Group Representation - Examples

Examples

Consider the complex number u = e2πi / 3 which has the property u3 = 1. The cyclic group C3 = {1, u, u2} has a representation ρ on C2 given by:

\rho \left( 1 \right) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \rho \left( u \right) = \begin{bmatrix} 1 & 0 \\ 0 & u \\ \end{bmatrix} \qquad \rho \left( u^2 \right) = \begin{bmatrix} 1 & 0 \\ 0 & u^2 \\ \end{bmatrix}.

This representation is faithful because ρ is a one-to-one map.

An isomorphic representation for C3 is

\rho \left( 1 \right) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \rho \left( u \right) = \begin{bmatrix} u & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \rho \left( u^2 \right) = \begin{bmatrix} u^2 & 0 \\ 0 & 1 \\ \end{bmatrix}.

The group C3 may also be faithfully represented on R2 by

\rho \left( 1 \right) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \rho \left( u \right) = \begin{bmatrix} a & -b \\ b & a \\ \end{bmatrix} \qquad \rho \left( u^2 \right) = \begin{bmatrix} a & b \\ -b & a \\ \end{bmatrix}

where and .

Read more about this topic:  Group Representation

Famous quotes containing the word examples:

    “It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
    —G.C. (Georg Christoph)

    “There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
    —Bernard Mandeville (1670–1733)

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)