In virtually all circumstances, the symmetry of some structural property of a structure at a given k point is defined through the same general means:
The structural property we want to analyze is first described as a complex vector field which has periodicity exactly equal to the Bravais lattice, and which is nonzero only at the discrete set of sites in the lattice.
For instance, a discussion of the symmetry of any property in a honeycomb structure would begin by consideration of a infinite vector field of the following form: (each label shown represents a delta function term scaled by that value)
t s t s t s t s t
s t s t s t s t s
t s t s t s t s t
s t s t s t s t s
t s t s t s t s t
s t s t s t s t s
t s t s t s t s t
where s and t are appropriately-chosen vectors in an appropriately chosen basis,
depending on what property we're looking at.
Regardless of what property is under consideration (it does not even need to be
wavelike), "symmetry of _ at the k point" is typically understood to mean that we
take each unit cell image and scale it by exp(i k.R) (R being the bravais lattice vector).
For instance, when discussing symmetry at the K point (= [1/3 1/3 0]) of a honeycomb lattice,
we will ultimately be considering a field like this:
t sφφ tφφ sφ tφ s t sφφ tφφ
sφ tφ s t sφφ tφφ sφ tφ s
t sφφ tφφ sφ tφ s t sφφ tφφ
sφ tφ s t sφφ tφφ sφ tφ s
t sφφ tφφ sφ tφ s t sφφ tφφ
sφ tφ s t sφφ tφφ sφ tφ s
t sφφ tφφ sφ tφ s t sφφ tφφ
where φ = exp(i * 2 pi/3).
Given this formulation, a symmetry operation is then an operation in 3D space which transforms the vector field into itself, up to constant phase;
ct csφφ ctφφ csφ ctφ cs ct csφφ
csφ ctφ cs ct csφφ ctφφ csφ ctφ
ct csφφ ctφφ csφ ctφ cs ct csφφ
csφ ctφ cs ct csφφ ctφφ csφ ctφ
ct csφφ ctφφ csφ ctφ cs ct csφφ
csφ ctφ cs ct csφφ ctφφ csφ ctφ
ct csφφ ctφφ csφ ctφ cs ct csφφ
for some complex c of absolute value 1.
When discussing the symmetry of the structure itself, for a structure with S distinct species,
then the vector space we consider is S-dimensional.
For graphene (which only has a single element), s = t = 1.
At the K point for graphene, one even sees something like this:
φφ φ φ 1 1 φφ φφ φ φ
1 1 φφ φφ φ φ 1 1 φφ
φφ φ φ 1 1 φφ φφ φ φ
1 1 φφ φφ φ φ 1 1 φφ
φφ φ φ 1 1 φφ φφ φ φ
1 1 φφ φφ φ φ 1 1 φφ
φφ φ φ 1 1 φφ φφ φ φ
which is for instance not symmetric under inversion because different unit cells will invariably be shifted by different phase amounts.
For symmetry of vibrational modes, each of the N sites in the unit cell receives its own
3D subspace of a 3N-dimensional vector field.
A normal mode in graphene described by 3D polarization vectors u and v (one for each site)
would have s = [ux uy uz 0 0 0], t = [0 0 0 vx vy vz].
There is also a simpler way to view vibrational modes, in terms of "symmetry of sites"
(I made this term up because I can't remember what it's usually called).
The vector field for vibrational modes is just the direct product of 3D cartesian space
with some sort of N-dimensional "unit cell site space", which can be described in the honeycomb lattice
by the vector field s = [1 0], t = [0 1].