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I have been reading about symmetries of systems' actions, e.g. the Polyakov action, and I have encountered Lorentz transformations of the form: $\Lambda^{\mu}_{\nu} X^{\nu}$. I am moderately familiar with $\Lambda$, the Lorentz matrix. If the indices are in superscript then it is the inverse of $\Lambda_{\mu \nu}$. However, what is $\Lambda^{\mu}_{\nu}$ in terms of the Lorentz matrix?

I have chosen mathematical physics as the tag, as I do not think any discipline of pure mathematics is appropriate based on the context. Please correct me if I am wrong.

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Let's recap: upper indices are vectors ($x^\mu$), the inner product on Minkowski space is given by $g_{\mu \nu}$ so "dual vectors" have lower indices $x_\nu = g_{\nu \rho} x^\rho.$

Then you see that a matrix (in the sense of linear map between vectors) has one upper and one lower index, because it maps a vector to another vector:

$$x^\mu \mapsto A^\mu{}_\nu x^\nu.$$

So actually $\Lambda^\mu {}_\nu$ is a very natural object, it's a matrix that rotates/Lorentz boosts vectors.

If you prefer working with the tensor $\Lambda_{\mu \nu}$, then the link between both tensors is given by

$$\Lambda^\mu{}_\nu = g^{\mu \rho} \Lambda_{\rho \nu} \Leftrightarrow \Lambda_{\mu \nu} = g_{\mu \sigma} \Lambda^\sigma {}_\nu.$$

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