1. ## Index notation meaning

I'm studying Group Theory by myself and I face the basic concept regarding superscript and subscript indices on the Kronecker delta tensor:

$\displaystyle \delta^{\alpha\beta}$

$\displaystyle \delta^{\alpha}_{\beta}$

What is the meaning (or difference) of these symbols?

2. Originally Posted by intervoxel
I'm studying Group Theory by myself and I face the basic concept regarding superscript and subscript indices on the Kronecker delta tensor:

$\displaystyle \delta^{\alpha\beta}$

$\displaystyle \delta^{\alpha}_{\beta}$

What is the meaning (or difference) of these symbols?
I thought I knew what the "Kroneker delta" was but it certainly isn't a tensor! $\delta_{\alpha\beta}$ is the array that is 1 if $\alpha= \beta$ and 0 otherwise. That doesn't depend on any coordinate system and so does not transform as a tensor would under a change of coordinate systems.

3. Actually what I'm trying to understand are the following expressions which appeared when studying the SU(2) group:

$\displaystyle X^{\alpha}_{~\alpha'}X^{\beta}_{~\beta'}\delta^{\a lpha'}_{ \beta'} = \delta^{\alpha}_{ \beta}$

$\displaystyle X^{\alpha}_{~\alpha'}X^{\beta}_{~\beta'}\delta^{\a lpha'\beta'} \neq \delta^{\alpha\beta}$