how do i test a stone to see if it is yttrium aluminium garnet,gadolinium gallium garnet or strantium titanat?

Answer 1

Build it into a laser and see what colour light you get?

If those three materials are the only possibilities then an optical/UV transmission spectrum or emission from a flame would probably resolve it through detecting the absence or presence of the various metals.

One of the best ways to identify a crystalline material is to do an X-ray diffraction pattern and compare lattice parameters with known materials. In this case you only need to check against the three possibles.

Answer 2

Use a Scanning Electron Microscope with 'backscatter' and this would indicate what material the sample is made from.

This would be by far the simplest method.