I am a beginner piano player here. I have been studying Barry Harris’s maj 6/ diminished chord approach. It seems that often you can have a shell in you left hand and different chord in the right hand (often a diminished, major 6 or minor 6) in order to get a particular sound. For example (I think) C#dim/C will give you a C dominant 7 sound. Depending on the situation in a given tune you could use a different inversion of C#dim. (And yes I know that inversion of a diminished chord are just a different diminished chord)
My question, is there a notation for quickly indicating what inversion of a chord to play? I guess I could do something like C#dim(iii)/C where the iii tells me to use the third inversion of C#dim played in the right hand over a C shell in the base but I'd rather not reinvent the wheel if a common notation already exists.