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For a rank-two covariant tensor, the derivative commutator contributes one curvature action on each index:

Pairing residues at opposite poles can produce an alternating series that converges in a larger half-plane than the corresponding absolutely summed residue series.

Residues at symmetry-related poles can occur with opposite signs and cancel in the contour sum.

Suppose a contour follows the real axis, closes in the upper half-plane, and avoids each simple real pole $a_j$ by a small semicircle above it. Each indentation is clockwise and tends toIf the large arc vanishes and there are no enclosed poles away from the real axis, the Cauchy principal value is therefore

Ifwith $g$ holomorphic at $a$, then

An indecomposable representation is nonzero and cannot be expressed as a direct sum of two nonzero invariant subspaces. It is an indecomposable module over the associated group algebra or representing algebra.

A representation factors through a finite quotient exactly when its kernel contains the defining kernel of that quotient.

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