The thermodynamical limit of general gl(N) spin chains: the vacuum state and densities

We study the vacuum state of spin chains where each site carries an arbitrary representation. We prove that the string hypothesis usually used to solve the Bethe ansatz equations is valid for representations characterized by rectangular Young tableaux. In these cases, we obtain the density of the centre of the strings for the vacuum. We work through different examples and, in particular, that for spin chains with a periodic array of impurities.