Article

# 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

${displaystyle rho colon Gto GL(V)}$

ρ
:
G

G
L
(
V
)

{displaystyle rho colon Gto GL(V)} a representation of a group

${displaystyle G}$

G

{displaystyle G} on a

${displaystyle mathbb {C} }$

C

{displaystyle mathbb {C} } -Hilbert space

${displaystyle V}$

V

{displaystyle V} with scalar product

${displaystyle langle cdot ,cdot rangle }$

,

{displaystyle langle cdot ,cdot rangle } For every two vectors

${displaystyle v,win V}$

v
,
w

V

{displaystyle v,win V} one defines the matrix coefficient

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

c

v
,
w

:
G

C

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

## Reconstruction of the representation from its matrix coefficients

After selecting a base

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

{

e

i

}

i

I

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

${displaystyle V}$

V

{displaystyle V} any

${displaystyle rho (g)}$

ρ
(
g
)

{displaystyle rho (g)} for

${displaystyle gin G}$

g

G

{displaystyle gin G} from the matrix coefficients

determine

## Schur orthogonality

Be

${displaystyle G}$

G

{displaystyle G} a compact group with hair measure

${displaystyle dg}$

d
g

{displaystyle dg} normalized to

${displaystyle int _{G}dg=1}$

G

d
g
=
1

{displaystyle int _{G}dg=1} and be

${displaystyle dim(V)

dim

(
V
)
<

{displaystyle dim(V)<infty } .
Then

for all

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

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

${displaystyle L^{2}(G)}$

L

2

(
G
)

{displaystyle L^{2}(G)} lie. It is called tempered if the matrix coefficients in

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

L

2
+
ϵ

(
G
)

{displaystyle L^{2+epsilon }(G)} in favour of a

${displaystyle epsilon >0}$

ϵ
>
0

{displaystyle epsilon >0} lie.