Authoritative group-theory resources

Choose the resource by the mathematical output you need. A character table answers many decomposition questions without providing generator matrices; a matrix file fixes a coefficient field and a basis; a Lie branching rule also fixes an embedding. Begin with the representation-category checklist, then use the guide below. These are external research resources, while the site’s supported workbenches calculate locally.

Start from the task

Task	Resource and entry point	Scope to check before reuse
Ordinary or Brauer character table	GAP Character Table Library and GAP Reference Manual, chapter 71	Table identifier, characteristic, class ordering, power maps, and whether irreducible values are actually available
Explicit finite-group generator matrices or permutations	ATLAS of Group Representations and its AtlasRep interface	Exact group or cover, generator convention, field, representation dimension, and individual record availability
Identify a small finite group and compare names	GroupNames	Name aliases, order, presentation, and SmallGroup identifier when supplied
Cross-reference groups, subgroups, conjugacy classes, and characters	LMFDB abstract groups	Separate coverage of groups and their computed properties; consult its completeness statement
Compute finite-dimensional Lie characters and tensor decompositions	SageMath Weyl character rings	Cartan type, node numbering, weight-coordinate style, and the chosen character ring
Restrict a Lie character along an embedding	SageMath branching rules or LieART 2.0, §§3.1–3.3	Actual torus or weight map, abelian normalization, global kernel, and software version

GAP’s table objects and its group representations are distinct objects. CharacterTable and Irr concern characters; IrreducibleRepresentations constructs representatives by available algorithms. The latter has field and algorithm restrictions and can be expensive. Moreover, its ordering need not agree with Irr, and its matrices need not preserve the standard unitary form in the returned basis. These distinctions are documented in GAP §§71.3-1, 71.8-2, and 71.14-4.

AtlasRep provides generating matrices and permutations for many almost simple groups and related records; it does not promise every representation of every finite group. The ATLAS A5 page makes the category visible: it lists characteristic-zero three-dimensional matrices and characteristic-two two-dimensional matrices over $\mathbb F_4$, as well as separate records for the double cover $2.A_5$. The two-dimensional modular representation is not an ordinary complex irrep of $A_5$.

GroupNames describes its bounded small-order collection on its About page. LMFDB’s coverage is broader in several directions, but individual properties are subject to computation and storage limits. In particular, its completeness statement explains omissions of large character tables and some subgroup data. An absent table does not mean that the group lacks representations, and a group record does not imply that every property has been computed.

Worked example: a character is not a matrix realization

For $S_3$, GroupNames and LMFDB record 6.1, “Character theory” identify the group of order six and give the ordinary two-dimensional character. In class order identity, transpositions, three-cycles,

$$\chi=(2,0,-1),\qquad |C_i|=(1,3,2).$$

Its norm is $(4+3\cdot0+2\cdot1)/6=1$, proving irreducibility in the ordinary complex category. Character multiplication also gives $\chi^2=\chi_{\mathbf1}+\chi_{\mathrm{sgn}}+\chi$. These facts are enough for $\mathbf2\otimes\mathbf2$ multiplicities in the finite-character workbench. They do not specify how the generators act on a chosen vector.

To supply that additional data, take $r=(123)$ and $s=(13)$ acting on the sum-zero subspace of $\mathbb C^3$, with column-vector basis $f_1=e_1-e_2$, $f_2=e_2-e_3$. Directly applying the permutations gives

$$R=\rho(r)=\begin{pmatrix}0&-1\\1&-1\end{pmatrix}, \qquad S=\rho(s)=\begin{pmatrix}0&-1\\-1&0\end{pmatrix}.$$

Now check the presentation and traces:

$$R^3=S^2=I,\qquad SRS=R^{-1},\qquad \operatorname{tr}I=2,\quad\operatorname{tr}S=0,\quad \operatorname{tr}R=-1.$$

The six matrices $I,R,R^2,S,RS,R^2S$ are distinct, so the realization is faithful. Its invariant positive form is the Gram matrix of this basis,

$$H=\begin{pmatrix}2&-1\\-1&2\end{pmatrix},\qquad R^\dagger H R=S^\dagger H S=H.$$

These matrices are unitary with respect to $H$, although $R$ is not unitary with respect to the identity matrix. A change of basis conjugates the matrices and transforms the Gram matrix, leaving the character unchanged. This independently constructed realization illustrates what to verify when importing a matrix file: relations, traces, field, kernel, and preserved form. It is not a transcription of a provider’s matrix basis.

