Lie groups · workbench

Global forms & representations

Change the global group. See which highest weights descend.

Supported: SU(2), SO(3), SU(3), PSU(3) · highest-weight labels 0–30 · complex representations

Highest-weight resultRepresentation descends

\begin{aligned}G&=\mathrm{SU}(3),\quad\lambda=(1,0)\\\dim R&=3,\quad C_2(R)=\frac{4}{3},\quad T(R)=\frac{1}{2}\\t&=1\pmod{3},\quad\text{descends: yes}\end{aligned}

Complex dimension: 3
Quadratic Casimir C₂: $\frac{4}{3}$
Quadratic index T(R): $\frac{1}{2}$
Charge on cover: $1\pmod{3}$
Conjugate weight: $(0, 1)$
Reality type: Complex type

This highest weight defines an ordinary representation of SU(3). The simply connected group admits every nonnegative integral highest weight.

Group center: $\mathbb Z_{3}$
Fundamental group π₁: $0$
Simply connected cover: $\mathrm{SU}(3)$

The cover charge is the action of Z(SU(3)), distinct from the center of the selected quotient group. Zero charge does not imply real type. For example, $(3,0)$ descends to PSU(3) and has complex type.

Representation browser

Complex dimension ≤

17 representations of SU(3) · complete for complex dimension ≤ 27. Conjugate representations remain separate.

Complete dimension-bounded representation list
Highest weight	Dimension	C₂	T(R)	Cover charge
$(0, 0)$	1	$0$	$0$	$0\pmod{3}$
$(0, 1)$	3	$\frac{4}{3}$	$\frac{1}{2}$	$2\pmod{3}$
$(1, 0)$	3	$\frac{4}{3}$	$\frac{1}{2}$	$1\pmod{3}$
$(0, 2)$	6	$\frac{10}{3}$	$\frac{5}{2}$	$1\pmod{3}$
$(2, 0)$	6	$\frac{10}{3}$	$\frac{5}{2}$	$2\pmod{3}$
$(1, 1)$	8	$3$	$3$	$0\pmod{3}$
$(0, 3)$	10	$6$	$\frac{15}{2}$	$0\pmod{3}$
$(3, 0)$	10	$6$	$\frac{15}{2}$	$0\pmod{3}$
$(0, 4)$	15	$\frac{28}{3}$	$\frac{35}{2}$	$2\pmod{3}$
$(1, 2)$	15	$\frac{16}{3}$	$10$	$2\pmod{3}$
$(2, 1)$	15	$\frac{16}{3}$	$10$	$1\pmod{3}$
$(4, 0)$	15	$\frac{28}{3}$	$\frac{35}{2}$	$1\pmod{3}$
$(0, 5)$	21	$\frac{40}{3}$	$35$	$1\pmod{3}$
$(5, 0)$	21	$\frac{40}{3}$	$35$	$2\pmod{3}$
$(1, 3)$	24	$\frac{25}{3}$	$25$	$1\pmod{3}$
$(3, 1)$	24	$\frac{25}{3}$	$25$	$2\pmod{3}$
$(2, 2)$	27	$8$	$27$	$0\pmod{3}$

Method, conventions & sources

SU(2)/SU(3) highest-weight formulas and trivial action of the quotient kernel; exhaustive dimension-bounded enumeration.

Finite-dimensional complex representations; Hermitian generators; tr_fund(T^a T^b) = delta_ab/2; T(R) dim(g) = C2(R) dim(R).

Calculation version: groups-design-1.0.0

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