I am going through the lecture note of Andrew Childs on Nonabelian Fourier analysis. I would like to quote from the note:
My question:
Why does it have to be weighted superposition and not equal superposition of the irreps?
2026-04-02 06:10:17.1775110217
Why is the Fourier transform of a non-Abelian finite group the weighted superposition over all irreps?
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The answer to virtually any question of the form "why is this construction defined this way" is "because doing that gives it nice properties."
I can guess, for example, that this is needed in equation 10 to make the transformation unitary.
Later on between equations 16 and 17 it mentions another property about the way it decomposes the regular representations, and quite probably different choices of the definition you're asking about would not have the same property.
I would encourage you to take whatever alternative definition you have in mind and test it to see if it has either of these features.