# Counting independent components

1. Sep 16, 2010

### haushofer

Hi,

I have a question about counting (how difficult should that be ;) )

I have the set of tensors in D dimensions

$$\{h_{\mu\nu}, H^{\mu\nu}, t_{\mu}, T^{\mu}\}$$

with the relations

$$H^{\mu\nu} h_{\nu\rho} = \delta^{\mu}_{\rho} - T^{\mu}t_{\rho}$$

$$T^{\mu}t_{\mu} = 1$$

$$H^{\mu\nu}t_{\nu} = h_{\mu\nu}T^{\nu} = 0$$

and h and H are symmetric tensors of rank (D-1).

The question now is: how many independent components does this set of fields constitute? Mathematica gives as answer 1\2D(D+1), the same amount as for a symmetric rank D tensor, but how can I derive this analytically?

2. Sep 17, 2010

### haushofer

Maybe this thread is better off in another place here? :)