E (or e in
the case of vibrational modes) stands for a 2-fold degenerate
representation of a
symmetry group such as, for example, that of the triangle
D3, or the cubic group.
T (or t in the case of vibrational modes, sometimes
also indicated as F or f) stands for a 3-fold degenerate
representation of a symmetry group such as, for example, the cubic
group, or the group of the icosahedron
Ih.
H (or
h in the case of vibrational modes)
stands for a 5-fold degenerate
representation. Among finite point groups, such a large degeneracy
appears only in the icosahedral
Ih group. If the icosahedral symmetry is perturbed by a
cubic field, an H representation splits
into E+T, much like a d (i.e. L=2) atomic level.
A group (either finite of infinite) is said simply reducible when the direct product of any two
irreducible
representations (irreps) decomposes into different irreps. It
means that each irrep appearing in the decomposition is there only once.
For example, the irreps of the group of rotations in 3 dimensions (SO(3)),
labeled by a "angular momentum" label L, combine according to what
the physicists know as theory of the composition of angular momentum: as
our memories of undergraduate quantum mechanics tell us, the direct product
L1 × L2 decomposes into
(|L1 - L2|) +
(|L1 - L2|+1) +
(|L1 - L2|+2) +
(|L1 - L2|+3) + ... +
(L1 + L2 -2) +
(L1 + L2 -1) +
(L1 + L2), each representation
appearing only once. So the group SO(3) is simply reducible, as the cubic
group, and a large fraction of the point groups.
Vice versa, many groups are not simply
reducible, meaning that in at least one irrep product decomposition
there appears the same irrep at least twice. For example, the icosahedral
group Ih is not simply reducible since, say, the product
H × H (5 × 5 = 25 states) decomposes into
A + T + T + G + G + H
+ H (1+3+3+4+4+5+5 = 25 states).
the concept of parallel
transport belongs to differential geometry. Given a "path" on a
differential manifold, for each
point along the path one can choose a vector in the tangent space to the
manifold. It may happen that the vector at any two close points are "as
parallel as possible" to each other. More precisely, that (to leading
order) the two tangent vectors differ only by an amount proportional to the
square of the distance between the points along the path. If this is the
case along all the path, then the tangent vector is said to be
parallel-transported along it.
