Finite groups Schur orthogonality relations

1 finite groups

1.1 intrinsic statement
1.2 coordinates statement
1.3 example of permutation group on 3 objects
1.4 direct implications

finite groups
intrinsic statement

the space of complex-valued class functions of finite group g has natural inner product:

⟨
α
,
β
⟩

:=


1

|
g
|




∑

g
∈
g


α
(
g
)



β
(
g
)

¯




{\displaystyle \left\langle \alpha ,\beta \right\rangle :={\frac {1}{\left|g\right|}}\sum _{g\in g}\alpha (g){\overline {\beta (g)}}}

where






β
(
g
)

¯




{\displaystyle {\overline {\beta (g)}}}

means complex conjugate of value of



β


{\displaystyle \beta }

on g. respect inner product, irreducible characters form orthonormal basis space of class functions, , yields orthogonality relation rows of character table:

⟨

χ

i


,

χ

j


⟩

=


{



0



 if 

i
≠
j
,




1



 if 

i
=
j
.








{\displaystyle \left\langle \chi _{i},\chi _{j}\right\rangle ={\begin{cases}0&{\mbox{ if }}i\neq j,\\1&{\mbox{ if }}i=j.\end{cases}}}

for



g
,
h
∈
g


{\displaystyle g,h\in g}

orthogonality relation columns follows:

∑


χ

i





χ

i


(
g
)




χ

i


(
h
)

¯


=


{




|

c

g


(
g
)
|

,



 if 

g
,
h

 are conjugate 





0



 otherwise.









{\displaystyle \sum _{\chi _{i}}\chi _{i}(g){\overline {\chi _{i}(h)}}={\begin{cases}\left|c_{g}(g)\right|,&{\mbox{ if }}g,h{\mbox{ conjugate }}\\0&{\mbox{ otherwise.}}\end{cases}}}

where sum on of irreducible characters




χ

i




{\displaystyle \chi _{i}}

of g , symbol




|

c

g


(
g
)
|



{\displaystyle \left|c_{g}(g)\right|}

denotes order of centralizer of



g


{\displaystyle g}

.

the orthogonality relations can aid many computations including:

decomposing unknown character linear combination of irreducible characters;
constructing complete character table when of irreducible characters known;
finding orders of centralizers of representatives of conjugacy classes of group; and
finding order of group.

coordinates statement

let




Γ

(
λ
)


(
r

)

m
n




{\displaystyle \gamma ^{(\lambda )}(r)_{mn}}

matrix element of irreducible matrix representation




Γ

(
λ
)




{\displaystyle \gamma ^{(\lambda )}}

of finite group



g
=
{
r
}


{\displaystyle g=\{r\}}

of order |g|, i.e., g has |g| elements. since can proven matrix representation of finite group equivalent unitary representation, assume




Γ

(
λ
)




{\displaystyle \gamma ^{(\lambda )}}

unitary:

∑

n
=
1



l

λ






Γ

(
λ
)


(
r

)

n
m


∗




Γ

(
λ
)


(
r

)

n
k


=

δ

m
k




all


r
∈
g
,


{\displaystyle \sum _{n=1}^{l_{\lambda }}\;\gamma ^{(\lambda )}(r)_{nm}^{*}\;\gamma ^{(\lambda )}(r)_{nk}=\delta _{mk}\quad {\hbox{for all}}\quad r\in g,}

where




l

λ




{\displaystyle l_{\lambda }}

(finite) dimension of irreducible representation




Γ

(
λ
)




{\displaystyle \gamma ^{(\lambda )}}

.

the orthogonality relations, valid matrix elements of irreducible representations, are:

∑

r
∈
g



|

g

|





Γ

(
λ
)


(
r

)

n
m


∗




Γ

(
μ
)


(
r

)


n
′


m
′



=

δ

λ
μ



δ

n

n
′




δ

m

m
′







|

g

|



l

λ




.


