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We show how to perform exact diagonalizations of $\mathrm{SU}(N)$Fermi-Hubbard models on L-sites clusters separately in each irreduciblerepresentation ({\it irrep}) of $\mathrm{SU}(N)$. Using the representationtheory of the unitary group $\mathrm{U}(L)$, we demonstrate that a convenientorthonormal basis, on which matrix elements of the Hamiltonian are very simple,is given by the set of {\it semi-standard Young tableaux} (or equivalently theGel'fand-Tsetlin patterns) corresponding to the targeted irrep. \,As anapplication, we study the robustness of some $\mathrm{SU}(N)$ phases predictedin the Heisenberg limit upon decreasing the on-site interaction on variouslattices of size $L \leq 12$ and for $2 \leq N \leq 6$. In particular, we showthat a long range color ordered phase emerges for moderate U for N=4 at filling$1/4$ on the triangular lattice.

Exact Diagonalization of $\mathrm{SU}(N)$ Fermi-Hubbard Models