# How to determine the irreducible representation of an atomic orbital at a K point?

In this paper "Unified theory of direct or indirect band-gap nature of conventional semiconductors", the authors calculated the irreducible representations of atomic orbitals at three K points, as shown in Table 1

How such irreducible representations are determined? For example, I calculate electronic band structures using VASP, is there any software that can calculate the irreducible representations, particularly, the irreducible representations of atomic orbitals from the WAVECAR file?

For example Table C10, which I reproduce here with slight edits for notation I'm more accustom to using for molecular point groups, gives the character table for $$C_{3v}$$ at the point $$L$$/$$\Lambda$$.
$$\begin{array}{c|ccc|c} \hline C_\mathrm{3v}(\Lambda) & E & 2C_3 & 3\sigma_\mathrm{v} & \text{Basis} \\ \hline \mathrm{\Lambda_1} & 1 & 1 & 1 & 1, z, x^2+y^2, z^2 \\ \mathrm{\Lambda_2} & 1 & 1 & -1 \\ \mathrm{\Lambda_3} & 2 & -1 & 0 & (x,y), (x^2-y^2,xy), (xz,yz) \\ \hline \end{array}$$
You can then determine the irrep of an orbital based on the symmetry of the basis elements needed to represent it. For example, an $$s$$ orbital is formed from a totally symmetric function, so it has the totally symmetry irrep $$\Lambda_1$$. $$p_x$$, $$p_y$$, and $$p_z$$ have a similar totally symmetric component, but also include factors of $$x$$, $$y$$, and $$z$$ respectively, so the $$p$$ orbitals are $$\Lambda_1\oplus\Lambda_3$$, with the $$\Lambda_1$$ due to $$p_z$$ and the $$\Lambda_3$$ from the combination of $$p_x$$ and $$p_y$$.
For VASP, you may try irvasp or irRep. Or for Quantum espresso, you can directly read the group representations from the output of bands.x.