{\displaystyle \sum _{r\in g}^{|g|}\;\gamma ^{(\lambda )}(r)_{nm}^{*}\;\gamma ^{(\mu )}(r)_{n m }=\delta _{\lambda \mu }\delta _{nn }\delta _{mm }{\frac {|g|}{l_{\lambda }}}.}

here




Γ

(
λ
)


(
r

)

n
m


∗




{\displaystyle \gamma ^{(\lambda )}(r)_{nm}^{*}}

complex conjugate of




Γ

(
λ
)


(
r

)

n
m





{\displaystyle \gamma ^{(\lambda )}(r)_{nm}\,}

, sum on elements of g. kronecker delta




δ

λ
μ




{\displaystyle \delta _{\lambda \mu }}

unity if matrices in same irreducible representation




Γ

(
λ
)


=

Γ

(
μ
)




{\displaystyle \gamma ^{(\lambda )}=\gamma ^{(\mu )}}

. if




Γ

(
λ
)




{\displaystyle \gamma ^{(\lambda )}}

,




Γ

(
μ
)




{\displaystyle \gamma ^{(\mu )}}

non-equivalent zero. other 2 kronecker delta s state row , column indices must equal (



n
=

n
′



{\displaystyle n=n }

,



m
=

m
′



{\displaystyle m=m }

) in order obtain non-vanishing result. theorem known great (or grand) orthogonality theorem.

every group has identity representation (all group elements mapped onto real number 1). irreducible representation. great orthogonality relations imply that

∑

r
∈
g



|

g

|





Γ

(
μ
)


(
r

)

n
m


=
0


{\displaystyle \sum _{r\in g}^{|g|}\;\gamma ^{(\mu )}(r)_{nm}=0}

for



n
,
m
=
1
,
…
,

l

μ




{\displaystyle n,m=1,\ldots ,l_{\mu }}

, irreducible representation




Γ

(
μ
)





{\displaystyle \gamma ^{(\mu )}\,}

not equal identity representation.

example of permutation group on 3 objects

the 3! permutations of 3 objects form group of order 6, commonly denoted




s

3




{\displaystyle s_{3}}

(symmetric group). group isomorphic point group




c

3
v




{\displaystyle c_{3v}}

, consisting of threefold rotation axis , 3 vertical mirror planes. groups have 2-dimensional irreducible representation (l = 2). in case of




s

3




{\displaystyle s_{3}}

1 labels representation young tableau



λ
=
[
2
,
1
]


{\displaystyle \lambda =[2,1]}

, in case of




c

3
v




{\displaystyle c_{3v}}

1 writes



λ
=
e


{\displaystyle \lambda =e}

. in both cases representation consists of following 6 real matrices, each representing single group element:

(



1


0




0


1



)





(



1


0




0


−
1



)





(



−


1
2







3

2









3

2






1
2





)





(



−


1
2




−



3

2






−



3

2






1
2





)





(



−


1
2







3

2






−



3

2




−


1
2





)





(



−


1
2




−



3

2









3

2




−


1
2





)




{\displaystyle {\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}\quad {\begin{pmatrix}1&0\\0&-1\\\end{pmatrix}}\quad {\begin{pmatrix}-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}\\{\frac {\sqrt {3}}{2}}&{\frac {1}{2}}\\\end{pmatrix}}\quad {\begin{pmatrix}-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}\\-{\frac {\sqrt {3}}{2}}&{\frac {1}{2}}\\\end{pmatrix}}\quad {\begin{pmatrix}-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}\\-{\frac {\sqrt {3}}{2}}&-{\frac {1}{2}}\\\end{pmatrix}}\quad {\begin{pmatrix}-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {\sqrt {3}}{2}}&-{\frac {1}{2}}\\\end{pmatrix}}}

the normalization of (1,1) element:

∑

r
∈
g


6



Γ
(
r

)

11


∗



Γ
(
r

)

11


=

1

2


+

1

2


+


(
−



1
2



)


2


+


(
−



1
2



)


2


+


(
−



1
2



)


2


+


(
−



1
2



)


2


=
3.


