A commuting tuple of operators (S₁,..., S_n-1,P), defined on a Hilbert space H, for which the closed symmetrized polydisc is a spectral set, is called a
Γ_n-contraction. To every Γ_n-contraction, there is a unique operator tuple
(A₁,...,A_n-1), defined on the closure of Ran(I-P*P), such that
S_i - S_n-i*P=D_PA_iD_P, D_P=(I - P*P)^1/2, i=1,..., n-1.
This is called the fundamental operator tuple or F_O-tuple associated with the
Γ_n-contraction. The F_O-tuple of a Γ_n-contraction completely determines the structure of a Γ_n-contraction and provides operator model and complete unitary invariant for them. In this note, we analyze the F_O-tuples and find some intrinsic properties of them. Given a Γ_n-contraction (S₁,...,S_n-1,P) and n-1 operators A₁,..., A_n-1 defined on the closure of Ran(D_P), we provide a necessary and sufficient condition under which (A₁,..., A_n-1) becomes the F_O-tuple of
(S₁,...,S_n-1,P). Also for given tuples of operators (A₁,..., A_n-1) and (B₁,..., B_n-1), defined on a Hilbert space E, we find a necessary condition and a sufficient condition under which there exist a Hilbert space H and a Γ_n-contraction
(S₁,..., S_n-1,P) on H such that (A₁,..., A_n-1) becomes the F_O-tuple of
(S₁,..., S_n-1,P) and (B₁,..., B_n-1) becomes the F_O-tuple of the adjoint
(S₁*,..., S_n-1*,P*).

The first named author is supported by a Ph.D fellowship of the University Grants Commissoin (UGC). The second named author is supported by the Seed Grant of IIT Bombay with Grant No. RD/0516-IRCCSH0-003 (15IRCCSG032), the INSPIRE Faculty Award (Award No. DST/INSPIRE/04/2014/001462) of DST, India and the MATRICS Grant (Award No. MTR/2019/001010) of Science and Engineering Research Board (SERB) of DST, India.