  It has been observed that photons in the prompt emission of some gamma-ray
bursts (GRBs) are highly polarized. The high polarization is used by some
authors to give a strict constraint on the Lorentz invariance violation (LIV).
If the Lorentz invariance is broken, the polarization vector of a photon may
rotate during its propagation. The rotation angle of polarization vector
depends on both the photon energy and the distance of source. It is believed
that if high polarization is observed, then the relative rotation angle
(denoted by $\alpha$) of polarization vector of the highest energy photon with
respect to that of the lowest energy photon should be no more than $\pi/2$.
Otherwise, the net polarization will be severely suppressed, thus couldn't be
as high as what was actually observed. In this paper, we will give a detailed
calculation on the evolution of GRB polarization arising from LIV effect
duration the propagation. It is shown that the polarization degree rapidly
decrease as $\alpha$ increases, and reaches a local minimum at $\alpha\approx
\pi$, then increases until $\alpha\approx 3\pi/2$, after that decreases again
until $\alpha \approx 2\pi$, etc. The polarization degree as a function of
$\alpha$ oscillates with a quasi-period $T\approx \pi$, while the oscillating
amplitude gradually decreases to zero. Moreover, we find that a considerable
amount (more than $60\%$ of the initial polarization) of polarization degree
can be conserved when $\alpha\approx \pi/2$. The polarization observation in a
higher and wider energy band, a softer photon spectrum, and a higher redshift
GRB is favorable in order to tightly constrain LIV effect.
