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I found a footnote on page 140 of this book that reads:

For a general $d$-dimensional quasicrystal, there is more than one $d$-dimensional irreducible representation. For the case of two-dimensional pentagonal symmetry, which describes the original Penrose tilings, there are two distinct two-dimensional representations of the pentagonal group.

From which my take-home message is: for regular crystals in $d$ dimensions, there is only one $d$ dimensional irreducible representation.

But then why does the dihedral group of order $n$ (even, for instance) have $(n-2)/2$ two-dimensional irreducible representations, as listed here?

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