Mathematics for Physics and Astronomy

Group axioms

1. Closure: For all elements f,\, g ∈ G, it is the case that the element f ∗g ∈ G.

2. Identity: There exists an element e\in G such that: e*g=g*e=e.

3. Inverse: For all g ∈ G there exists a element g^{−1} ∈ G with the property that g ∗g^{−1} = g^{−1} ∗g =e.

4. Associativity: For all elements f, \,g,\, h ∈ G, it holds that (f ∗g)∗h = f ∗(g∗h).

Order of a group

The number of elements in a finite group, where |G| is the order of G.

Order of elements

The order of g\in G is given by |g|=k, where k is the smallest value for which g^k = e.

Cyclic group

The group of rotations of a regular n-sided polygon,

C_n =\{e,r,r^2,...,r^{n−1}\}, which has |C_n|=n.

• Is an Abelian group.

Dihedral group

The group of rotations and reflections of a regular n-sided polygon (n ≥ 3), D_n =\{e,r,r^2,...,r^{n−1},s,rs,r^2s,...,r^{n−1}s\}, where r is anticlockwise rotation by 2π/n and s is a reflection.

• Has has order |D_n|=2n.

Abelian group

Groups for which the operation ∗ is commutative, i.e. f ∗ g = g ∗ f for all f,\,g ∈ G.

The multiplication table of an Abelian group is symmetric about the leading diagonal.

Subgroup

If a subset H forms a group under the same group operation as G, then H is a subgroup of G, and we write H ≤ G.

To test for a subgroup, ask:

• Does H contains the identity element of G?

• For all g,h\in H, is gh\in H?

• For all g\in H, is g^{-1}\in H?

Lagrange’s theorem

If G is a finite group and H is a subgroup of G, then |H| divides |G|.

Homomorphism

A mapping φ : G → G' is called a group homomorphism if

φ(xy) = φ(x)φ(y)

for all x,y ∈ G.

In other words, any two elements x,y ∈ G have the same multiplication properties in G as their images do in G'.

Properties of a homomorphism

• The identity element of G is mapped to the identity element of G', i.e., φ(e_G)=e_{G’}.

• The inverse of x ∈ G is mapped to the inverse of ϕ(x) ∈ G', i.e., φ(x^{−1}) = φ(x)^{−1}.

Bijective function

A function between two sets such that each element of the second set is the image of exactly one element of the first set, i.e., it gives a one-to-one correspondence.

Isomorphism

A bijective homomorphism is called an isomorphism.

G and H are isomorphic if there is a bijection f : G→ H such that for all x,y\in G we have f(xy) = f(x)f(y).

If two groups have identical multiplication tables, up to a relabelling of group elements, they are said to be isomorphic. If two groups G and G' are isomorphic, we write G \cong G'.

Matrix group

A group whose elements are invertible square matrices, equipped with the group operation of matrix multiplication.

An example of an infinite matrix group is the General Linear Group: GL(n):=\{A∈\text{Mat}_{n×n}(\mathbb{R}):\text{det}(A)\neq0\}

Representation

A n-dimensional representation of a finite group G is a homomorphism ρ : G → M, where M is a matrix group of n×n matrices.

Faithful representation

A representation is said to be faithful if ρ : G → M is an isomorphism.

Reducible representation

An n-dimensional representation ρ : G → M of a finite group G is said to be reducible if there exists an invertible matrix S such that

S^{−1}ρ(g)S = \begin{pmatrix}ρ_1(g)\,\,\,\,\,\,\,\,\ 0 \\0 \,\,\,\,\,\,\,\,\,\ ρ_2(g) \end{pmatrix}\qquad \text{for each } g∈G

where ρ_1 and ρ_2 are n_1 ×n_1 and n_2 ×n_2 dimensional representations of G respectively (satisfying n_1 +n_2 =n, and at least one of n_1,\,n_2 is non-zero).

Reducible representations all share the same eigenvectors.

Differentiable function (complex)

A complex function f(z) is said to be differentiable at the point z = z_0 if the limit f(z_0) = \lim_{z\rightarrow z_0} \frac{f(z)- f(z_0)}{ z -z_0} exists, that is to say it is independent of the direction of approach of z\rightarrow z_0 in the complex plane.

A function is analytic in a region D if it is differentiable at every point z\in D.

Fundamental theorem of calculus

If f(z) is analytic everywhere in an open region D that encloses a piecewise-smooth curve C, and a function F(z), satisfying F'(z) = f(z), can be found that is also analytic everywhere in D, then

\int_C f(z)\, \mathrm{d}z = F(z_B)- F(z_A)

where z_A and z_B are the locations of the start and end points of C.

