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A group representation over a field is absolutely irreducible if it remains irreducible after every field extension. For a finite-dimensional representation it is enough to check an algebraic closure. Irreducibility over the original field alone is weaker: the two-dimensional real rotation representation of the cyclic group of order three splits into two lines over the complex numbers.

A real group representation is absolutely irreducible if its complexification is irreducible. For a finite group this is equivalent to having only scalar real endomorphisms commuting with the representation. Independent coordinate sign changes force a commuting matrix to be diagonal; a transitive cyclic permutation of the coordinates then forces its entries equal. Alternatively, the coordinate projections and the cyclic permutation directly prove irreducibility over either field.

Every complex character of a representation of a finite group $G$ is a rational linear combination of characters of induced representations from one-dimensional group representations of cyclic subgroups $C_j$. One-dimensionality follows because the irreducible representations of a finite cyclic group over the complex numbers are one-dimensional. The rational coefficients distinguish this statement from Brauer induction, which permits integer coefficients by allowing more general subgroups. For a rational-valued character, another form of Artin induction uses rational combinations of induced representations of trivial group representations of cyclic subgroups.

The branching of a group representation $V$ to a subgroup $H$ is its decomposition into irreducible representations of $H$, when that restriction of a representation is a semisimple representation: $V{|}_H={⨁}_{\lambda }{m}_{\lambda }{W}_{\lambda }$. The nonnegative integers $m_{\lambda }$ are the branching multiplicities. For finite groups over the complex numbers, they are the character inner products of the restricted character with the irreducible characters of $H$. Restriction to an arbitrary subgroup need not be completely reducible.

A branching rule decomposes a representation into irreducible representations after restriction from a Lie group or Lie algebra to a subgroup or subalgebra.

A branching rule gives the irreducible representation decomposition after restricting a group representation to a subgroup, or a Lie algebra representation to a Lie subalgebra. For the standard block embedding $SU(5)\subset SU(6)$, $6\downarrow =5\oplus 1$ and ${\Lambda }^{2}6\downarrow =10\oplus 5$.

Every complex character of a representation of a finite group $G$ is an integer linear combination of characters of induced representations from one-dimensional group representations of subgroups $H_j\le G$. Negative coefficients are allowed. Unlike Artin induction, the subgroups need not be cyclic groups. Applied to an Artin L-function, compatibility with induced representations turns this character identity into a product of one-dimensional Artin L-functions with integer exponents, reducing the behavior at $s=1$ to abelian extensions.

For a compact torus $T\cong U(1{)}^r$, its continuous one-dimensional characters form a free abelian group of rank $r$. They are precisely $z\mapsto z_1^{m_1}\cdots z_r^{m_r}$ with integer exponents. This follows from the classification of representations of the circle group, applied to each factor. These characters are the integral group weights occurring upon restricting a group representation to a maximal torus; global integrality must be distinguished from integrality against coroots for a simply connected semisimple covering group.

A symmetry transformation acts on localized atomic orbitals both by moving their centres and by transforming their angular shapes. Only centres left in place can contribute to the trace of the corresponding group representation. At each fixed centre, sum the diagonal coefficients of the orbital transformation: a reversed p orbital contributes $-1$, whereas a fixed s orbital contributes $+1$. For Cartesian displacement vectors at a fixed atom, use the trace of the full polar-vector matrix, not merely the number of fixed components. This rule generates characters of a representation without writing its entire matrix.

For finite-dimensional group representations, the character of their tensor product is the pointwise product of their characters. In a tensor-product basis, the diagonal entries of $A\otimes B$ are $A_{ii}{B}_{jj}$, so their sum is $(\mathrm{tr}A)(\mathrm{tr}B)$. This gives a direct route from a character table to tensor-product decompositions.

The charge-conjugate field of a fermion field transforms in the complex-conjugate internal group representation. With a compatible Dirac adjoint and charge conjugation matrix convention it is ${\psi }^c=C{\bar{\psi }}^T$, up to a conventional constant phase. In four spacetime dimensions charge conjugation exchanges the chirality of the field; a right-handed quark can therefore be represented by a left-handed conjugate field when listing chiral unified matter. In two-dimensional Minkowski spacetime, charge conjugation can instead preserve chirality, permitting a Majorana-Weyl spinor.

