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Exact calculations · explicit conventions

Calculate with the group.

From representations to fields and interactions. Start with a precise mathematical object and carry its conventions into your research.

Research workbench IV

From a field inventory to every allowed monomial and its exact invariant count, with optional global charge selection rules.

What changed
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.

Inspect one monomial

How many contractions
does a monomial admit?

Specify scalar and Weyl fields, gauge representations, and flavors. Count local invariants with Lorentz indices and Bose/Fermi statistics included.

Explore local operators
THE WEINBERG SECTOR · THREE LEPTON FLAVORS
L2H2,Δ=5L^2H^2,\qquad \Delta=5
6

independent complex invariants

A symmetric flavor tensor. One Higgs doublet. No derivatives.

Tensors, branching, and global groups.

Research workbench II
01 · Model building

SU(N) tensor products

Decompose compact SU(N) representations with exact Littlewood–Richardson coefficients and explicit Young-diagram conventions.

Finite-dimensional continuous complex SU(N) representations; 2≤N≤6.
Open workbench 02 · Symmetry breaking

Branching with U(1) charges

Restrict an SU(N) representation along a specified block map, keeping both nonabelian labels and the commuting U(1) charge.

SU(p) × SU(N−p) × U(1) mapped into SU(N); 3≤N≤6, 1≤p<N.
Open workbench 03 · Global gauge structure

Product gauge groups & matter

Find the common central kernel of matter, then test an explicitly generated quotient of a product gauge group.

One or two SU(N) factors, 2≤N≤8, and one compact U(1) with period 2π.
Open workbench
A connected research workflow

Keep the charges. Check the group.

Branch an SU(5) multiplet into color, weak, and hypercharge representations, then send those exact multiplets to the product-group workbench. The transfer retains the quotient generator and converts the charge convention explicitly.

Start with the SU(5) 10

Characters and highest weights.

Research workbench I
01 · Finite groups

Finite characters & tensor products

Decompose tensor, symmetric, and exterior powers using exact ordinary complex characters.

Seven curated groups; base representations of complex dimension ≤60; multiplicities ≤12; powers of degree 0–12.
Open workbench 02 · Compact Lie groups

Highest weights & Weyl dimensions

Calculate exact dimensions and quadratic Casimirs from a highest weight, with the root conventions visible.

A₁–A₈, B₂–B₈, C₂–C₈, D₄–D₈, E₆, E₇, E₈, F₄, G₂; Dynkin labels 0–30. No noncompact unitary-dual calculation.
Open workbench 03 · Compact Lie groups

SU(3) tensor products

Resolve a product into irreducible highest weights, retaining conjugates and repeated multiplets.

Each Dynkin label of either input is 0–6. Complete decomposition within that input domain; output labels may be larger.
Open workbench 04 · Symmetry breaking

SU(3) branching by a chosen map

Restrict any supported SU(3) highest weight along the regular or principal SU(2) map.

SU(3) labels p,q from 0–6. Complete SU(2) decomposition; the principal map has kernel Z₂.
Open workbench 05 · Gauge theory

Matter & global gauge structure

Find the common central kernel of matter multiplets and every compatible central quotient of SU(N).

2≤N≤12; at most 16 multiplets; each Dynkin label 0–30. Kinematic descent only.
Open workbench

Focused calculations.

The original five tools
G / Γ

Global forms & representations

Change the global group. See which highest weights descend.

SU(2), SO(3), SU(3), PSU(3); each highest-weight input 0–30. Complete dimension-bounded browser: A1 up to 30, A2 up to 100.
R ⊗ S

Tensor products

Decompose two SU(2) representations with exact multiplicities.

Two SU(2) highest-weight inputs 0–30; output highest weights can reach 60.
Inv(R)

Invariant couplings

Compare distinct fields with one identical commuting scalar.

The A4 real triplet, tensor or symmetric powers, degrees 0–12. No fermion or derivative operator engine.
G → H

Branching & embeddings

Follow the regular or principal map into SU(3).

Only SU(3) representations 3 and 8 along the named regular and principal maps.
G / H

Symmetry breaking & topology

Explore the homotopy sequence for two spherical vacuum orbits.

Only the specified SU(2)/U(1) and SO(3)/SO(2) fibrations with declared angular conventions.

Understand the result.

Learning paths
Build the interaction space

From fields to interactions

Turn a field inventory into an exhaustive list of monomials and distinguish a sector from its invariant multiplicity.

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.

Mathematical method

Exact operator enumeration

Occupation bounds, charge filters, completeness, resource limits, and reproducible enumeration files.

The field representation

Fields, Lorentz indices, and statistics

Build the complete single-field representation before imposing Bose or Fermi symmetry.

Neutrino masses

The Weinberg operator

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

Model building

Yukawa and four-fermion interactions

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

Mathematical method

How local operators are counted

Exact graded characters, Weyl projection, selection rules, and the meaning of the count.

Representation products

Young diagrams in a model-building calculation

Use SU(N) highest weights, determinant columns, conjugation, and exact tensor multiplicities.

Charged branching

From SU(5) to Standard Model multiplets

Follow an explicit homomorphism, keep hypercharge, and identify the quotient subgroup.

Global structure

Which Standard Model gauge group?

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

Finite symmetries

Finite tensor products, with statistics

Use characters to distinguish a tensor product from a symmetric or exterior power.

Lie representations

Highest weights in practice

Read the Dynkin labels, choose the global group, and check a dimension.

Global gauge structure

Which global forms admit the matter?

Derive the common-kernel criterion and use it in worked SU(4) and SU(6) examples.

Color and flavor

The SU(3) tensor-product dictionary

Track conjugates, multiplicities, triality, and the two octets in 8 ⊗ 8.

A catalogue record and an executable capability carry different promises. Each calculator lists its precise category and bounds. Compare reference and computational coverage.