Non-associative behavior for group action on KVec

[thchr]

A consequence of having SymOperations act via their transpose on a KVec is that their action is not associative:

kv  = KVec("α, α")
kv′ = KVec("α, -2α") # a k-vector in the star of `kv`
sg = spacegroup(15,2)
lg′ = littlegroup(sg, kv′) # the little group associated with `kv′`
h′  = lg′[2]
g   = S"-y,x-y"
@assert g*kv == kv′

# observe non-associative behavior
h′ * (g * kv)  # correct
(h′ * g) * kv  # wrong

Fundamentally, we end up with an issue of the kind that transpose(rotation(h′ * g)) is in general not equal to transpose(rotation(h′)) * transpose(rotation(g)).

I'm not sure what could or should be done about this - since the transpose is required and reasonable for other reasons - but it is an awful footgun (as evidenced e.g., by musings in #14).

[thchr]

It seems I have previously wrestled with this:

Crystalline.jl/src/symops.jl

Lines 391 to 425 in b9f25a9

	# ----- LITTLE GROUP OF 𝐤 -----
	# A symmetry operation g acts on a wave vector as (𝐤′)ᵀ = 𝐤ᵀg⁻¹ since we
	# generically operate with g on functions f(𝐫) via gf(𝐫) = f(g⁻¹𝐫), such that
	# the operation on a plane wave creates exp(i𝐤⋅g⁻¹𝐫); invariant plane waves
	# then define the little group elements {g}ₖ associated with wave vector 𝐤.
	# The plane waves are evidently invariant if 𝐤ᵀg⁻¹ = 𝐤ᵀ, or since g⁻¹ = gᵀ
	# (orthogonal transformations), if (𝐤ᵀg⁻¹)ᵀ = 𝐤 = (g⁻¹)ᵀ𝐤 = g𝐤; corresponding
	# to the requirement that 𝐤 = g𝐤). Because we have g and 𝐤 in different bases
	# (in the direct {𝐑} and reciprocal {𝐆} bases, respectively), we have to take
	# a little extra care here. Consider each side of the equation 𝐤ᵀ = 𝐤ᵀg⁻¹,
	# originally written in Cartesian coordinates, and rewrite each Cartesian term
	# through basis-transformation to a representation we know (w/ P(𝐗) denoting
	# a matrix with columns of 𝐗m that facilitates this transformation):
	# 𝐤ᵀ = [P(𝐆)𝐤(𝐆)]ᵀ = 𝐤(𝐆)ᵀP(𝐆)ᵀ (1)
	# 𝐤ᵀg⁻¹ = [P(𝐆)𝐤(𝐆)]ᵀ[P(𝐑)g(𝐑)P(𝐑)⁻¹]⁻¹
	# = 𝐤(𝐆)ᵀP(𝐆)ᵀ[P(𝐑)⁻¹]⁻¹g(𝐑)⁻¹P(𝐑)⁻¹
	# = 𝐤(𝐆)ᵀ2πg(𝐑)⁻¹P(𝐑)⁻¹ (2)
	# (1+2): 𝐤′(𝐆)ᵀP(𝐆)ᵀ = 𝐤(𝐆)ᵀ2πg(𝐑)⁻¹P(𝐑)⁻¹
	# ⇔ 𝐤′(𝐆)ᵀ = 𝐤(𝐆)ᵀ2πg(𝐑)⁻¹P(𝐑)⁻¹[P(𝐆)ᵀ]⁻¹
	# = 𝐤(𝐆)ᵀ2πg(𝐑)⁻¹P(𝐑)⁻¹[2πP(𝐑)⁻¹]⁻¹
	# = 𝐤(𝐆)ᵀg(𝐑)⁻¹
	# ⇔ 𝐤′(𝐆) = [g(𝐑)⁻¹]ᵀ𝐤(𝐆) = [g(𝐑)ᵀ]⁻¹𝐤(𝐆)
	# where we have used that P(𝐆)ᵀ = 2πP(𝐑)⁻¹ several times. Importantly, this
	# essentially shows that we can consider g(𝐆) and g(𝐑) mutually interchangeable
	# in practice.
	# By similar means, one can show that
	# [g(𝐑)⁻¹]ᵀ = P(𝐑)ᵀP(𝐑)g(𝐑)[P(𝐑)ᵀP(𝐑)]⁻¹
	# = [P(𝐆)ᵀP(𝐆)]⁻¹g(𝐑)[P(𝐆)ᵀP(𝐆)],
	# by using that g(C)ᵀ = g(C)⁻¹ is an orthogonal matrix in the Cartesian basis.
	# [ *) We transform from a Cartesian basis to an arbitrary 𝐗ⱼ basis via a
	# [ transformation matrix P(𝐗) = [𝐗₁ 𝐗₂ 𝐗₃] with columns of 𝐗ⱼ; a vector
	# [ v(𝐗) in the 𝐗-representation corresponds to a Cartesian vector v(C)≡v via
	# [ v(C) = P(𝐗)v(𝐗)
	# [ while an operator O(𝐗) corresponds to a Cartesian operator O(C)≡O via
	# [ O(C) = P(𝐗)O(𝐗)P(𝐗)⁻¹

That reasoning appears careful and rigorous. But it is not - afaict - what I actually implemented in compose(::SymOperation, ::KVec) - there's no inverse there. I.e., maybe it should act inversely? It seems that would also fix the associativity issue by flipping the order twice.