Is it permissible to purchase goods for a certain amount and inform the partner of a higher price, taking the difference as personal profit, then dividing the remaining profit, and what is the expiation for one who does that?

It is not permissible for an agent appointed to purchase something to sell what he bought for his principal at a higher price than he bought it for, because he is an agent for him. It is not valid for an agent to sell to himself or buy from himself for his principal except with the latter's permission. Furthermore, your not taking a wage for your work does not permit you to deceive and consume wealth unlawfully. You must repent and return what you took unjustly.