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docs/src/advanced/symmetries.md

@@ -51,24 +51,22 @@ one can find a reduced set of ``k``-points
 (the *irreducible ``k``-points*) such that the eigenvectors at the
 reducible ``k``-points can be deduced from those at the irreducible ``k``-points.
 
-### Symmetrization
+## Symmetrization
 Quantities that are calculated by summing over the reducible ``k`` points can be
-calculated by first summing over the irreducible ``k`` points and then symmetrizing.
-In this subsection, we denote by ``S`` the combined rotation and fractional transformation.
-If ``S`` is the symmetry of the system, ``f(Sk) = S(f(k))`` holds, where ``S(f)`` is the
-symmetry transformation of ``f`` which is a linear function and depends on the quantity
-``f``. Then, we find
+calculated by first summing over the irreducible ``k`` points and then symmetrizing. Let ``f`` be a ``k``-dependent quantity to sum (for instance, energies, densities, forces, etc). ``f`` transforms in a particular way under symmetries: ``f(Sk) = S(f(k))`` where the (linear) action of ``S`` on ``f`` depends on the particular ``f``. Let ``\mathcal S`` the group of all crystal symmetries that leave the Brillouin zone mesh invariant.
 ```math
+\begin{aligned}
 \sum_{k\ \mathrm{reducible}} f(k)
-= \sum_{k\ \mathrm{irreducible}} \sum_{S\text{ that maps $k$ to reducible $k$}} S(f(k))
-= \sum_{k\ \mathrm{irreducible}} \frac{N_S}{N_{S,k}} \sum_{S} S(f(k))
-= S \left( \sum_{k\ \mathrm{irreducible}} \frac{N_S}{N_{S,k}} f(k) \right).
+&= \sum_{k\ \mathrm{irreducible}} \sum_{S\text{ mapping $k$ to a reducible point}} S(f(k)) \\
+&= \sum_{k\ \mathrm{irreducible}} \frac{1}{N_{S,k}} \sum_{S \in \mathcal S} S(f(k))\\
+&= \frac 1 {N_S} \sum_{S \in \mathcal S} \left(\sum_{k\ \mathrm{irreducible}} \frac{N_S}{N_{S,k}} f(k) \right)
+\end{aligned}
 ```
-Here, ``N_S`` and ``N_{S,k}`` are the total number of symmetry operations and the
+Here, ``N_S = |\mathcal S|`` and ``N_{S,k}`` are the total number of symmetry operations and the
 number of operations such that ``k=Sk``, respectively. The latter operations form
-a subgroup of the group of all symmetry operations. This subgroup is often called
+a subgroup of the group of all symmetry operations, sometimes called
 the "small/little group of ``k``".
-The factor ``\frac{N_S}{N_{S,k}}`` determines the weight of each ``k`` point.
+The factor ``\frac{N_S}{N_{S,k}}``, also equal to the ratio of number of reducible points encoded by this particular irreducible ``k`` to the total number of reducible points, determines the weight of each irreducible ``k`` point.
 
 ## Example
 ```@setup symmetries