Lie theory · Representation

$C_{2}:\quad V_{(1,0)}$ Defining representation of Sp(2) = USp(4)

Highest weight (1,0) of the specified simply connected compact group: complex dimension 4, quaternionic type, with exact dimension, Casimir, and center checks.

Executable coverageHighest weightCompactIrreducible

Definition

The finite-dimensional continuous complex irreducible representation of Sp(2) = USp(4) with Dynkin labels (1,0) in the displayed Bourbaki/Sage node order.

Definition sources

Complex dimension: $4$
Coefficient field: ℂ; characteristic zero

Representation identity

Domain

\mathrm{Sp}(2)

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	$4$	Sources
Coefficient field	ℂ; characteristic zero	Sources
Quadratic Casimir C₂	$5/4$	Sources
Quadratic index T(R)	$1/2$	Sources
Reality type	Quaternionic	Sources
Conjugate highest weight	(1,0)	Sources
Descends to the adjoint global form	No	Sources
Order of central kernel	$1$	Sources

Highest weight and conventions

Nodes 1,…,r run along the chain; node r is long. Sp(r) denotes the compact quaternionic group. 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

$\mathrm{Sp}(2)$
Compact symplectic group of quaternionic rank two
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 Sp(2) = USp(4) with Dynkin labels (1,0) 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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2-weight-1-0#definition

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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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2 · source:workbench-etingof-lie · source:workbench-sage-roots · source:workbench-sage-c · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:sp-2-weight-1-0#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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2-weight-1-0#field

Link to claim Suggest a correction

Quadratic Casimir C₂

5/4

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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2 · source:workbench-etingof-lie · source:workbench-sage-roots · source:workbench-sage-c · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:sp-2-weight-1-0#casimir

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Quadratic index T(R)

1/2

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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2 · source:workbench-etingof-lie · source:workbench-sage-roots · source:workbench-sage-c · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:sp-2-weight-1-0#index

Link to claim Suggest a correction

Reality type

Quaternionic

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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2-weight-1-0#reality

Link to claim Suggest a correction

Conjugate highest weight

(1,0)

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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2-weight-1-0#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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2 · source:workbench-etingof-lie · source:workbench-sage-roots · source:workbench-sage-c · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:sp-2-weight-1-0#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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2 · source:workbench-etingof-lie · source:workbench-sage-roots · source:workbench-sage-c · source:sage · source:conrad-centers · source:workbench-adams-indicator

representation:sp-2-weight-1-0#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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2-weight-1-0#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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2-weight-1-0#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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2-weight-1-0#physics-1

Link to claim Suggest a correction

Representation domain

group:sp-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: Root system data for type CAmbientSpace.simple_root(): final node is long.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:sp-2-weight-1-0#relation-1

Link to claim Suggest a correction

C2:V(1,0)C_{2}:\quad V_{(1,0)}C2​:V(1,0)​Defining representation of Sp(2) = USp(4)

Representation identity

Reference properties

Highest weight and conventions

Global form and center

Use in a calculation

Related objects and maps

Claims and sources

$C_{2}:\quad V_{(1,0)}$ Defining representation of Sp(2) = USp(4)