Representation categories at a glance

“The representations of a group” is not a complete specification. The answer changes with the coefficient field, characteristic, continuity, dimension, and whether multiplication holds exactly or only up to phase. Even after these choices, a classification of irreducibles supplies neither a preferred basis nor explicit matrices. This page identifies the category needed before choosing an atlas record or a calculator.

Specify the question

An ordinary representation is a homomorphism $\rho:G\to GL_k(V)$. Record:

1. Object: the exact group, including its global form, or a specified Lie algebra. For branching, also give the homomorphism being restricted along.
2. Scalars: the field $k$, its characteristic, and any field extension allowed. “Over $\mathbb R$” differs from “a complex representation of real type.”
3. Regularity: for a Lie group, continuous finite-dimensional representations or strongly continuous unitary representations on a Hilbert space. Algebraic representations of an algebraic group are a further specified category.
4. Size and structure: irreducible or arbitrary, finite-dimensional or unrestricted, ordinary or projective, and any invariant positive inner product required.
5. Requested output: equivalence classes, characters, decomposition multiplicities, generator matrices, or explicit intertwining tensors.

For a Hilbert-space representation, strong continuity means that $g\mapsto\rho(g)v$ is continuous for each vector $v$. Unitarity refers to a positive-definite Hermitian inner product; preserving an indefinite spacetime metric does not establish it. These definitions follow Kowalski, §§2.1, 3.2–3.3.

Which decomposition theorem applies?

Here “semisimple” means that every representation in the indicated finite-dimensional category is a direct sum of irreducibles.

Category	Decomposition and labels	What must remain explicit
Finite $G$, ordinary complex, characteristic zero	Semisimple; finitely many irreducibles, classified by their characters	The group, conjugacy-class order, and full character values
Finite $G$ over $k$ with $\operatorname{char}k\nmid\lvert G\rvert$	Semisimple by Maschke’s theorem	The field need not split all irreducibles; complex labels cannot simply be reused
Finite $G$ in characteristic $p\mid\lvert G\rvert$	Generally not semisimple; indecomposable extensions matter	Field, prime, and modular data; Brauer characters concern $p$-regular elements
Connected compact Lie $G$, continuous finite-dimensional complex	Unitarizable and semisimple	Highest weights for the simply connected semisimple cover, plus descent and any torus charges
Complex semisimple Lie algebra, finite-dimensional	Semisimple; dominant integral highest weights	Integration to a specified group is an additional condition
Noncompact real Lie group	Finite-dimensional field representations and unitary Hilbert-space representations are different problems	Real form, cover, continuity, unitarity, and any energy or spectral restrictions
Complex projective representations	Multiplication holds up to a scalar cocycle	Multiplier class or central extension; ordinary tables alone do not specify it

Maschke’s averaging argument requires division by $|G|$: Etingof et al., §4.1, Theorem 4.1.1 and Proposition 4.1.2. For a compact Lie group, averaging a positive Hermitian form with normalized Haar measure instead gives

$$\langle v,w\rangle_G =\int_G\langle\rho(g)v,\rho(g)w\rangle_0\,dg, \qquad \int_Gdg=1.$$

The form is positive and invariant, so an invariant subspace has an invariant orthogonal complement. For semisimple Lie algebras, the corresponding complete-reducibility theorem assumes characteristic zero. See Etingof, §35.2, Proposition 35.5 and Corollary 35.6; §26.10, Theorem 26.11.

Neither conclusion extends to every representation of every Lie algebra. For example, a one-dimensional abelian Lie algebra may act by a nonzero nilpotent Jordan block. Its invariant line has no invariant complement, despite the algebra being reductive. Restricting to representations whose center acts semisimply is a consequential extra assumption.

Worked example: all representations of SU(2)

For continuous finite-dimensional complex irreducibles of the compact, simply connected group $SU(2)$, the answer is the infinite family

$$V_n=\operatorname{Sym}^n\mathbb C^2, \qquad n=0,1,2,\ldots,\qquad \dim V_n=n+1,\qquad j=n/2.$$

With torus element $\operatorname{diag}(e^{i\theta},e^{-i\theta})$, the monomials $x^{n-r}y^r$ have weights $n-2r$, so

$$\chi_n(\theta)=\sum_{r=0}^n e^{i(n-2r)\theta}.$$

The weights give the dimension at $\theta=0$ and the central action $\rho_n(-I)=(-1)^nI$. Thus only even $n$ descend to $SO(3)$. The highest-weight construction is developed in Etingof et al., §2.15 and our irreducible-representation lesson. Use the highest-weight tool for a bounded calculation, and covers and global forms for the descent condition.

