Group Theory 3.1 Representation


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.



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.


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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