  We investigate the structural and magnetic properties of Mn$_{2}$FeGa for
different phases(cubic, hexagonal and tetragonal) reported experimentally using
density functional theory. The relative structural stabilities, and the
possible phase transformation mechanisms are discussed using results for total
energy, electronic structure and elastic constants. We find that the phase
transformation form hexagonal to ground state tetragonal structure would take
place through a Heusler-like phase which has a pronounced electronic
instability. The electronic structures, the elastic constants and the
supplementary phonon dispersions indicate that the transition from the
Heusler-like to the tetragonal phase is of pure Jahn-Teller origin. We also
describe the ground state magentic structures in each phase by computations of
the exchange interactions. For Heusler-like and tetragonal phases, the
ferromagnetic exchange interactions associated with the Fe atoms balance the
dominating antiferromagnetic interactions between the Mn atoms leading to
collinear magnetic structures. In the hexagonal phase, the direction of atomic
moment are completely in the planes with a collinear like structure, in stark
contrast to the well known non-collinear magnetic structure in the hexagonal
phase of Mn$_{3}$Ga, another material with similar structural properties. The
overwhelmingly large exchange interactions of Fe with other magnetic atoms
destroy the possibility of magnetic frustration in the hexagonal phase of
Mn$_{2}$FeGa. This comprehensive study provides significant insights into the
microscopic physics associated with the structural and magnetic orders in this
compound.
