  We report the structural, magnetic and transport properties of
polycrystalline CrVTiAl alloy along with first principles calculations. It
crystallizes in the LiMgPdSn type structure with lattice parameter 6.14 \AA\ at
room temperature. Absence of (111) peak along with the presence of a weak (200)
peak indicates the antisite disorder of Al with Cr and V atoms. The
magnetization measurements reveal a ferrimagnetic transition near 710 K and a
coercive field of 100 Oe at 3 K. Very low moment and coercive field indicate
fully compensated ferrimagnetism in the alloy. Temperature coefficient of
resistivity is found to be negative, indicating a characteristic of
semiconducting nature. Absence of exponential dependence of resistivity on
temperature indicates a gapless/spin-gapless semiconducting behaviour.
Electronic and magnetic properties of CrVTiAl for three possible
crystallograpic configurations are studied theoretically. All the three
configurations are found to be different forms of semiconductors. Ground state
configuration is a fully compensated ferrimagnet with band gaps 0.58 eV and
0.30 eV for up and down spin bands respectively. The next higher energy
configuration is also ferrimagnetic, but has spin-gapless semiconducting
nature. The highest energy configuration corresponds to a non-magnetic gapless
semiconductor. The energy differences among these configurations are quite
small ($<$ 1 $\mathrm{mRy/atom}$) which hints that at finite temperatures, the
alloy exists in a disordered phase, which is a mixture of the three
configurations. By taking into account the theoretical and the experimental
findings, we conclude that CrVTiAl is a fully compensated ferrimagnet with
predominantly spin gap-less semiconductor nature.