{\displaystyle \sum _{r\in g}^{6}\;\gamma (r)_{11}^{*}\;\gamma (r)_{11}=1^{2}+1^{2}+\left(-{\tfrac {1}{2}}\right)^{2}+\left(-{\tfrac {1}{2}}\right)^{2}+\left(-{\tfrac {1}{2}}\right)^{2}+\left(-{\tfrac {1}{2}}\right)^{2}=3.}

in same manner 1 can show normalization of other matrix elements: (2,2), (1,2), , (2,1). orthogonality of (1,1) , (2,2) elements:

∑

r
∈
g


6



Γ
(
r

)

11


∗



Γ
(
r

)

22


=

1

2


+
(
1
)
(
−
1
)
+

(
−



1
2



)


(



1
2



)

+

(
−



1
2



)


(



1
2



)

+


(
−



1
2



)


2


+


(
−



1
2



)


2


=
0.


{\displaystyle \sum _{r\in g}^{6}\;\gamma (r)_{11}^{*}\;\gamma (r)_{22}=1^{2}+(1)(-1)+\left(-{\tfrac {1}{2}}\right)\left({\tfrac {1}{2}}\right)+\left(-{\tfrac {1}{2}}\right)\left({\tfrac {1}{2}}\right)+\left(-{\tfrac {1}{2}}\right)^{2}+\left(-{\tfrac {1}{2}}\right)^{2}=0.}

similar relations hold orthogonality of elements (1,1) , (1,2), etc. 1 verifies in example sums of corresponding matrix elements vanish because of orthogonality of given irreducible representation identity representation.

direct implications

the trace of matrix sum of diagonal matrix elements,

tr
⁡


(


Γ
(
r
)


)


=

∑

m
=
1


l


Γ
(
r

)

m
m


.


{\displaystyle \operatorname {tr} {\big (}\gamma (r){\big )}=\sum _{m=1}^{l}\gamma (r)_{mm}.}

the collection of traces character



χ
≡
{
tr
⁡


(


Γ
(
r
)


)




|


r
∈
g
}


{\displaystyle \chi \equiv \{\operatorname {tr} {\big (}\gamma (r){\big )}\;|\;r\in g\}}

of representation. 1 writes trace of matrix in irreducible representation character




χ

(
λ
)




{\displaystyle \chi ^{(\lambda )}}

χ

(
λ
)


(
r
)
≡
tr
⁡

(

Γ

(
λ
)


(
r
)
)

.


{\displaystyle \chi ^{(\lambda )}(r)\equiv \operatorname {tr} \left(\gamma ^{(\lambda )}(r)\right).}

in notation can write several character formulas:

∑

r
∈
g



|

g

|




χ

(
λ
)


(
r

)

∗




χ

(
μ
)


(
r
)
=

δ

λ
μ



|

g

|

,


{\displaystyle \sum _{r\in g}^{|g|}\chi ^{(\lambda )}(r)^{*}\,\chi ^{(\mu )}(r)=\delta _{\lambda \mu }|g|,}

which allows check whether or not representation irreducible. (the formula means lines in character table have orthogonal vectors.) and

∑

r
∈
g



|

g

|




χ

(
λ
)


(
r

)

∗



χ
(
r
)
=

n

(
λ
)



|

g

|

,


{\displaystyle \sum _{r\in g}^{|g|}\chi ^{(\lambda )}(r)^{*}\,\chi (r)=n^{(\lambda )}|g|,}

which helps determine how irreducible representation




Γ

(
λ
)




{\displaystyle \gamma ^{(\lambda )}}

contained within reducible representation



Γ



{\displaystyle \gamma \,}

character



χ
(
r
)


{\displaystyle \chi (r)}

.

for instance, if

n

(
λ
)




|

g

|

=
96


{\displaystyle n^{(\lambda )}\,|g|=96}

and order of group is

|

g

|

=
24



{\displaystyle |g|=24\,}

then number of times




Γ

(
λ
)





{\displaystyle \gamma ^{(\lambda )}\,}

contained within given reducible representation



Γ



{\displaystyle \gamma \,}

is

n

(
λ
)


=
4

.


{\displaystyle n^{(\lambda )}=4\,.}

see character theory more group characters.