- #1
filip97
- 31
- 0
- Homework Statement
- .
- Relevant Equations
- .
Group action on ##2##x##2## complex matrices of group ##C_{3v}## for all matrices from ##C^{22}##, for all ##g## from ##C_{3v}## is given by:
##D(g)A=E(g)AE(g^{-1}), A=\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}##,
Where ##E## is ##2D## irrep of ##C_{3v}##. I think that representation of rep ##D## is two dimensional, because act on ##2##x##2## matrix ##A##.
when I calculate characters
##\chi_D(g)## by
##tr(D(g)A)=\sum_{i,j=1,1}^{2,2} D(g)^{\dagger}_{ij}(g)A_{ij}=##
##=aD^{*}_{11}(g)+bD^{*}_{21}(g)+cD^{*}_{12}(g)+dD^{*}_{22}(g)##.
I always get for all ##g, D^{*}_{11}(g)=D^{*}_{11}(g)=1## and for
##D^{*}_{21}(g)=D^{*}_{12}(g)=0##, and ##\chi_D(g)=2## for all ##g##. And ##D=A\oplus A##
Main question is how decompose representation ##D## in irreducible components of group ##C_{3v}##
##D(g)A=E(g)AE(g^{-1}), A=\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}##,
Where ##E## is ##2D## irrep of ##C_{3v}##. I think that representation of rep ##D## is two dimensional, because act on ##2##x##2## matrix ##A##.
when I calculate characters
##\chi_D(g)## by
##tr(D(g)A)=\sum_{i,j=1,1}^{2,2} D(g)^{\dagger}_{ij}(g)A_{ij}=##
##=aD^{*}_{11}(g)+bD^{*}_{21}(g)+cD^{*}_{12}(g)+dD^{*}_{22}(g)##.
I always get for all ##g, D^{*}_{11}(g)=D^{*}_{11}(g)=1## and for
##D^{*}_{21}(g)=D^{*}_{12}(g)=0##, and ##\chi_D(g)=2## for all ##g##. And ##D=A\oplus A##
Main question is how decompose representation ##D## in irreducible components of group ##C_{3v}##