Quick question about contravariant and covariant tensors I have seen many different notations to denote contravariant/covariant and mixed tensors.  For example, I think the notation $\omega^{v}_{\,\,\,\mu}$ stands for a mixed tensor, where one index transforms contravariantly and the other covariantly.   What then would the notation $\omega^{v}_{\mu}$ denote?

A covariant vector transforms like $a'_{\mu} = \beta_{\mu}^{\,\,\,v}a_v$, but what is the difference between this and writing $a'_{\mu} = \beta_{\mu}^{v}a_v$ for example?

The only instance where I have seen an equivalence between the two notations is for the Kronecker delta, $\delta^v_{\,\,\,\mu} = \delta_{\mu}^{\,\,\,v} = \delta^v_{\mu}$, but the reason for this is not quite clear.

Many thanks.
In Einstein summation convention, repeated indices represent summation. Depending on the range of indices ($\nu=1$ to n), you would have a sum of n terms, and the contravariant index would cancel out the covariant index, leaving just a covariant $a'_\mu$
In Einstein summation convention, repeated indices represent summation. Depending on the range of indices ($\nu=1$ to n), you would have a sum of n terms, and the contravariant index would cancel out the covariant index, leaving just a covariant $a'_\mu$