Lie theory · Representation

$D_{4}:\quad V_{(0,0,0,1)}$ Other half-spin representation of Spin(8)

Highest weight (0,0,0,1) of the specified simply connected compact group: complex dimension 8, real type, with exact dimension, Casimir, and center checks.

Executable coverageHighest weightCompactIrreducible

Definition

The finite-dimensional continuous complex irreducible representation of Spin(8) with Dynkin labels (0,0,0,1) in the displayed Bourbaki/Sage node order.

Definition sources

Complex dimension: $8$
Coefficient field: ℂ; characteristic zero

Representation identity

Domain

\mathrm{Spin}(8)

Categoryfinite-dimensional-continuous

Coefficient field

\mathbb C

Conventionlie-bourbaki-long-root-2-v1

This record specifies an abstract representation. A matrix basis and normalized invariant tensors are separate choices.

Reference properties

Property	Value	Evidence
Complex dimension	$8$	Sources
Coefficient field	ℂ; characteristic zero	Sources
Quadratic Casimir C₂	$7/2$	Sources
Quadratic index T(R)	$1$	Sources
Reality type	Real	Sources
Conjugate highest weight	(0,0,0,1)	Sources
Descends to the adjoint global form	No	Sources
Order of central kernel	$2$	Sources

Highest weight and conventions

The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant. [sources]

Global form and center

The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives no. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel. [sources]

Use in a calculation

Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners. [sources]

Related objects and maps

$\mathrm{Spin}(8)$
Simply connected compact spin group in eight dimensions
Representation domain

Claims and sources

Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.

Definition

The finite-dimensional continuous complex irreducible representation of Spin(8) with Dynkin labels (0,0,0,1) in the displayed Bourbaki/Sage node order.

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:spin-8-weight-0-0-0-1#definition

Link to claim Suggest a correction

Complex dimension

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

Derivation: Exact Weyl dimension product over positive coroots

Premises: group:spin-8 · source:workbench-etingof-lie · source:workbench-sage-roots · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:spin-8-weight-0-0-0-1#dimension

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Coefficient field

ℂ; characteristic zero

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:spin-8-weight-0-0-0-1#field

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Quadratic Casimir C₂

7/2

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

Derivation: (lambda,lambda+2rho)/2 with squared long-root length two

Premises: group:spin-8 · source:workbench-etingof-lie · source:workbench-sage-roots · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:spin-8-weight-0-0-0-1#casimir

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Quadratic index T(R)

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

Derivation: T(R) dim(g) = C2(R) dim(R)

Premises: group:spin-8 · source:workbench-etingof-lie · source:workbench-sage-roots · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:spin-8-weight-0-0-0-1#index

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Reality type

Real

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:spin-8-weight-0-0-0-1#reality

Link to claim Suggest a correction

Conjugate highest weight

(0,0,0,1)

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:spin-8-weight-0-0-0-1#conjugate

Link to claim Suggest a correction

Descends to the adjoint global form

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

Derivation: Highest weight lies in the root lattice exactly when the full center acts trivially

Premises: group:spin-8 · source:workbench-etingof-lie · source:workbench-sage-roots · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:spin-8-weight-0-0-0-1#adjoint-descent

Link to claim Suggest a correction

Order of central kernel

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

Derivation: Order of the center divided by the order of this central character

Premises: group:spin-8 · source:workbench-etingof-lie · source:workbench-sage-roots · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:spin-8-weight-0-0-0-1#central-kernel-order

Link to claim Suggest a correction

Highest weight and conventions

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:spin-8-weight-0-0-0-1#representations-1

Link to claim Suggest a correction

Global form and center

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:spin-8-weight-0-0-0-1#global-form-1

Link to claim Suggest a correction

Use in a calculation

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:spin-8-weight-0-0-0-1#physics-1

Link to claim Suggest a correction

Representation domain

group:spin-8

Lie Groups and Lie Algebras ISections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.Open source
SageMath: Cartan matricesFinite Cartan matrices and root-system conventions; documentation checked 2026-11-08.Open source
SageMath · Weyl charactersCoroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.Open source
Math 210C. Centers of simply connected semisimple groupsPages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.Open source
The Real Chevalley InvolutionLemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.Open source

representation:spin-8-weight-0-0-0-1#relation-1

Link to claim Suggest a correction

D4:V(0,0,0,1)D_{4}:\quad V_{(0,0,0,1)}D4​:V(0,0,0,1)​Other half-spin representation of Spin(8)

Representation identity

Reference properties

Highest weight and conventions

Global form and center

Use in a calculation

Related objects and maps

Claims and sources

$D_{4}:\quad V_{(0,0,0,1)}$ Other half-spin representation of Spin(8)