Class labels must also be matched. Two providers can present equivalent character tables with different row and column orders. Matching only the row number or only the dimension is insufficient; match class invariants, generator choices, and character values. For more complicated groups, even element order alone may not identify a conjugacy class.

Worked example: record the Lie conventions with the result

Suppose an external system reports

$$\mathbf3\otimes\overline{\mathbf3}=\mathbf1\oplus\mathbf8.$$

A reusable specification is: finite-dimensional complex representations of simply connected compact $SU(3)$; Cartan type $A_2$; simple-coroot Dynkin labels $(1,0)$ and $(0,1)$; result $(0,0)\oplus(1,1)$. In the site’s convention,

$$A=\begin{pmatrix}2&-1\\-1&2\end{pmatrix},\qquad (\alpha_{\mathrm{long}},\alpha_{\mathrm{long}})=2, \qquad C_2(\lambda)=\tfrac12(\lambda,\lambda+2\rho).$$

Sage’s Weyl-character tutorial, “Coroot notation” explains its style="coroots" input; default ambient weight coordinates must not be silently read as Dynkin labels. LieART’s §3.1 and §5.3, equation (13) likewise distinguish representation labels and weights. A dimension printed as “3” alone does not distinguish a fundamental from its conjugate.

An independent derivation identifies $\mathbb C^3\otimes(\mathbb C^3)^*\cong\operatorname{End}(\mathbb C^3)$. Conjugation preserves the scalar matrices and the traceless matrices:

$$\operatorname{End}(\mathbb C^3)=\mathbb C I\oplus\mathfrak{sl}_3(\mathbb C), \qquad 9=1+8.$$

The traceless summand is the adjoint. The center acts trivially on both summands, agreeing with total triality $1+2=0\pmod3$. Both outputs therefore descend to $PSU(3)$, even though the two input factors individually do not. Reproduce the decomposition in the SU(3) tensor tool, and consult the formula reference when comparing Casimirs or indices.

For branching, add the map to this specification. Under $U\mapsto\operatorname{diag}(U,1)$, the $SU(3)$ fundamental restricts to $\mathbf2\oplus\mathbf1$. Under $U\mapsto\operatorname{Sym}^2U$, it restricts to the spin-one $\mathbf3$. The second map has kernel $\{\pm I\}$. A result labelled only “$SU(3)\to SU(2)$” has omitted decisive data. Sage’s branching tutorial, opening “Branching rules” section explains its use of encoded torus embeddings. The embedding lesson and local SU(3) map reference spell out these two maps; neither map by itself retains a commuting $U(1)$ charge.

Preserve a result that another researcher can reproduce

Keep a short record with the calculation:

• Identity and category: stable group/table identifier, exact cover or quotient, field and characteristic, and representation labels.
• Input convention: generator words and class order for finite groups; Cartan matrix, node order, coordinate style, embedding, and abelian period for Lie groups.
• Execution: system and package versions, exact commands or notebook, input files, full output, and any resource limits or warnings. A cited manual is not evidence that its software executed a particular calculation.
• Independent check: dimensions, character inner products, generator relations, central phases, or a second mathematical construction appropriate to the result.
• Provenance and terms: source URL, record identifier, retrieval date or release version, upstream contributors, and the applicable terms for the material actually redistributed.

The Character Table Library’s project page identifies multiple data sources and points to per-table InfoText credits. Retain those specific origins, rather than attributing every table only to the surrounding software. A software license, documentation license, and dataset permission can cover different material; verify the relevant provider terms when copying files or substantial tables. This guide does not assign a common license to the linked resources.

For this site’s own exports, use the reproducibility guide, convention profiles, and source records. The atlas coverage view separates a sourced record from an implemented operation, and both from independently reviewed mathematics.