# conceptual problem about the energy band along the High symmetry point in the Brillouin zone

When we calculate the band structure of certain material, we only have to calculate the value along the high symmetry point which enclose the Irreducible Brillouin Zone. Why the information lie in the IBZ is enough for us to know the information of entire BZ ? How to find the IBZ given a reciprocal lattice vector in K-space without checking the table? Is there any proof ? Ive google it, but they just give you the table without telling you how to find out these High symmetry point.

For example, in a 2D material, the dispersion relation is $$\epsilon\left(k_x,k_y\right)$$ and we differentiate this with respect to a vector $$\vec{k} = \vec{k}_x\sin{\alpha} + \vec{k}_y \cos{\alpha}$$ for multiple $$\alpha$$. The magnitude and sign of the derivative are a periodic function whose period is defined by the symmetry of the high-symmetry point. Thus, extrema tend to be a high-symmetry points.