# Decomposing a tensor into symmetric and anti-symmetric components

I was looking at one of my textbooks and saw terms like $X_{(\alpha\beta)[\dot{\alpha}\dot{\beta}]}$ when I suddenly realised I dont actually know how to write this out. My first guess was $$X_{(\alpha\beta)[\dot{\alpha}\dot{\beta}]}=\frac{1}{2}(X_{\alpha\beta\dot{\alpha}\dot{\beta}}+X_{\beta\alpha\dot{\alpha}\dot{\beta}})+\frac{1}{2}(X_{\alpha\beta\dot{\alpha}\dot{\beta}}-X_{\alpha\beta\dot{\beta}\dot{\alpha}})$$ But I don't think this is the case. So how is this actually defined?

You can figure it out by just doing each expansion in turn. Here, I'll start with the antisymmetrization, which gives me two terms that are symmetrized on their first pairs of indices. In that first expansion, I leave the first pairs of indices with their symmetrization marks because they are irrelevant to the antisymmetrization of the second pair. But then I can go on to expand them as well; each of these gives me another two terms with no explicit symmetrization at the end: \begin{align} X_{(\alpha\beta)[\dot{\alpha}\dot{\beta}]} &= \frac{1}{2} \left\{ X_{(\alpha\beta)\dot{\alpha}\dot{\beta}} - X_{(\alpha\beta)\dot{\beta}\dot{\alpha}} \right\} \\ &= \frac{1}{2} \left\{ \frac{1}{2} \left[ X_{\alpha\beta\dot{\alpha}\dot{\beta}} + X_{\beta\alpha\dot{\alpha}\dot{\beta}} \right] - \frac{1}{2} \left[ X_{\alpha\beta\dot{\beta}\dot{\alpha}} + X_{\beta\alpha\dot{\beta}\dot{\alpha}} \right] \right\} \\ &= \frac{1}{4} \left\{ X_{\alpha\beta\dot{\alpha}\dot{\beta}} + X_{\beta\alpha\dot{\alpha}\dot{\beta}} - X_{\alpha\beta\dot{\beta}\dot{\alpha}} -X_{\beta\alpha\dot{\beta}\dot{\alpha}} \right\}. \end{align} You can check that the final result has the desired properties: $$X_{(\alpha\beta)[\dot{\alpha}\dot{\beta}]} = X_{(\beta\alpha)[\dot{\alpha}\dot{\beta}]}$$ because $$\frac{1}{4} \left\{ X_{\alpha\beta\dot{\alpha}\dot{\beta}} + X_{\beta\alpha\dot{\alpha}\dot{\beta}} - X_{\alpha\beta\dot{\beta}\dot{\alpha}} -X_{\beta\alpha\dot{\beta}\dot{\alpha}} \right\} = \frac{1}{4} \left\{ X_{\beta\alpha\dot{\alpha}\dot{\beta}} + X_{\alpha\beta\dot{\alpha}\dot{\beta}} - X_{\beta\alpha\dot{\beta}\dot{\alpha}} -X_{\alpha\beta\dot{\beta}\dot{\alpha}} \right\},$$ and $$X_{(\alpha\beta)[\dot{\alpha}\dot{\beta}]} = -X_{(\alpha\beta)[\dot{\beta}\dot{\alpha}]}$$ because $$\frac{1}{4} \left\{ X_{\alpha\beta\dot{\alpha}\dot{\beta}} + X_{\beta\alpha\dot{\alpha}\dot{\beta}} - X_{\alpha\beta\dot{\beta}\dot{\alpha}} -X_{\beta\alpha\dot{\beta}\dot{\alpha}} \right\} = -\frac{1}{4} \left\{ X_{\alpha\beta\dot{\beta}\dot{\alpha}} + X_{\beta\alpha\dot{\beta}\dot{\alpha}} - X_{\alpha\beta\dot{\alpha}\dot{\beta}} -X_{\beta\alpha\dot{\alpha}\dot{\beta}} \right\}.$$