Permutation groups are applied to analyze the symmetries of pentaquark states. All possible quark configurations of the color, flavor, spin and spatial degrees of freedom are worked out in the language of permutation groups, and the corresponding wave functions are constructed systematically in the form of Yamanouchi basis. The pentaquark spatial wave functions of various symmetries, which are derived in the harmonic-oscillator interaction, are applied as complete bases to evaluate the low-lying light q 4 q pentaquark mass of all configurations, where the Cornell-like potential is employed.

We study pentaquark states of both light q 4q and hidden heavy q 3 QQ (q = u,d,s quark in SU(3) flavor symmetry; Q = c, b quark) systems with a general group theory approach in the constituent quark model, and the spectrum of light baryon resonances in the ansatz that the l = 1 baryon states may consist of the q 3 as well as q 4q pentaquark component. The model is fitted to ground state baryons and light baryon resonances which are believed to be normal three-quark states. The work reveals that the N (1535)1/2 − and N (1520)3/2 − may consist of a large q 4q component while the N (1895)1/2 − and N (1875)3/2 − are respectively their partners, and the N + (1685) might be a q 4q state. By the way, a new set of color-spin-flavor-spatial wave function for q 3 QQ systems in the compact pentaquark picture are constructed systematically for studying hidden charm pentaquark states. *

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We present an analysis of strong two-body decays of the a1(1420) with J P C = 1 ++ recently reported by the COMPASS Collaboration at CERN. Following the interpretation of the COMPASS Collaboration that the a1 is an unusual state with a four-quark qqss structure we consider two possible configurations for this state -hadronic molecular and color diquark-antidiquark structures. We find that the dominant decay mode of the a1 is the decay into K and K * . In particular, we calculate that the four decay modes a1 → V P with V P = K * ± K ∓ , K * 0K 0 ,K * 0 K 0 together give a dominant contribution to the measured total width of about 150 MeV. The observational mode a1 → f0(980) + π 0 is significantly suppressed by one order of magnitude.