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We investigate the ground state phase diagram of an SU($N$)-symmetricantiferromagnetic spin model on a square lattice where each site hosts anirreducible representation of SU($N$) described by a square Young tableau of$N/2$ rows and $2S$ columns. We show that negative sign free fermion MonteCarlo simulations can be carried out for this class of quantum magnets at any$S$ and even values of $N$. In the large-$N$ limit, the saddle pointapproximation favors a four-fold degenerate valence bond solid phase. In thelarge $S$-limit, the semi-classical approximation points to N\'eel state. On aline set by $N=8S + 2$ in the $S$ versus $N$ phase diagram, we observe avariety of phases proximate to the N\'eel state. At $S = 1/2$ and $3/2$ weobserve the aforementioned four fold degenerate valence bond solid state. At$S=1$ a two fold degenerate spin nematic state in which the C$_4$ latticesymmetry is broken down to C$_2$ emerges. Finally at $S=2$ we observe a uniqueground state that pertains to a two-dimensional version of theAffleck-Kennedy-Lieb-Tasaki state. For our specific realization, this symmetryprotected topological state is characterized by an SU(18), $S=1/2$ boundarystate, that has a dimerized ground state. These phases that are proximate tothe N\'eel state are consistent with the notion of monopole condensation of theantiferromagnetic order parameter. In particular one expects spin disorderedstates with degeneracy set by $\text{mod}(4,2S)$.

Phase diagram of the SU($N$) antiferromagnet of spin $S$ on a square lattice