The system Ag[Li$_{1/3}$Mn$_{2/3}$]O$_{2}$ belongs to a quaternary 3R-delafossite family and crystallizes in a monoclinic symmetry with space group $C,2/m$ and the magnetic Mn$^{4+}$($S=3/2$) ions form a honeycomb network in the $ab$-plane. An anomaly around 50 K and the presence of antiferromagnetic (AFM) coupling (Curie-Weiss temperature $theta_{CW}sim-51$ K) were inferred from our magnetic susceptibility data. The magnetic specific heat clearly manifests the onset of magnetic ordering in the vicinity of 48,K and the recovered magnetic entropy, above the ordering temperature, falls short of the expected value, implying the presence of short-range magnetic correlations. The (ESR) line broadening on approaching the ordering temperature $T_{{rm N}}$ could be described in terms of a Berezinski-Kosterlitz-Thouless (BKT) scenario with $T_{{rm KT}}=40(1)$ K. $^{7}$Li NMR line-shift probed as a function of temperature tracks the static susceptibility (K$_{iso}$) of magnetically coupled Mn$^{4+}$ ions. The $^{7}$Li spin-lattice relaxation rate (1/$T$$_{1}$) exhibits a sharp decrease below about 50 K. Combining our bulk and local probe measurements, we establish the presence of an ordered ground state for the honeycomb system Ag$_{3}$LiMn$_{2}$O$_{6}$.Our ab-initio electronic structure calculations suggest that in the $ab$-plane, the nearest neighbor (NN) exchange interaction is strong and AFM, while the next NN and the third NN exchange interactions are FM and AFM respectively. In the absence of any frustration the system is expected to exhibit long-range, AFM order, in agreement with experiment.
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