¯

=

|

G

|

⁢

δ

i

⁢

k

.

Replacing

g

i

or

g

k

with any conjuagate will not change the expression above.
thus, if our two elements are not

conjugate

, we obtain that

∑

χ

χ

⁢

(

g

)

⁢

χ

⁢

(

g

′

)

¯

=

0

.
On the other hand, if

g

∼

g

′

, then

i

=

k

in the sum above, which

reduced

to the expression
we desired.
∎

A special case of this result, applied to

1

is that

|

G

|

=

∑

χ

χ

⁢

(

1

)

2

, that is, the sum of the squares of the

dimensions

(

http://planetmath.org/Dimension

) of the irreducible representations of any finite group is the order of the group.

Title

orthogonality relations

Canonical name

OrthogonalityRelations

Date of creation

2013-03-22 13:21:27

Last modified on

2013-03-22 13:21:27

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

16

Author

mhale (572)

Entry type

Theorem

Classification

msc 20C15

Generated on Fri Feb 9 19:21:46 2018 by

LaTeXML