Versioned convention profiles

A numerical answer can change its meaning when its conventions change. Dynkin labels require an ordered root system; a quadratic Casimir requires an invariant form; branching requires a homomorphism; a finite character table requires a class ordering and a choice of exact number field. This edition gives those choices stable identifiers and versions.

The machine-readable convention manifest contains ten profiles, all at version 1.0.0. The nine Workbench I–III additions include the applicable profiles in their calculation records. The five inherited calculators retain their original convention text and algorithm version. A profile describes mathematical choices; it is distinct from the application, dataset, and algorithm versions described in reproducibility.

The profiles at a glance

Identifier	Applied to	Consequential choices
finite-ordinary-complex-v1	Finite characters, products, and powers	Ordinary complex representations, characteristic zero, named classes, exact character fields, class power maps
lie-bourbaki-long-root-2-v1	Weyl dimensions and Lie representation invariants	Ordered simple coroots, displayed Cartan matrix, long roots of squared length 2, simply connected compact integration
su3-dynkin-pq-v1	SU(3) tensors and branching	Fundamental $(1,0)$, conjugate $(0,1)$, triality $p+2q\pmod 3$
su3-regular-su2-v1	Regular SU(3) restriction	Injective map $U\mapsto\operatorname{diag}(U,1)$, index 1
su3-principal-su2-v1	Principal SU(3) restriction	Spin-one map, kernel $\mathbb Z_2$, image $SO(3)$, index 4
su-n-dynkin-center-v1	Matter and SU(N) quotients	Ordered Dynkin labels, specified central generator, subgroup order $d$, charge-divisibility test
su-n-young-lr-v1	SU(N) tensor products	Canonical polynomial Young diagrams, Littlewood–Richardson multiplicities, determinant-column removal only on restriction to SU(N)
su-n-block-u1-v1	Charged block restriction	Primitive $2\pi$ circle, ordered block map, finite kernel, determinant powers retained until charges are assigned
product-su-u1-center-v1	Product groups and matter	Specified SU centers, integer U(1) charges, rational circle phases, and the actual embedded quotient generators
four-d-local-polynomial-v1	Local scalar/Weyl operators	Four-dimensional Lorentzian spacetime; complete spin × gauge × flavor representations; joint Bose/Fermi statistics; complex invariant dimension at a fixed monomial without derivatives

Ordinary finite characters

The coefficient field of a representation is $\mathbb C$. A character value can lie in a smaller exact algebraic number field; this is a property of the table, not a switch to a modular or real representation category. Every downloaded table states its field relation and encodes algebraic coefficients exactly.

Rows enumerate all ordinary complex irreducible characters of the selected group. Columns specify conjugacy classes, representatives, sizes, element orders, and the power maps used by symmetric and exterior powers. A label such as “the 5-cycle class” needs the table’s convention when several classes have that order. In particular, the two order-five classes in $A_5$ must be paired with the character values and power maps consistently.

The Frobenius–Schur indicator

$$\nu(\chi)=\frac{1}{|G|}\sum_{g\in G}\chi(g^2)$$

classifies an irreducible complex representation as real, complex, or quaternionic for values $+1,0,-1$. A quaternionic complex irrep remains a complex representation with the displayed complex dimension. These conventions agree with the categories and class-function operations described in the GAP Reference Manual, Chapter 71. The finite character methods page explains the exact arithmetic and reconstruction checks.

Highest weights and invariant forms

For numbered simple roots $\alpha_i$, the Dynkin labels are

$$a_i=\langle\lambda,\alpha_i^\vee\rangle, \qquad A_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle=\frac{2(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}.$$

The full displayed Cartan matrix fixes the convention, including its orientation for multiple bonds. The numbering follows the Bourbaki/Sage finite types. For the exceptional $E_r$ diagrams, the chain is $1,3,4,5,\ldots,r$ with node 2 attached to node 4. In $F_4$, nodes 1 and 2 are long and nodes 3 and 4 are short. In $G_2$, node 1 is short and node 2 is long. The root-system export gives the complete matrices and lengths, so a picture is not needed to reconstruct the labels. Compare Sage’s Cartan-matrix conventions.