Cauchy’s residue theorem

If a function f(z) is analytic and single-valued everywhere in a region R containing a closed contour C, except at a finite number of isolated singularities inside C (at z_1,...,z_N), then

\oint_Cf(z)\,\mathrm{d}z=2\pi i \sum_{j=1}^N\text{Res}\{f(z);z=z_j\}

Equivalent/similar groups

Two representations \rho,\psi are equivalent if they map to the same matrix group and there exists an invertible matrix A such that for all g\in G we have

A\rho(g)A^{-1} = \psi(g)

If two representations are equivalent, they will have the same eigenvalues for each g\in G.

(Equivalent representations must have the same dimension.)

Cauchy-Riemann equations

A function f(z)=u(x,y)+iv(x,y) is analytic in the region which it satisfies

u_x =v_y\quad \text{and}\quad v_x = -u_y.

Harmonic function

A twice-differentiable function \phi(x,y) that satisfies Laplace's equation, that is

\nabla^2 \phi=\phi_{xx}+\phi_{yy}=0

One-dimensional representations

• Are always irreducible.

• As 1×1 matrices commute, they are equivalent for \rho_2(g)=S^{-1}\rho_1(g)S=S^{-1}S\rho_1(g)=\rho_1(g), and so two representations are only equivalent if they commute.

Reducibility and eigenvectors

A 2- or 3-dimensional representation \rho is reducible if and only if the matrices \{\rho(g):g\in G\} all share a common eigenvector.

Conjugacy class

\text{Given }\, x,y ∈ G : x∼y \,\text{ if there exists }\,b ∈ G\,\text{ such that } \,y = b^{−1}xb

Under this equivalence relation, the equivalence class [g] of an element g ∈ G is called the conjugacy class of g.

Note the only conjugacy class which is also a subgroup is \{e\}.

Conjugacy class for an abelian group

The conjugacy classes are

[g] = \{g\}\quad \text{for each} \quad g ∈ G.

Note that this means all conjugacy classes for an abelian group are one-dimensional, and as the number of conjugacy classes is equal to the number of irreducible representations, all these irreducible representations are one-dimensional.

Number of conjugacy classes

The number of conjugacy classes in a group G is equal to the number of irreducible representations of that group, up to equivalence.

Order of a permutation in disjoint cycle form?

The order is the lowest common multiple of the lengths of its cycles.

Isolated singularity

A function f(z) is said to have an isolated singularity at z = z_0 if there exists an \varepsilon> 0 such that the singularity at z = z_0 is the only one within the circle |z- z_0| < \varepsilon.

An example of a non-isolated singularity is \frac{1}{\sin(1/z)}

Essential singularity

An isolated singularity at z = z_0 is said be an essential singularity if the principal part of the Laurent expansion valid in the immediate vicinity of z_0 is infinite.

E.g. e^{1/z}

Pole of order m

An isolated singularity at z = z_0 is said to be a pole of order m if the principal part of the Laurent expansion valid in the immediate vicinity of z_0 has a finite number of terms and has its lowest non-zero coefficient for (z- z_0)^{-m}, that is, A_n = 0 for all n < -m.

A pole of order one is said to be a simple pole.

Residue

The residue of an isolated singularity of an analytic function f(z) at z = z_0 is the coefficient (A_{-1}) of the (z-z_0)^{-1} term in the Laurent series valid in the immediate vicinity of the singularity.

A_{-1}=\text{Res} \{f(z); z =z_0\} =\frac{1}{(m-1)!}\left[\frac{d^{(m-1)}}{dz^{(m-1)}}\left((z-z_0)^m f(z)\right) \right]_{z=z_0}

Methods of finding whether a representation is reducible

• Finding eigenvectors: if every rep. shares a common eigenvector, then that rep. is reducible.

• Using character theory: a rep. \rho is reducible if \frac{1}{|G|}\sum_{g\in G}\chi_\rho^*(g)\chi_\rho(g)\neq 1

If a rep. \rho: G\rightarrow M is reducible, there exists an invertible matrix such that

S^{-1}\rho(g)S= \begin{pmatrix}\rho_1(g)\ \,\,\,\,0\,\,\,\, \\ \,\,\,\,0\,\,\,\,\rho_2(g)\end{pmatrix}

Jordan’s lemma

Consider the function

g(R) = \max_{\theta\in[0,\pi]}|f(Re^{i\theta})|

If g(R)\rightarrow 0 as R\rightarrow\infty then

\lim_{R\rightarrow\infty}\int_{C_R} f(z)e^{iaz} \mathrm{d}z = 0

where a > 0 and C_R is the path from R to -R along the half of circle |z|= R in \text{Im}(z)\geq 0.