The two-dimensional representation of acting on has

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Representations, Schur's lemma and characters

Problem

The two-dimensional representation

E

C_{3v}

acting on

(x,y)

has, for the mirror plane containing the

x

-axis, the matrix

D(\sigma_v) = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},

and for the

120^\circ

rotation,

D(C_3) = \begin{pmatrix} -\tfrac12 & -\tfrac{\sqrt3}{2} \\ \tfrac{\sqrt3}{2} & -\tfrac12 \end{pmatrix}.

Someone argues: "

D(\sigma_v)

is diagonal, with zero off-diagonal entries, so

(1,0)

and

(0,1)

are invariant one-dimensional subspaces for the whole representation, and

E

is reducible into two one-dimensional pieces." ((a)) Check whether

(1,0)

and

(0,1)

are eigenvectors of

D(C_3)

as well as of

D(\sigma_v)

. ((b)) Say what is wrong with the argument, and state whether

E

is in fact reducible.