groups.qft.org
Search⌘ K

Find the exact object.

Names, aliases, Unicode, and common TeX notation. Open full search

First public edition · 266 records · 15 workbenchesExplore the coverage
An atlas for theoretical physicists

Groups
for Physics.

Find a group, choose its representations, and trace the consequences for your theory.

Explore the atlas Open a workbench
FINITE & LIE GROUPS · REPRESENTATIONS · TOPOLOGY
A change of global formTry the comparison
PSU⁡(3)\PSU(3)
SU⁡(3)/Z3\SU(3)/\mathbb Z_3 · adjoint global form
Lie algebra
su(3)\mathfrak{su}(3)
Center
11
Fundamental group
Z3\mathbb Z_3
Selected irreps of the simply connected cover
11allowed33does not descend3‾\overline3does not descend66does not descend88allowed1010allowed2727allowed

Only zero-triality representations descend.

Global form and sourcesExplore the calculation

Finite groups and Lie groups.

Browse by theme

Finite groups

Characters & invariants
A₄Alternating group on four lettersorder 122TBinary tetrahedral grouporder 24A₅Alternating group on five lettersorder 60Q₈Quaternion grouporder 8

Lie groups

Global forms & maps
SU(2)Special unitary groupdim 3SU(3)Special unitary groupdim 8SL(2,ℂ)Lorentz spin groupdim 6SU(2,2)Indefinite special unitary group of signature (2,2)dim 15

Classification, structure, and global form.

Classification and conventions
Finite

Simple groups

Families, exceptional identifications, 26 sporadic groups, and the Tits group.

Local structure

Lie algebras

Complex types and actual real algebras have distinct records.

Relationships

Maps & covers

Domain, image, and kernel are part of every specified map.

Start with a physical question.

All workbenches
01

Which tensor channels?

Resolve SU(N) products into highest weights, conjugates, and multiplicities.

02

How does it branch?

Keep nonabelian labels and U(1) charges along an explicit block homomorphism.

03

Which global gauge group?

Test matter against an embedded central quotient of a product group.

New · research workbench IV

From a field inventory
to allowed interactions.

Choose fields, a dimension range, and optional charge selection. Enumerate every allowed monomial and calculate its exact complex invariant count.

Enumerate operators
ONE HIGGS DOUBLET · WITH ITS CONJUGATE
H†H,(H†H)2,(H†H)3H^\dagger H,\quad (H^\dagger H)^2,\quad (H^\dagger H)^3
DIMENSION 2
1
DIMENSION 4
1
DIMENSION 6
1

Three monomial sectors. Three independent invariants. Exact field statistics, no derivatives.

Follow a learning path.

Paths and prerequisites

Finite symmetries

From an S₃ doublet to characters, tensor channels, and an A₄ scalar model.

Lie groups and topology

From roots and weights to global forms, explicit maps, and bundle topology.

Spacetime and fields

From Lorentz spinors to conformal primaries, AdS symmetry, and local interactions.

Go from labels to a result.

Representation records
Finite groups

Use a character table

Resolve tensor products and find invariant channels with exact character data.

Lie groups

Work with highest weights

Read labels, calculate dimensions, and keep the global group in view.

Understand the calculation.

Physics guides
A worked field inventory

Standard Model operator sectors

Read the Higgs, Yukawa, Weinberg, and four-fermion sectors with explicit conjugates and baryon/lepton selection rules.

Read the guide Neutrino masses

The Weinberg operator

Derive the unique dimension-five field structure and its symmetric flavor coefficients.

Read the guide Model building

Yukawa and four-fermion interactions

Track chirality, gauge charges, flavor, and index identities in concrete Standard Model sectors.

Read the guide Global structure

Which Standard Model gauge group?

Check the diagonal center against matter and distinguish the four compatible global forms.

Read the guide
Exact results within a stated scopeGlobal forms and maps made explicitConventions and sources beside the result