Wave functions of multiquark hadrons from representations of the symmetry groups $S_n$

Construction of the wave functions of multiquark hadrons by traditional method based on tensor products of colors, flavors, spins (and orbital) parts becomes quite complex when quark numbers grow $n=5,\mathrm{6...12}$, as it gets difficult to satisfy requirements of Fermi statistics. Our novel approach is focused directly on representations of the permutation symmetry generators. After showing how $C_3$ is manifested in the wave functions of (excited) baryons, we use it to construct the wave functions for a set of pentaquarks and hexaquarks (n=5,6). We also have some partial results for larger systems, with $n=9$ and 12, and even beyond that as far as $n=24$.