Matrix coefficient

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

For example, one can by choice of a base in the representation space , the representation by the group elements associated matrices describe whose individual entries are matrix coefficients as defined in the general definition.

definition

Let be a representation of a group on a - Hilbert space with a scalar product . ${\displaystyle \rho \colon G\to GL(V)}$ ${\displaystyle G}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle V}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

For any two vectors to define the matrix coefficients by ${\displaystyle v,w\in V}$ ${\displaystyle c_{v,w}\colon G\to \mathbb {C} }$

${\displaystyle c_{v,w}(g)=\langle \rho (g)v,w\rangle }$.

Reconstruction of the representation from its matrix coefficients

After choosing a basis of , each of the matrix coefficients can be found for ${\displaystyle \left\{e_{i}\right\}_{i\in I}}$ ${\displaystyle V}$ ${\displaystyle \rho (g)}$ ${\displaystyle g\in G}$

${\displaystyle \left\{c_{e_{i},e_{j}}(g)\colon i,j\in I\right\}}$

determine.

Schur orthogonality

Be a compact group with hair measure , normalized to , and be . Then ${\displaystyle G}$ ${\displaystyle dg}$ ${\displaystyle \int _{G}dg=1}$ ${\displaystyle \dim(V)<\infty }$

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

for everyone . ${\displaystyle v_{1},w_{1},v_{2},w_{2}\in V}$

Classes of representations

A representation is called discrete if all matrix coefficients square integrable are so in fall. It is called tempered if the matrix coefficients are in for one . ${\displaystyle L^{2}(G)}$ ${\displaystyle L^{2+\epsilon }(G)}$ ${\displaystyle \epsilon >0}$