I am going to try and reinterpret some of that paper, see if we can get some kind of answer started. Please comment and contribute, I think this is an interesting physical system.
It would seem that they are using the fact that there are both boson (symmetric) and fermion (antisymmetric) representations of $SU(N)$ to generalize the usual $SU(2)$ magnet. The Schwinger representation of $SU(2)$ gives a state of spin $S=n_c/2$ with $n_c$ bosons at each site. If you allow $N$ bosons (rather then 1 with up and down states), the state transform under an $SU(N)$ representation. These are symmetric, with a Young's tableau of a single row with $n_c$ boxes.
This same representation can be described by fermionic operators. These are totally antisymmetric so they are a Young's tableau with a column of $m$ boxes (for $m$ fermions.)
For full generality, you want to describe a representation of arbitrary symmetry; a Young's tableau with arbitrary rows and columns. The author says specifically they are interested in the case when the two sublattices have conjugate representations. This is to match the results with the usual $SU(2)$ case.
The results are that in the semiclassical limit (bosons $n_c\to \infty$ with constant $N$ and fermions $m$) the expectation value of the spin of the coherent states has an $U(N)/((U(m)\times U(N-m))$ symmetry. So the fermion content determines the symmetry of the resulting classical system. The integer $N$ is something like a generalized spin quantum number where normally $N=2$ for up and down.