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The character of a finite-dimensional representation $\rho$ is the class function ${\chi }_{\rho }(g)=\mathrm{tr}(\rho (g))$. Isomorphic representations have the same character.

The character table of a finite group lists its irreducible complex characters on its conjugacy classes. The number of rows equals the number of conjugacy classes, and the squared row degrees sum to the group order.

Irreducible characters are orthonormal for $⟨\chi ,\psi ⟩=|G{|}^{-1}{\sum }_g\chi (g)\overline{\psi (g)}$, and ${\sum }_{\chi }\chi (1{)}^2=|G|$.

A complex representation is unitary when every representing matrix preserves a positive-definite Hermitian inner product.

Averaging any object over a finite group, $|G{|}^{-1}{\sum }_{g\in G}g\cdot x$, produces a group-invariant object whenever the relevant operations are linear.

Newton's identities applied to the eigenvalues of $g$ give

For any finite-dimensional complex representation,

A complex representation of a group $G$ is a homomorphism $\rho :G\to \mathrm{GL}(V)$ for a complex vector space $V$.

Every group representation of a finite group over a field whose characteristic does not divide the number of elements of the group is semisimple. Given an invariant subspace, averaging any projection onto it over the group produces an equivariant projection, whose kernel is an invariant complement.

A group representation is a group homomorphism from a group to the general linear group of a vector space. It realizes abstract group elements as invertible linear transformations.

The defining identities are $\rho (e)=I$ and $\rho (gh)=\rho (g)\rho (h)$, so an inverse element is represented by a matrix inverse. A choice of basis gives a matrix representation, but the underlying action does not depend on that choice. The action need not distinguish every group element: the trivial representation sends all of them to $I$.

A gauge field is a connection associated with a local symmetry in a gauge theory. Its gauge potential defines the internal-symmetry contribution to a covariant derivative. For an Abelian gauge field, the gauge field strength is $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$.

A free Dirac field has Lagrangian density $\mathcal{L}=i\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -m\bar{\psi }\psi$ and obeys the Dirac equation.

A complex scalar field has distinct particle and antiparticle excitations and a global phase symmetry.

A free real scalar field obeys the Klein-Gordon equation and creates neutral spin-zero particles.

Perpendicular fields with $\mathbf{E}\cdot \mathbf{B}=0$ and $E^2-c^2{B}^2=0$ form a null electromagnetic field. The constant-field Lorentz-force generator is nilpotent, so its exponential terminates as a polynomial.

Both invariants of the electromagnetic field tensor vanish for this field class.

A particle released from rest in a constant electric field parallel to its motion follows a spacetime hyperbola and has rapidity proportional to proper time.

This force law realizes hyperbolic motion with constant proper acceleration.

Lorentz boosts mix transverse electric and magnetic field components.

The four-potential combines the electric potential and magnetic vector potential as $A^{\mu }=(\varphi /c,\mathbf{A})$. The electromagnetic four-potential $A_{\mu }$ transforms underwhile the antisymmetric field tensor $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ is unchanged.

For metric signature $(-,+,+,+)$, a particle of mass $m$ and charge $q$ coupled to a four-potential has the reparametrization-invariant action

Every real number in $[0,1]$ is a sum ${\sum }_{j\ge 1}{\epsilon }_j{2}^{-j}$ with binary digits ${\epsilon }_j\in \{0,1\}$.