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In band diagrams (e.g. GaAs below), symmetry points are sometimes indicated with a numerical index. I first thought it was the band index in the Bloch function (so two electrons with the same wave vector but different band index would have different energies), but the same point appears at different energies...

What is this index really corresponding to ?

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Those are the labels of irreducible representations corresponding to the wave-function on that symmetry point, derived via group theory. They classify how wave-function at that symmetry point evolves under crystal symmetry operations, similar as the classification of odd and even function $f(x)$ by whether it changes sign under inversion operation of $x$. Here is a pretty nice introduction to this topic of group theory and band structure (if you have access to it):

https://link.springer.com/content/pdf/10.1007%2F978-3-642-00710-1_2.pdf

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