Conventions

Every calculation has a representation category, an exact group or algebra identity, and consequential mathematical conventions. The ten Research workbench I–IV additions attach stable convention profile identifiers to their exports.

Coefficient field and representation category

Finite-group calculations use ordinary complex representations in characteristic zero. “Real”, “complex”, and “quaternionic” describe the Frobenius–Schur type of a complex irrep; they do not change the declared coefficient field. Projective representations and modular characters require additional input and methods. See finite character methods and GAP’s character-table documentation.

The highest-weight and SU(3) tools use finite-dimensional continuous complex representations of compact groups. The highest-weight engine integrates to the simply connected compact form by default. Noncompact unitary representation theory is outside this calculation category; spacetime representations explains why this distinction matters.

Dynkin labels and the Cartan matrix

The label $a_i=\langle\lambda,\alpha_i^\vee\rangle$ is paired with the numbered simple coroot. The displayed matrix uses $A_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle=2(\alpha_i,\alpha_j)/(\alpha_i,\alpha_i)$. Bourbaki/Sage numbering is fixed by the complete matrix; for example, $G_2$ has short node 1 and $A=\bigl(\begin{smallmatrix}2&-3\\-1&2\end{smallmatrix}\bigr)$. The root-system download records the node order, squared root lengths, and all positive coroot coordinates.

The SU(2) label is the integer $n=2j$. SU(3) uses $(p,q)$, with $3=(1,0)$ and $\overline{3}=(0,1)$; conjugation exchanges the two labels. Read highest weights in practice before translating labels from a source with another numbering convention.

Quadratic Casimir and trace index

Long roots have squared length $2$. The reported quadratic Casimir and trace index satisfy

$C_2(\lambda)=\frac{(\lambda,\lambda+2\rho)}{2},\qquad T(R)\dim\mathfrak g=C_2(R)\dim R.$

This gives $C_2(\mathrm{adj})=h^\vee$ and, for $SU(N)$, $T(\mathrm{fund})=\tfrac12$. A defining representation of another type need not have index $\tfrac12$: in this convention the defining vector of $SO(M)$, $M\ge5$, has index $1$. “Fundamental representation” may also mean any highest weight $\omega_i$, so its index must be read from the selected weight. The Weyl reference derives the weight-space formula and states the convention when comparing sources.

Global groups and central characters

A dominant integral weight defines a representation of the simply connected compact group. It descends to a quotient only if the subgroup being divided out acts trivially. The Weyl tool’s adjoint descent test concerns the full center and is equivalent to $\lambda$ lying in the root lattice. It does not test every intermediate global form. For a chosen $SU(N)/\mathbb Z_d$, use the matter quotient workbench.

For $SU(N)$, fix $z=e^{2\pi i/N}I$ and $q=\sum_{i=1}^{N-1}i a_i\pmod N$. The subgroup $\mathbb Z_d$ is generated by $z^{N/d}$, so a representation descends precisely when $d$ divides its charge. This kinematic criterion does not choose line operators, discrete theta data, or a vacuum; see matter and global form.

Homomorphisms in branching

The regular map is $U\mapsto\operatorname{diag}(U,1)$ and is injective. The principal map is $U\mapsto\operatorname{Sym}^2(U)$; it has kernel $\{\pm I\}$ and image $SO(3)$. Its restricted irreps therefore have even $n=2j$. Their trace embedding indices in the stated normalization are 1 and 4. The map reference specifies both torus maps. This dedicated SU(3) tool forgets the commuting charge. The charged block workbench separately retains the primitive $U(1)$ and its finite kernel.

For $N=p+q$ and $g=\gcd(p,q)$, the block map is $\operatorname{diag}(z^{q/g}A,z^{-p/g}B)$ with a $2\pi$ circle. Its kernel has order $pq/g$. The color-first $SU(5)$ example uses $Y=-X/6$; the product-group tool uses integer $Q_Y=6Y=-X$ and retains the actual embedded quotient generators. Its input key ranks means SU factor sizes $N_i$, not Lie-algebra ranks.

Groups, real algebras, and notation

A compact group, its real Lie algebra, and its complexification are distinct records. Dimensions state whether they are real or complex. “Compact simple group” means a compact connected group with simple Lie algebra here and can allow a nontrivial center. Compact Sp(n) has quaternionic defining dimension n and complex defining dimension 2n; USp(2n) is an alias under this convention. Sp(2n,R) is the real symplectic matrix group. D₈ is the dihedral group of order eight; some sources call it D₄.

Statistics, bases, and topological generators

Tensor products describe independent factors. Symmetric powers select commuting copies; exterior powers select alternating internal tensors. A full local fermionic operator must also include Lorentz indices, species, derivatives, and any relations imposed. The character workbench returns multiplicities, without choosing a Clebsch–Gordan basis. See finite tensors and statistics.

The local-operator workbench explicitly combines the Lorentz, gauge, and flavor representation of each field before applying its statistics. It counts complex local polynomials at a fixed monomial in four-dimensional Lorentzian spacetime, without derivatives. The finite-dimensional Lorentz spin algebra is complexified for character projection; it is not identified with the physical compact Euclidean rotation group. Fields and statistics and the counting method state the contract and its limits.

$U(1)$ weights are integers after fixing a $2\pi$ period. The topology examples retain their stated subgroup embeddings and oriented winding generators. Physical magnetic-charge units additionally depend on the gauge-field normalization. The topology guide separates the homotopy calculation from dynamical existence and stability.