A cube's four body diagonals can be represented by direction vectors with first…

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

Invariance, Ch Groups: the algebra of symmetry operations

Problem

A cube's four body diagonals can be represented by direction vectors with first coordinate +1+1: d1=(1,1,1)d_1=(1,1,1), d2=(1,1,−1)d_2=(1,1,-1), d3=(1,−1,1)d_3=(1,-1,1), d4=(1,−1,−1)d_4=(1,-1,-1) (each line is unaffected by reversing the sign of its vector). The face rotation by 90∘90^\circ about the zz axis is, from Chapter 1's construction, R=(0−10100001),R(x,y,z)=(−y,x,z).\mathsf{R} = \begin{pmatrix}0&-1&0\\1&0&0\\0&0&1\end{pmatrix},\qquad \mathsf{R}(x,y,z)=(-y,x,z). ((a)) Apply R\mathsf{R} to each did_i, and identify which diagonal (up to an overall sign) each image lies on. ((b)) Write the resulting permutation of {d1,d2,d3,d4}\{d_1,d_2,d_3,d_4\} in cycle notation. ((c)) State its parity, and say what this exercise illustrates about the cube's rotation group.

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