  $P_c$ resonances are studied in the approach of quark model and group theory.
It is found that there are totally 17 possible pentaquark states with the quark
contents $q^3Q \bar Q$ ($q$ are $u$ and $d$ quarks; $Q$ is $c$ quark) in the
compact pentaquark picture, where the hidden heavy pentaquark states may take
the color singlet-singlet ($[111]_{{qqq}}\otimes [111]_{{c \bar c}}$) and color
octet-octet ($[21]_{{qqq}}\otimes [21]_{{c \bar c}}$) configurations. The
partial decay widths of hidden heavy pentaquark states are calculated for all
possible decay channels. The results show that the $pJ/\psi$ is the dominant
decay channel for both the spin $3/2$ and $1/2$ pentaquark states, and indicate
that the $P_c(4440)$ may not be a compact pentaquark state while $P_c(4312)$
and $P_c(4457)$ could be the spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$
pentaquark states, respectively.
