We systematically study hidden-charm pentaquark currents with the quark configurations $$[\bar{c} u][u d c]$$
[
c
¯
u
]
[
u
d
c
]
, $$[\bar{c} d][u u c]$$
[
c
¯
d
]
[
u
u
c
]
, and $$[\bar{c} c][u u d]$$
[
c
¯
c
]
[
u
u
d
]
. Some of their relations are derived using the Fierz rearrangement of the Dirac and color indices, and the obtained results are used to study strong decay properties of $$P_c$$
P
c
states as $$\bar{D}^{(*)} \varSigma _c$$
D
¯
(
∗
)
Σ
c
hadronic molecules. We calculate their relative branching ratios for the $$J/\psi p$$
J
/
ψ
p
, $$\eta _c p$$
η
c
p
, $$\chi _{c0} p$$
χ
c
0
p
, $$\chi _{c1} p$$
χ
c
1
p
, $$\bar{D}^{(*)0} \varLambda _c^+$$
D
¯
(
∗
)
0
Λ
c
+
, $$\bar{D}^{0} \varSigma _c^{+}$$
D
¯
0
Σ
c
+
, and $$\bar{D}^{-} \varSigma _c^{++}$$
D
¯
-
Σ
c
+
+
decay channels. We propose to search for the $$P_c(4312)$$
P
c
(
4312
)
in the $$\eta _c p$$
η
c
p
channel and the $$P_c(4440)/P_c(4457)$$
P
c
(
4440
)
/
P
c
(
4457
)
in the $$\bar{D}^{0} \varLambda _c^+$$
D
¯
0
Λ
c
+
channel.