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Consider the atomic wavefunction: $$\newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\f}[2]{\frac{ #1}{ #2}} \newcommand{\l}[0]{\left(} \newcommand{\r}[0]{\right)} \newcommand{\mean}[1]{\langle #1 \rangle}\newcommand{\e}[0]{\varepsilon} \newcommand{\ket}[1]{\left|#1\right>} \ket{\psi}=\ket{L,M_L}\ket{S,M_S}$$ let us assume that $\ket{L,0}$ is has a certain symmetry on swapping electrons $i$ and $j$ such that the overall wavefunction $\ket{\psi}=\ket{L,0}\ket{S,M_S}$ is anti-symmetric then will the wavefunction $\ket{L,M_L}$ (for the same $L$) have the same symmetry on swapping electrons $i$ and $j$? i.e. if $\ket{L,0}\ket{S,M_S}$ is anti-symmetric is $\ket{L,M_L}\ket{S,M_S}$ always anti-symmetric for a given $L$, $S$ and $M_S$?

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The symmetry does not depend on $M_L$. The simplest way to see this is as follows. Start with $\vert L,M_L\rangle$ as a product state.

If it is a product of only two states it would be written as $$ \vert L M_L\rangle = \sum_{m_1m_2}C_{\ell m_1;\ell m_2}^{LM_L} \vert \ell m_1\rangle \vert \ell m_2\rangle \tag{1} $$ where $C_{\ell m_1,\ell m_2}^{LM_L}$ is a Clebsch-Gordan coefficient. Permute particles $1$ and $2$, which is same as permuting $m_1$ and $m_2$ and you get $$ C_{\ell m_2;\ell m_1}^{L M_L}=(-1)^{2\ell-L} C_{\ell m_1;\ell m_2}^{L M_L} $$ showing that the phase $(-1)^{2\ell -L}$ and thus the symmetry character does not depend on $M_L$.

If you have more than two particles the job is a little more complicated. Start with $\vert L,L\rangle$ as a product state, generalizing (1) to more than one constituent. The coefficients in the linear combinations are no longer CGs but this doesn't matter for now.

From the state $\vert L,M_L\rangle$ you reach the state $\vert L,M_L-1\rangle$ by application of the lowering operator $$ \hat L_-= \sum_i \hat L_{i,-}= \hat L_{1,-}+\hat L_{2,-}+\hat L_{3,-}\ldots $$ Note that this sum is symmetric under permutation of particle numbers, so that, for instance: \begin{align} P_{12}\vert L,M_L-1\rangle &= P_{12}{\cal N}_{M_L}\left(\hat L_{1,-}+\hat L_{2,-}+\ldots \right)\vert L,M_L\rangle\, ,\\ &={\cal N}_{M_L}\left(\hat L_{1-}+\hat L_{2-}+\ldots\right)P_{12}\vert L,M_L\rangle \end{align} where ${\cal N}_{M_L}$ is a normalization constant. This shows that the symmetry under permutation of $\vert L,M_L-1\rangle$ is that of $\vert L,M_L\rangle$, and independent of this $M_L$.

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