If the question instead asks for all finite-dimensional representations, one must allow every finite direct sum $\bigoplus_n m_nV_n$, with $m_n\in\mathbb Z_{\ge0}$. If it asks for strongly continuous unitary Hilbert-space representations, Hilbert direct sums are allowed. Compactness still forces every irreducible unitary representation to be finite-dimensional; this is Kowalski, §5.4, Corollary 5.4.2.

Changing the coefficient field changes the bookkeeping again. The invariant antisymmetric form on $\mathbb C^2$ induces a bilinear form on $V_n$ with exchange sign $(-1)^n$. Accordingly $V_n$ is of real type for even $n$ and quaternionic type for odd $n$. An odd-$n$ complex irrep does not have a real form of real dimension $n+1$; its underlying real representation has dimension $2(n+1)$. See reality types. This does not mean that its ordinary complex representation matrices have changed coefficient field.

Why an ordinary table does not solve a modular problem

Take $G=C_3=\langle g\mid g^3=1\rangle$. Over $\mathbb C$, its three irreducibles send $g$ to $1,\omega,\omega^2$, where $\omega^3=1$. Over a field $k$ of characteristic three,

$$x^3-1=(x-1)^3.$$

For any nonzero finite-dimensional module, $\rho(g)-I$ is nilpotent and has a nonzero kernel. On a simple module that invariant kernel is the whole space, so the only simple module is the one-dimensional trivial module. This is the $p=3$ case of Etingof et al., §4.1, Example 4.1.3, p. 63.

Nevertheless, the two-dimensional representation

$$\rho(g)=J=\begin{pmatrix}1&1\\0&1\end{pmatrix}, \qquad J^3=I$$

is not the direct sum of two trivial modules. Its fixed space is only $k e_1$. If an invariant complementary line were spanned by $ae_1+be_2$ with $b\ne0$, its generator would have eigenvalue one, which would force $b=0$ from $(J-I)(ae_1+be_2)=be_1$. Thus the extension does not split. The matrices $J^r$ all have trace two, just as two trivial summands do: traces do not recover this extension.

Brauer characters use the elements whose orders are prime to $p$, with eigenvalues lifted to characteristic zero. For $C_3$ at $p=3$, only the identity is $p$-regular. A Brauer character therefore cannot distinguish the extension above from its two composition factors. GAP distinguishes ordinary tables from this modular input in Reference Manual §71.3-2, BrauerTable. The site’s finite-character workbench implements the ordinary complex category only.

Projective and noncompact questions

For unitary projective representations over $\mathbb C$, a choice of representatives obeys

$$\rho(g)\rho(h)=\omega(g,h)\rho(gh),\qquad \omega(g,h)\in U(1).$$

Associativity imposes $\omega(g,h)\omega(gh,k)=\omega(h,k)\omega(g,hk)$. Changing representatives by phases changes $\omega$ by a coboundary. The resulting map to the projective unitary group is the intrinsic object; a globally continuous choice of representatives need not exist. A suitable central extension can turn such a projective action into an ordinary one, with specified action of the central kernel. For example, the $SU(2)$ doublet induces a projective action of $SO(3)$ on rays, although it fails ordinary descent because $-I$ acts as $-1$.

For noncompact groups, finite dimension does not imply unitarizability. In the defining representation of $SL(2,\mathbb R)$, the element $\operatorname{diag}(e^t,e^{-t})$ has eigenvalues off the unit circle for $t\ne0$, ruling out any invariant positive-definite Hermitian form. Conversely, noncompactness does not force all irreducibles to be infinite-dimensional: the additive group $\mathbb R$ has continuous unitary characters $x\mapsto e^{ipx}$ for every real $p$. For the physical distinction between covariant field indices and unitary particle states, continue to spacetime representations, CFT representations, and the spacetime learning path.

What the site’s tools actually return

Tools	Representation category and output
SU(2) coupling, SU(3) products, SU(N) products	Finite-dimensional complex compact-group irreducibles; decomposition multiplicities
Highest weights	Dominant integral weights in selected simple types; dimensions, Casimirs, indices, conjugates, and adjoint descent
Finite characters	Ordinary complex characters of seven finite groups; exact character operations and multiplicities
Selected branching, SU(3) branching, charged block branching	Restriction along specified maps, with the charges retained by the chosen tool
Global forms, matter quotients, product gauge groups	Central actions and descent for specified compact groups and matter
Tensor invariants, local operators, operator enumeration	Supported complex invariant spaces; local-field tools additionally impose Lorentz and exchange statistics
Topology	Selected groups and homogeneous spaces with fixed maps; this is not an irrep classifier

The coverage policy supplies the numerical bounds; the conventions fix labels and normalizations. A character determines an ordinary complex finite-group representation up to equivalence, but it does not select matrices. A tensor multiplicity counts intertwiners without selecting their normalized components. Use the external-resource guide when explicit generator matrices, modular modules, or broader computations are needed.