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A universal $k^{-4}$ decay of the large-momentum tails of the momentumdistribution, fixed by Tan's contact coefficients, constitutes a directsignature of strong correlations in a short-range interacting quantum gas. Herewe consider a repulsive multicomponent Fermi gas under harmonic confinement, asin the experiment of Pagano et al. [Nat. Phys. {\bf 10}, 198 (2014)], realizinga gas with tunable $SU(\kappa)$ symmetry. We exploit an exact solution atinfinite repulsion to show a direct correspondence between the value of theTan's contact for each of the $\kappa$ components of the gas and the Youngtableaux for the $S_N$ permutation symmetry group identifying the magneticstructure of the ground-state. This opens a route for the experimentaldetermination of magnetic configurations in cold atomic gases, employing onlystandard (spin-resolved) time-of-flight techniques. Combining the exact resultwith matrix-product-states simulations, we obtain the Tan's contact at allvalues of repulsive interactions. We show that a local density approximation(LDA) on the Bethe-Ansatz equation of state for the homogeneous mixture is inexcellent agreement with the results for the harmonically confined gas. Atstrong interactions, the LDA predicts a scaling behavior of the Tan's contact.This provides a useful analytical expression for the dependence on the numberof fermions, number of components and on interaction strength. Moreover, usinga virial approach in the limit of infinite interactions, we show that thecontact increases with the temperature and the number of components. At zerotemperature, we predict that the weight of the momentum distribution tailsincreases with interaction strength and the number of components if thepopulation per component is kept constant. This latter property wasexperimentally observed in Ref.~[Nat. Phys. {\bf 10}, 198 (2014)].

High-momentum tails as magnetic structure probes for strongly-correlated $SU(\kappa)$ fermionic mixtures in one-dimensional traps