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.