Compact Lie groups · research workbench

Highest weights & Weyl dimensions

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

Scope: Simply connected compact forms of the selected simple Lie types. A₁–A₈, B₂–B₈, C₂–C₈, D₄–D₈, E₆, E₇, E₈, F₄, G₂; Dynkin labels 0–30. No noncompact unitary-dual calculation.

Exact resultExact checks passed

\begin{aligned}\lambda&=(1,1)\\\dim V_\lambda&=8\\C_2(V_\lambda)&=3\\T(V_\lambda)&=3\end{aligned}

Compact simply connected group: SU(3)
Complex dimension: $8$
Conjugate weight: $(1,1)$
Reality type: Real
Full-center quotient descent: Allowed
Center-character order: $1$

Node order fixes the Dynkin labels. Edges show adjacency and multiplicity; squared root lengths in this order are 2, 2. The Cartan matrix below specifies the orientation.

Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). The descent test here is for the adjoint global form, the quotient by the entire center. Other central quotients require their own kernel test. Check matter-compatible SU(N) quotients.

Root conventions and exact Weyl factors

A=\begin{pmatrix}2&-1\\-1&2\end{pmatrix}

A_ij = <alpha_j, alpha_i^vee>. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. The index satisfies T(R) dim(g) = C₂(R) dim(R).

Weyl dimension factors in the simple-coroot basis
Positive coroot coefficients	Numerator	Denominator
$(0,1)$	$2$	$1$
$(1,0)$	$2$	$1$
$(1,1)$	$4$	$2$

✓ positive coroot count✓ dimension integral✓ weight gram symmetric✓ central character order divides center

Try a worked example

Method & interpretation

The Weyl product

The dimension is an exact product over positive coroots. Casimirs use long roots of squared length two. Root-lattice membership tests descent to the adjoint global form.

Read the worked guide · Mathematical method

Download script for an independent environment; not executed by this website. Requires SageMath; verifies Dynkin conventions, dimension, conjugate, reality, Casimir, index, and full-center descent.

Sources, conventions, and version

Computed locally with exact integer or algebraic arithmetic. Independent human expert review is pending. The export records the conventions and implementation version.

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.
SageMath: The Weyl Character RingCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.
Centers of Simply Connected Semisimple GroupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.

Convention: lie-bourbaki-long-root-2-v1
Algorithm: lie-representations-1.0.0

Convention profiles · Reproduce and cite

How calculation links, versions, and exports work