Long roots have squared length 2. The website reports

$$C_2(\lambda)=\frac12(\lambda,\lambda+2\rho), \qquad T(R)=\frac{C_2(R)\dim R}{\dim\mathfrak g}.$$

Thus $C_2(\mathrm{adj})=h^\vee$. For $SU(N)$ this gives $T(\mathbf N)=\tfrac12$; for the defining vector of $SO(M)$, $M\ge5$, the index is 1. A source that calls $(\lambda,\lambda+2\rho)$ the Casimir is using twice the displayed value. Always compare the invariant form and the operator normalization together. The Weyl formula reference derives the implementation from positive coroots and gives worked non-simply-laced examples; see also Etingof’s Lie theory notes, Sections 25.4 and 28.1.

Every dominant integral weight integrates to the simply connected compact group. Descent to the adjoint quotient requires trivial action of the full center, equivalently membership in the root lattice. A failure of this test can coexist with descent to an intermediate quotient: the vector of $\operatorname{Spin}(10)$ descends to $SO(10)$, while its highest weight is not in the $D_5$ root lattice. The distinction follows from the relation between the center and weight/root lattices; see Conrad, Centers of simply connected semisimple groups.

SU(3) labels and the two restriction maps

The $SU(3)$ fundamental is $3=(1,0)$; its complex conjugate is $\overline3=(0,1)$. Conjugation sends $(p,q)$ to $(q,p)$. The chosen central generator $e^{2\pi i/3}I$ acts with charge $p+2q\pmod3$. A product decomposition retains the difference between conjugate irreps and the multiplicity of each summand. It does not select Clebsch–Gordan coefficients or a basis for multiple copies of an irrep.

For the regular inclusion, the torus map is

$$\operatorname{diag}(z,z^{-1}) \longmapsto \operatorname{diag}(z,z^{-1},1).$$

The principal homomorphism is the symmetric square of the defining doublet, with torus map

$$\operatorname{diag}(z,z^{-1}) \longmapsto \operatorname{diag}(z^2,1,z^{-2}).$$

It kills $-I$ and has image $SO(3)$. In the $SU(2)$ output convention $n=2j$, every principal summand therefore has even $n$. The trace embedding indices are 1 and 4 in the stated normalization. These are different group maps, even though both differentials embed $\mathfrak{su}(2)$ in $\mathfrak{su}(3)$. The branching-map reference gives the formulas, source locators, and the $3$, $8$, and $6$ examples. Regular branching here is not refined by a commuting $U(1)$ charge. The separate block-branching workbench retains that charge under the profile below.

Central quotients compatible with SU(N) matter

Fix

$$z=e^{2\pi i/N}I, \qquad q(\lambda)=\sum_{i=1}^{N-1} i a_i\pmod N.$$

For a divisor $d$ of $N$, the subgroup of order $d$ is generated by $z^{N/d}$. A representation descends to $SU(N)/\mathbb Z_d$ exactly when $d\mid q$. For matter charges $q_1,\ldots,q_s$, their common central kernel has order

$k=\gcd(N,q_1,\ldots,q_s),$

so every divisor $d$ of $k$ defines a compatible central quotient. With no matter, the common kernel is the whole center. Conjugation reverses each charge and leaves the compatible quotients unchanged. The derivation and the limitations of this kinematic test are developed in matter and global form, following the center action and gcd criterion in Cohen, arXiv:1407.4128.

Matter alone does not specify a complete gauge theory. Choices of line operators and discrete theta data can distinguish theories sharing local matter and a gauge algebra; see Aharony, Seiberg, and Tachikawa, Sections 1–2. The workbench labels its output as a descent condition and does not claim an anomaly test or a selection by vacuum dynamics.

Young diagrams and SU(N) tensor products

The canonical polynomial lift of Dynkin labels $a$ is

$$\lambda_i=\sum_{j=i}^{N-1}a_j,\qquad \lambda_N=0.$$

Schur multiplication produces $GL(N)$ partitions $\nu$ with $|\nu|=|\lambda|+|\mu|$. Restricting to determinant one removes $\nu_N$ full columns of height $N$, and the output highest weight has labels $b_i=\nu_i-\nu_{i+1}$. The unreduced partition remains in the calculation record because a general polynomial-character specialization need not have $x_1\cdots x_N=1$. Removing its determinant factor before that check would change the character. The Littlewood–Richardson reference states the exact reading convention and bounded computation, following SageMath’s Weyl-character conventions.

Conjugation reverses the Dynkin-label vector. Reality describes the type of a complex irreducible representation; it does not change the coefficient field. The decomposition retains abstract multiplicities, without choosing a basis of intertwiners or generally resolving permutation symmetry.

