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A one-dimensional character is a character of a representation arising from a one-dimensional representation, equivalently a homomorphism from a group to the multiplicative group of its scalar field.

The infinite dihedral group is $⟨r,t\mid t^2=1,tr{t}^{-1}=r^{-1}⟩$.

An irreducible group representation is nonzero and has no nonzero proper invariant subspace. These are the pieces that cannot be decomposed further by taking a smaller symmetry-preserved vector space.

The real two-dimensional rotation through $120$ degrees illustrates the role of the coefficient field. A one-dimensional real invariant subspace would supply a real eigenvector, but the characteristic polynomial ${\lambda }^2+\lambda +1$ has no real root. The representation is therefore irreducible over the real numbers. Over the complex numbers, that polynomial has distinct roots of unity $e^{2\pi i/3}$ and $e^{-2\pi i/3}$, giving two invariant eigenlines. Irreducibility over the real numbers does not imply absolute irreducibility.

Induction extends a representation of a subgroup to the whole group; Frobenius reciprocity computes its multiplicities.

For a representation on $V$, the dual action on $V^{\ast }$ is $(g\phi )(v)=\phi (g^{-1}v)$.

An intertwining operator between representations $\rho$ and $\sigma$ is a linear equivariant map $T$ satisfying $T\rho (g)=\sigma (g)T$ for every group element $g$.

For a finite $G$-set $X$, the character of $\mathbb{C}[X]$ at $g$ is $|X^g|$. Its invariant dimension is the number of $G$-orbits.

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.