Matrix coefficient

In the mathematical field of representation theory, matrix coefficients are certain functions on the group associated to a group representation.

For example, after choosing a basis in the representation space, one can describe the representation by matrices associated with the group elements, whose individual entries are matrix coefficients in the sense of the general definition.

Definition

Be

$$


ρ
:
G
→
G
L
(
V
)


{displaystyle rho colon Gto GL(V)}

a representation of a group

$$


G


{displaystyle G}

on a

$$



C



{displaystyle mathbb {C} }

-Hilbert space

$$


V


{displaystyle V}

with scalar product

$$


⟨
⋅
,
⋅
⟩


{displaystyle langle cdot ,cdot rangle }

For every two vectors

$$


v
,
w
∈
V


{displaystyle v,win V}

one defines the matrix coefficient

$$



c

v
,
w


:
G
→

C



{displaystyle c_{v,w}colon Gto mathbb {C} }

through

$$



c

v
,
w


(
g
)
=
⟨
ρ
(
g
)
v
,
w
⟩


{displaystyle c_{v,w}(g)=langle rho (g)v,wrangle }

.

Reconstruction of the representation from its matrix coefficients

After selecting a base

$$




{

e

i


}


i
∈
I




{displaystyle left{e_{i}right}_{iin I}}

from

$$


V


{displaystyle V}

any

$$


ρ
(
g
)


{displaystyle rho (g)}

for

$$


g
∈
G


{displaystyle gin G}

from the matrix coefficients

$$



{


c


e

i


,

e

j




(
g
)
:
i
,
j
∈
I

}



{displaystyle left{c_{e_{i},e_{j}}(g)colon i,jin Iright}}

determine

Schur orthogonality

Be

$$


G


{displaystyle G}

a compact group with hair measure

$$


d
g


{displaystyle dg}

normalized to

$$



∫

G


d
g
=
1


{displaystyle int _{G}dg=1}

and be

dim

(
V
)
<
∞


{displaystyle dim(V)<infty }

.
Then

$$



∫

G



f


v

1


,

w

1




(
g
)




f


v

2


,

w

2




(
g
)

¯



d
g
=


1

dim

(
V
)



⟨

v

1


,

w

1


⟩



⟨

v

2


,

w

2


⟩

¯




{displaystyle int _{G}f_{v_{1},w_{1}}(g){overline {f_{v_{2},w_{2}}(g)}},dg={frac {1}{dim(V)}}langle v_{1},w_{1}rangle {overline {langle v_{2},w_{2}rangle }}

for all

$$



v

1


,

w

1


,

v

2


,

w

2


∈
V


{displaystyle v_{1},w_{1},v_{2},w_{2}in V}

.

Classes of representations

A representation is called discrete if all matrix coefficients are square integrable, that is, in

$$



L

2


(
G
)


{displaystyle L^{2}(G)}

lie. It is called tempered if the matrix coefficients in

$$



L

2
+
ϵ


(
G
)


{displaystyle L^{2+epsilon }(G)}

in favour of a

$$


ϵ
>
0


{displaystyle epsilon >0}

lie.