LGIrreps at arbitrary k-points

[thchr]

This obviously requires that we fix #12 first, but after that, we should implement methods to:

1. Obtain the LGIrreps of a k-point k_α (D⁽ⁱ⁾_kα) from the LGIrreps of a "canonical" (i.e. referenced) k-point k in the star of k_α (D⁽ⁱ⁾_k), using the mapping D⁽ⁱ⁾_kα(h) = D⁽ⁱ⁾_k(g_α^-1hg_α), where h is an element of the little group of k_α, i.e. G_kα, and g_α is a coset representative that takes k to k_α via k_α = g_αk, such that g_α^-1hg_α is in the little group of k, i.e. G_k. (see e.g., Inui Eq. (11.52) as a reference for this)
2. After that, we could implement an interface over that utility, where the user would give an arbitrary k-point and then we'd match up to the relevant "maximal" k-point irrep in a given star and return the evaluated irreps via point 1.

[thchr]

For relevant literature that gives context to that formula, we can e.g. just think about how the "full" space group irreps are constructed from little group irreps. A good reference is Inui's book, Section 11.9 (and especially Eq. (11.52)).

[thchr]

I think all "missing" k-points from Φ-Ω should have been added in #30, so we should now in principle be able to "cover" all of the BZ with operations g_α from the coset.

[thchr]

There's a somewhat subtle point to add to bullet 1 in the original post, which is not obvious, I think.

In particular, we say to pick a an element of the little group coset g_α which sends k to k_α via:

k_α = g_αk

Very importantly, it is implicitly assumed that g_α here acts without any transposition, as we otherwise do when we compose symmetry operations with k-vectors. I.e., what we really mean is that if g_α = {R_α|t_α}, then (cf. Inui Eq. (11.49)):

k_α = R_αk

and not:

k_α = R_α^Tk

which is what e.g. compose(g_α, k_α) would have returned.

EDIT: Ugh, I somehow think this is still not right.