Lorentz boost tensor notation confusion I have been given this$$
\delta X^{\mu}=\omega_{\mu \nu}\left(M^{\mu \mu}\right)_{\sigma}^{\rho} X^{\sigma}
$$
and I think it should be equal to this but I'm confused if I'm doing it correctly
$$
\delta X^{\mu}=\omega_{\sigma}^{\rho} X^{\sigma}
$$
I just couldn't find a rule in the lowering and upping indices section in the wiki or my Tensor calc book.
 A: $M$ here represents the generators of the vector representation of the Lorentz group, and $\omega$ is an antisymmetric tensor parametrizing the group elements. In terms of these generators, a Lorentz group element in the vector representation is given by
$$\Lambda=\exp\left(\omega_{\mu\nu}M^{\mu\nu}\right),$$
where it is understood that $\Lambda$ and $M^{\mu\nu}$ are $d\times d$ matrixes in $d$ dimensions.
A finite transformation is given by $X^{\mu}\to\Lambda^{\mu}_{\,\,\,\nu}X^{\nu}$, which tells us that the infinitesimal transformation law is given by
$$\delta X^{\mu}=\omega_{\alpha\beta}(M^{\alpha\beta})^{\mu}_{\,\,\,\nu}X^{\nu}.$$
For an explicit form of $M$, consult basically any textbook on field theory (for instance, Chapter 3 of Peskin and Schroeder).
A: Well, there's a problem in your first formula because as far as Einstein's Summation Convention is concerned the symbol: 
$$ \omega_{\mu\nu}M^{\mu\mu}$$
is totaly meaningless, because you have sum only with two repeated indices.
Also, poincaré transformations are, in fact, and with more generality:
$$\delta X'^{\mu}= \Lambda^{\mu}_{\nu} \delta X^{\nu} + a^{\mu}$$
