Density matrix renormalization group simulations of the SU(<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>) Fermi-Hubbard chain implementing the full SU(<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><

We have developed an efficient method for performing density matrix renormalization group (DMRG) simulations of the SU($N$) Fermi-Hubbard chain, fully leveraging the SU($N$) symmetry of the problem. This method extends a previously developed approach for the SU($N$) Heisenberg model and relies on the systematic use of the semistandard Young tableaux (SSYT) basis in a DMRG algorithm ``\`a la White.'' Specifically, the method aligns the site-by-site growth process of the infinite-size part of the DMRG, in its original formulation, with the site-by-site construction of the SSYT (or Gelfand-like) basis, based on the chain of unitary subgroups $U(1)\ensuremath{\subset}U(2)\ensuremath{\subset}U(3)\ensuremath{\subset}U(4)\ensuremath{\cdots}$. We give special emphasis to the calculation of the symmetry-resolved reduced matrix elements of the hopping terms between the left and the right block, which makes direct use of the basis of SSYT and of the Gelfand-Tsetlin coefficients, offering a computational advantage in scaling with $N$ compared to alternative methods that rely on summing over Clebsch-Gordan coefficients. Focusing on the model with homogeneous hopping between nearest neighbors, we have calculated the ground-state energy as a function of $U$, i.e., the atom-atom interaction amplitude, up to $N=6$ for filling $1/N$ (one particle per site in average), and for one atom (respectively, hole) away from filling $1/N$. It allows us to compute the charge gaps and to estimate in the thermodynamical limit, the critical value ${U}_{c}$ (separating the Mott insulator from the metallic phase), which is proved to be finite (i.e., ${U}_{c}&gt;0$) and to increase with $N$. Central charges $c$ are also extracted from the entanglement entropy using the Calabrese-Cardy formula and are consistent with the theoretical predictions: $c=N\ensuremath{-}1$, expected from the $\mathrm{SU}{(N)}_{1}$ Wess-Zumino-Witten conformal field theory in the spin sector for the Mott phase, and $c=N$ in the metallic phase, reflecting the presence of one additional (charge) gapless critical mode.