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A vector $v$ is cyclic for a group representation $\rho :G\to GL(V)$ if the linear span of $\{\rho (g)v:g\in G\}$ is all of $V$. Equivalently $v$ generates $V$ as a module over the group algebra. Its projection to the invariant subspace generates that subspace, so a cyclic representation of a finite group has at most one copy of the trivial representation.

Every affine algebraic group has a faithful finite-dimensional rational group representation. Choose algebra generators of $k[G]$ and place them in a finite-dimensional subspace stable under right translations. An element acting trivially on that subspace has the same coordinate values as the identity and therefore is the identity.

The complexified Lorentz algebra splits into two commuting ${\mathfrak{sl}}_2$ factors. Extending two homogeneous polynomial representations of SU2 therefore gives every finite-dimensional irreducible complex group representation of the Lorentz spinor double cover. The kernel sign permits descent precisely when $j_L+j_R$ is an integer. These field group representations are generally nonunitary for a positive-definite inner product; infinite-dimensional unitary Lorentz group representations belong to a different construction.

A four-dimensional $\mathcal{N}=4$ supersymmetric gauge theory has one supersymmetric vector multiplet containing a gauge boson, four Weyl spinors, and six real scalar fields in the adjoint group representation. Its massless helicity multiplicities are $1,4,6,4,1$ from helicity $1$ to $-1$.

A Galois representation is a continuous group representation of an absolute Galois group, usually on a vector space or lattice over a local field. Examples include the cyclotomic character and the representations attached to modular forms. The coefficient topology is part of the definition.

A gauge group representation specifies how a matter field transforms under a gauge group. It fixes the gauge charges and whether gauge-invariant mass terms can pair a field with a field in the conjugate group representation.

The group norm element in the group algebra $kG$ is $N_G={\sum }_{g\in G}g$. It spans the invariant subspace of the left regular group representation, and $N_G^2=|G|N_G$. If the characteristic divides $|G|$, a projection from $kG$ onto $k{N}_G$ would send $N_G$ to both itself and zero, proving that $kG$ is not a semisimple algebra. Its action on indecomposable modules is described by group norm element detects the trivial projective cover.

An intertwiner between group representations $\rho$ on $V$ and $\sigma$ on $W$ is a linear map $A:V\to W$ satisfying $A\rho (x)=\sigma (x)A$ for every group element. The Schur lemma controls these maps between irreducible representations.

An invariant subspace of a linear map $A:V\to V$ is a vector subspace $W\subseteq V$ satisfying $A(W)\subseteq W$. For a group representation, invariance means $\rho (g)W\subseteq W$ for every represented group element $g$.

An invariant tensor is fixed by every element of the represented group, so it spans or belongs to a trivial group representation. The special unitary group preserves its identity tensor and the upper and lower volume tensors. Such tensors make tensor contraction equivariant and permit invariant inner products and real structures.

With Adjoint representation convention ${\mathrm{ad}}_X(Y)=[X,Y]$, the matrix exponential identity is $e^{-X}Y{e}^X=e^{-{\mathrm{ad}}_X}Y$. Inverse conjugation reverses product order, so it is a right action or antirepresentation, whereas ${\mathrm{Ad}}_g(Y)=gY{g}^{-1}$ is an ordinary left group representation. The first-order terms $Y\mp [X,Y]$ detect a sign mismatch immediately. Negating every adjoint generator without reversing the Lie bracket does not give a new Lie-algebra homomorphism.

A Lie-group representation is a smooth group representation $D:G\to GL(V)$.

When ${\partial }_{-}$ is invertible, divergence determines components with a plus index from $h_{--},h_{-I},h_{IJ}$. The trace constraint fixes the trace of $h_{IJ}$. The remaining transverse group representations are a symmetric traceless square, a vector representation, and a scalar.

Identify Minkowski vectors with Hermitian matrices $X=x^{0}I+x^a{\sigma }_a$. The action $X\mapsto SX{S}^{†}$ preserves $\det X=(x^0{)}^2-|\mathbf{x}{|}^2$. It gives a surjective homomorphism from $SL(2,\mathbb{C})$ to the Proper orthochronous Lorentz group with kernel $\{I,-I\}$. Polar decomposition retracts the covering group onto SU(2), making it simply connected. This cover permits half-integer spin group representations that do not descend to the Lorentz group itself.

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 matrix coefficient of a finite-dimensional group representation is a scalar function obtained by applying a linear functional to the translate of a fixed vector. In a chosen basis these functions include $x\mapsto \rho (x{)}_{ij}$. For a finite group, the Schur orthogonality relations make the scaled coefficients of all inequivalent unitary irreducible representations an orthonormal basis of its scalar functions.

A matrix representation is a group representation together with a basis, so each group element is represented by an invertible matrix and multiplication is represented by matrix multiplication.

The modular representation ring is the Grothendieck group of finite-dimensional group representations, with relations $[V]=[U]+[W]$ for every short exact sequence $0\to U\to V\to W\to 0$, and multiplication $[V][W]=[V{\otimes }_{k}W]$ using the tensor product of group representations. The Jordan–Hölder theorem makes the classes of simple modules a free integral basis. Tensoring over a field is exact, so this multiplication is well defined. When $k$ is a splitting field for finite group representations, Brauer characters identify $\mathbb{C}{\otimes }_{\mathbb{Z}}{R}_k(G)$ with the algebra of complex class functions on p-regular elements. The splitting hypothesis is essential: ${\mathbb{F}}_2{C}_3\cong {\mathbb{F}}_2\times {\mathbb{F}}_4$ has two simple modules, although $C_3$ has three $2$-regular conjugacy classes.