For a finite-dimensional complex group representation, combining inverse and conjugate transpose gives another representation: both operations reverse multiplication order. This is the complex conjugate of the algebraic dual representation in matrices. If $W$ is invariant under $D^{\#}$, its orthogonal complement is invariant under $D$, because $⟨w,D(g)v⟩=⟨D^{\#}(g^{-1})w,v⟩=0$ for $w\in W$, $v\in W^{\perp }$. Thus irreducibility is preserved. Applying the construction twice returns $D$, so the equivalence of irreducibility works in both directions for arbitrary groups in finite dimension.

For a complex group representation with matrices $R(g)$, its conjugate representation has matrices $\overline{R(g)}$ on the conjugate vector space. In a unitary representation it is equivalent to the dual representation, since $(R(g{)}^{-1}{)}^T=\overline{R(g)}$. On a diagonal maximal torus, its weights are the negatives of the original weights.

A continuous unitary character is a continuous group homomorphism from a topological group to the unit circle with multiplication. For abelian groups, multiplication of continuous unitary characters is pointwise. Such continuous unitary characters must be distinguished from the traces of higher-dimensional group representations, called characters of a representation, and from characters of an algebra.

For every integer $n$, the map ${\chi }_n(z)=z^n$ is a continuous unitary character of the circle group: ${\chi }_n(zw)={\chi }_n(z){\chi }_n(w)$ and $|{\chi }_n(z)|=1$. By contrast, the two-dimensional group representation $\rho (z)=\mathrm{diag}(z,z^{-1})$ has character of a representation $\mathrm{tr}\rho (e^{i\theta })=2\cos \theta$. Its value at the identity element is $2$, so this trace is not a continuous unitary character.

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.

For a group representation $\rho$ with character $\theta$, its determinant character sends $g$ to $\det \rho (g)$. Multiplicativity makes it a linear character, independent of the equivalent matrix realization of $\rho$.

The top exterior power of an $n$-dimensional group representation is one-dimensional, and $g$ acts on it by $\det R(g)$. It is trivial for the defining representation of $SU(n)$. This is why a full $n$-box column in an $SU(n)$ Young diagram can be removed without changing the representation.

For group representations $R:G\to GL(V)$ and $S:G\to GL(W)$, their direct sum acts on the direct sum of vector spaces by $(R\oplus S)(g)(v,w)=(R(g)v,S(g)w)$. Its character and Dynkin index, when defined, are the sums of those of the two summands.

Two finite-dimensional group representations $D,E$ over the same field are equivalent if there is one invertible linear map $S$ intertwining all their group actions, equivalently $E(g)=S^{-1}D(g)S$ for every $g$. The same change of basis must work simultaneously for all group elements. The cyclic property of the trace implies that equivalent representations have identical characters: $\mathrm{tr}(S^{-1}D(g)S)=\mathrm{tr}D(g)$. Consequently a difference at a single group element proves inequivalence. This implication does not need the group to be finite, and the reverse implication is not asserted for arbitrary groups or fields.

Let the Klein four-group have generators $X,Y$ and $Z=XY$, with $X^2=Y^2=I$ and $XY=YX$. For every real $p$, the matricesdefine a group representation. Indeed, $D(Y{)}^2=I_2$, and $D(X)$ commutes with every matrix, so the group relations hold.

Set $S=\left(\begin{array}{cc}1& -p/2\\ 0& 1\end{array}\right)$. A direct change of basis givesConsequently all these representations are isomorphic representations, with character values $(2,-2,0,0)$ on $(I,X,Y,Z)$. Each is a direct sum of the two one-dimensional representations whose values on $X$ are $-1$ and whose values on $Y$ are $+1$ and $-1$. A group representation need not be orthogonal; nonzero $p$ changes the basis rather than the represented group.

An equivariant dynamical system has a vector field commuting with the action of a symmetry group. Its flow map preserves every subgroup's fixed-point subspace, because equivariance and uniqueness keep initial states fixed by that subgroup. Polynomial normal forms are constrained by the weights and parity of the group representation.