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
a representation of a group
on a
-Hilbert space
with scalar product
For every two vectors
one defines the matrix coefficient
through
- .
Reconstruction of the representation from its matrix coefficients
After selecting a base
from
any
for
from the matrix coefficients
determine
Schur orthogonality
Be
a compact group with hair measure
normalized to
and be
Then
for all
.
Classes of representations
A representation is called discrete if all matrix coefficients are square integrable, that is, in
lie. It is called tempered if the matrix coefficients in
in favour of a
lie.
Category
- Representation theory of groups