Label the eight vertices of a cube of side 2

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Group theory and symmetry

Invariance, Ch Groups: the algebra of symmetry operations

Problem

Label the eight vertices of a cube of side 2, centred at the origin, by their sign patterns:

A=(1,1,1)

B=(1,1,-1)

C=(1,-1,1)

D=(1,-1,-1)

E=(-1,1,1)

F=(-1,1,-1)

G=(-1,-1,1)

H=(-1,-1,-1)

. The rotation by

240^\circ

about the body diagonal through

A

and

H

(a turn by

120^\circ

clockwise seen from

A

) is represented, using the standard rotation-matrix construction of Chapter 1, by the matrix that cyclically permutes coordinates,

\mathsf{R} = \begin{pmatrix} 0&1&0\\0&0&1\\1&0&0\end{pmatrix},\qquad \mathsf{R}\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}y\\z\\x\end{pmatrix}.

((a)) Using

\det \mathsf{R} = \epsilon_{ijk}\mathsf{R}_{1i}\mathsf{R}_{2j}\mathsf{R}_{3k}

(Chapter 1), find

\det\mathsf{R}

. ((b)) Apply

\mathsf{R}

to each of the eight vertices and write the resulting permutation of

\{A,\ldots,H\}

in cycle notation. ((c)) State whether this permutation is even or odd, and check that this agrees with

\det\mathsf{R}