I'm trying to compute the symmetrized version of a rank-4 tensor $F_{ijkl}$ associated to the following Young tableau:
The Young symmetrizer for this tableau should be: $$Y=Y_VY_H=[e-(13)][e-(24)][e+(12)][e+(34)]$$ Now I apply this symmetrizer to the tensor $F$ as follows:
- Applying the "horizontal" permutations: $$(Y_HF)_{ijkl} = F_{ijkl}+F_{jikl}+F_{ijlk}+F_{jilk}.$$
- Applying the "vertical" permutations: $$\begin{align} (Y_VY_HF)_{ijkl} =\ &F_{ijkl}-F_{kjil}-F_{ilkj}+F_{klij}\\&+F_{jikl}-F_{kijl}-F_{jlki}+F_{klji}\\&+F_{ijlk}-F_{ljik}-F_{iklj}+F_{lkij}\\&+F_{jilk}-F_{lijk}-F_{jkli}+F_{lkji} \end{align}$$
However, when I compare this to the result in Hamermesh' book "Group Theory and its Applications to Physical problems" I see that he obtains the following result:
$$\begin{align} (Y_VY_HF)_{ijkl} =\ &F_{ijkl}-F_{kjil}-F_{ilkj}+F_{klij}\\&+F_{jikl}-F_{jkil}-F_{likj}+F_{lkij}\\&+F_{ijlk}-F_{kjli}-F_{iljk}+F_{klji}\\&+F_{jilk}-F_{jkli}-F_{lijk}+F_{lkji} \end{align}$$
Now, although the term with a plus-sign are all the same, the ones with the minus-sign are clearly different. (He claims to use the same Young symmetrizer so up to there we agree.) However, he seems to apply the transpositions in the following way:
E.g. I expect $-(13)(12)=-(123)$ to give $$F_{ijkl}\longrightarrow -F_{kijl}$$ while he claims it to be $-F_{jkil}$ as if he did $i\leftrightarrow j$ and then $i\leftrightarrow k$ instead of applying the transposition $(13)$ to the new ordering.
Am I missing something or am I correct? I haven't found any worked out examples of $S_4$-diagrams online.
EDIT: The difference just boils down to which viewpoint I adopt. Active or passive, i.e. does the cycle (123) act as replacing the first index by the second index and so on, or does it act by taking the first index to the second one and so on.
Might it be that both these conventions lead to a sensible construction, in the sense that both of these conventions lead to a basis for the "tensor" space that we consider.