Primitive U(1) charges in block restriction

Write $N=p+q$ and $g=\gcd(p,q)$. The convention fixes

$$\iota(A,B,z)=\operatorname{diag}(z^{q/g}A,z^{-p/g}B), \qquad z=e^{i\theta},\quad \theta\sim\theta+2\pi.$$

The image is $S(U(p)\times U(q))$, and the kernel is cyclic of order $pq/g$. The Schur coproduct term $(\mu,\nu)$ has integer charge

$X=\frac{q|\mu|-p|\nu|}{g}.$

The complete $GL(p)$ and $GL(q)$ box counts enter this formula. Determinant columns of one block are trivial under its SU factor but can carry this U(1) charge. The block reference derives the map, kernel generator, and compatibility congruence from the Schur coproduct. An $SU(1)$ factor is explicitly trivial and has an empty label list.

For the color-first $3+2$ split of $SU(5)$, physical hypercharge is $Y=-X/6$ with $Q_{\rm em}=T_3+Y$. The product-group convention uses $Q_Y=6Y=-X$. The transfer between tools therefore reverses the integer charge and the circle coordinate; it does not conjugate the nonabelian representations. The SU(5) guide derives the field assignments.

Embedded quotients of product groups

For $\prod_i SU(N_i)\times U(1)$, the specified central generator in the $i$th factor is $z_i=e^{2\pi i/N_i}\mathbf1_{N_i}$. A matter multiplet has center charges $t_i=\sum_j j a_{ij}\pmod {N_i}$ and integer abelian charge $Q$. An entered generator $(z_1^{k_1},\ldots,z_r^{k_r},e^{2\pi i\theta})$ acts by

$$\exp\!\left[2\pi i\left(\sum_i\frac{t_i k_i}{N_i}+Q\theta\right)\right].$$

Here $\theta$ is a rational number of turns. Descent requires this phase to be one for every supplied generator. The abstract order of a subgroup alone does not specify these phases or the quotient topology. The center-lattice reference explains the finite and continuous kernels and the lift to the universal cover. Its input field ranks contains the factor sizes $N_i$, whereas the Lie-algebra ranks are $N_i-1$.

Standard Model examples order color before weak isospin, take $Q=6Y$, and use $\gamma=(\omega_3,-\mathbf1_2,e^{2\pi i/6})$. Matter compatibility with the diagonal $\mathbb Z_6$ permits its four subgroups; it does not select the full quantum gauge theory. See Tong, Introduction and Section 3 and the global-group guide.

Local fields, flavor, and statistics

The profile four-d-local-polynomial-v1 describes finite-dimensional field representations of the proper connected Lorentz spin group in four-dimensional Lorentzian spacetime. A scalar has label $(0,0)$; left and right Weyl fields have labels $(\tfrac12,0)$ and $(0,\tfrac12)$. Complexifying the Lorentz spin algebra allows invariant projection using characters of two compact $SU(2)$ factors. This algebraic calculation preserves the physical Lorentzian interpretation; it is not a classification of unitary particle representations.

For each species $i$, the complete component space is

$$R_i=L_i\otimes C_i\otimes W_i\otimes\mathbb C^{f_i}.$$

The last factor counts ungauged flavors. Repeating a scalar $n_i$ times uses $\operatorname{Sym}^{n_i}R_i$; repeating a Weyl field uses $\Lambda^{n_i}R_i$. These powers act on the entire space together. Applying separate exterior powers to the spin and gauge factors would impose a different permutation symmetry. The fields guide derives the distinction, using the spinor conventions of Dreiner–Haber–Martin, Sections 2 and 3.2.

The result is a complex invariant dimension at one fixed multidegree. It includes algebraic index identities, while derivatives, EOM, IBP, Hermiticity, and equivalence under flavor-basis changes are outside this release. Conjugate fields are separate polynomial generators. Gauge charges and the specified diagonal quotient use the product-group convention above. The counting reference develops the graded character method, following Marinissen–Rahn–Waalewijn, Section II A; physical setups makes every assumption explicit in the saved input.

Comparing another convention

Before comparing outputs, translate the numbered weights and identify the global group, coefficient field, invariant-form scale, center generator, and map. Relabeling a Dynkin diagram or replacing a center generator can alter the displayed coordinates while preserving the representation. Changing the quotient group or homomorphism can instead change which representations exist or how they restrict. Preserve the original profile with any exported result and record a translation when one is applied.

These profiles and source checks are part of the reproducibility record. Independent human mathematics and physics approval remains separately tracked in the review queue.