Consider the triangle rotation group \(\{\mathbf{E}, \mathbf{A}, \mathbf{B}\}\). The group member \(\mathbf{A}\) is a rotation by \(120^{\circ}\). Show that \(x^{2}-y^{2}, x y\) are basis functions for \(\mathbf{A}\), and develop a \(2 \times 2\) matrix representation for \(\mathbf{A}\). Compare with the representation given. Why is your result different?