The one dimensional representations gives us the coordinate functions $$ \psi^1=1, \psi^2=\sign~. $$ An irreducible two dimensional representation is given by the standard representation $\pi\mapsto P(\pi)$ of $S(3)$ in $\OO(3)$ restricted to $E=[e_1+e_2+e_3]^\perp$. An orthonormal basis for $E$ is $b_1=(e_2-e_1)/\sqrt2$ and $b_2\colon=(-e_1-e_2+2e_3)/\sqrt6$ and thus the matrices of the two dimensional irreducible representation are $(\la P(\pi)b_k,b_j\ra)_{j,k=1}^2$. For $\pi=C_3\colon=(231)$ we get the matrix $$ \left( \begin{array}{cc} -1/2&-\sqrt3/2\\ \sqrt3/2&-1/2 \end{array} \right) $$ and for $\pi=\s_1\colon=(132)$ we get: $$ \left( \begin{array}{cc} 1/2&\sqrt3/2\\ \sqrt3/2&-1/2 \end{array} \right) $$ We already computed these matrices in subsection. This gives the following matrices for the permutations $\s_3=\s_1C_3=(213),\s_2=C_3\s_1=(321),C_3^*=C_3^2=(312)$: $$ \left( \begin{array}{cc} -1&0\\ 0&1 \end{array} \right), \left( \begin{array}{cc} 1/2&-\sqrt3/2\\ -\sqrt3/2&-1/2 \end{array} \right), \left( \begin{array}{cc} -1/2&\sqrt3/2\\ -\sqrt3/2&-1/2 \end{array} \right), $$ and finally the table of coordinate functions: $$ \begin{array}{c|cccccc|c} C_{3v}&E&C_3&\s_1&\s_3&\s_2&C_3^*&\norm{.}_{L_2(S(3))}\\ \hline \psi^1&1&1&1&1&1&1&1\\ \psi^2&1&1&-1&-1&-1&1&1\\ \psi_{11}^3&1&-1/2&1/2&-1&1/2&-1/2&1/\sqrt2\\ \psi_{12}^3&0&-\sqrt3/2&\sqrt3/2&0&-\sqrt3/2&\sqrt3/2&1/\sqrt2\\ \psi_{21}^3&0&\sqrt3/2&\sqrt3/2&0&-\sqrt3/2&-\sqrt3/2&1/\sqrt2\\ \psi_{22}^3&1&-1/2&-1/2&1&-1/2&-1/2&1/\sqrt2 \end{array} $$