Group Theory 3.1 Representation

Gist

Representation is just a homomorphism mapping

(1)   \begin{equation*} g\in G \rightarrow U(g) \end{equation*}

where U(g) is an operator on linear vector space V, such that

(2)   \begin{equation*} U(g_1)U(g_2)=U(g_1g_2) \end{equation*}

Dimension of the representation: is the dimension of the vector space V.
Faithful: If the homomorphism is also an isomorphism.
Degenerate representation: The one which is not Faithful.
Matrix Representation: is the case that U(g) is a matrix D(g) and the matrix multiplication preserve the group multiplication.

 

Analysis

The most important concept is mapping. In fact, we device group representation for the  purpose of elaborating the operations(group) in a mathematical way, thus we can write it down and discussion it in math symbols.

Problems

3.1 Consider the six dihedral group D_3. Let V be the 2-dimensional Euclidean space spanned by \hat{e}_x and \hat{e}_y. Write down the matrix representation of elements of D_3 on V with respect to this Cartesian basis.

Just right down the matrix which can transform the \hat{e}_x and \hat{e}_y into \hat{e}'_x and \hat{e}'_y as what the transformation in D_3. That is the matrix representation is the matrix D(g) in

(3)   \begin{equation*} \left( \begin{array}{c} \hat{e}_x \\ \hat{e}_y \end{array} \right) = D(g) \left( \begin{array}{c} \hat{e}'_x \\ \hat{e}'_y \end{array} \right) \end{equation*}

Based on this Figure, we can have
D(e) = \left( \begin{array}{cc} 1 & 0\\ 0 & 1 \end{array} \right), D(12) = \left( \begin{array}{cc} \frac{1}{2} & -\frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \end{array} \right), D(23) = \left( \begin{array}{cc} -1 & 0\\ 0 & 1 \end{array} \right), D(31) = \left( \begin{array}{cc} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{array} \right), D(123) = \left( \begin{array}{cc} -\frac{1}{2} & -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{array} \right), D(321) = \left( \begin{array}{cc} -\frac{1}{2} & \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \end{array} \right)

One thought to “Group Theory 3.1 Representation”

  1. It seems that the solution to Problems 3.1 is wrong.
    The Eq.(3) should be

    (1)   \begin{equation*} \hat{e}'_i=\hat{e}_j D(g)_{ji} \end{equation*}

    where, we have written \hat{e}_x as \hat{e}_1, \hat{e}_y as \hat{e}_2. And the representation matrices should be
    D(e) = \left( \begin{array}{cc} 1 & 0\\ 0 & 1 \end{array} \right), D(12) = \left( \begin{array}{cc} \frac{1}{2} & -\frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \end{array} \right), D(23) = \left( \begin{array}{cc} -1 & 0\\ 0 & 1 \end{array} \right), D(31) = \left( \begin{array}{cc} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{array} \right), D(123) = \left( \begin{array}{cc} -\frac{1}{2} & \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} \end{array} \right), D(321) = \left( \begin{array}{cc} -\frac{1}{2} & -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{array} \right)
    However, the solution by Michael Aivazis has the same form in my previous solution.

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