Knowlpedia

Example

Weights of the defining sl₃-representation

The three multiplicity-one weights of the natural three-dimensional representation of sl3.

Example

Let V=C3V=\mathbb C^3 be the of sl3(C)\mathfrak{sl}_3(\mathbb C), and take the

h={diag⁡(h1,h2,h3):h1+h2+h3=0}.\mathfrak h=\left\{\operatorname{diag}(h_1,h_2,h_3): h_1+h_2+h_3=0\right\}.

Define εi∈h∗\varepsilon_i\in\mathfrak h^* by εi(diag⁡(h1,h2,h3))=hi\varepsilon_i(\operatorname{diag}(h_1,h_2,h_3))=h_i. Then the three of VV are

ε1,ε2,ε3,ε1+ε2+ε3=0,\varepsilon_1,\qquad\varepsilon_2,\qquad\varepsilon_3, \qquad \varepsilon_1+\varepsilon_2+\varepsilon_3=0,

each with multiplicity one. The standard basis vector eie_i spans the εi\varepsilon_i-weight space.

Simple-root coordinates

Choose the

α1=ε1−ε2,α2=ε2−ε3.\alpha_1=\varepsilon_1-\varepsilon_2, \qquad \alpha_2=\varepsilon_2-\varepsilon_3.

With the corresponding , the is

ω1=ε1.\omega_1=\varepsilon_1.

The three weights form the string

ω1,ω1−α1,ω1−α1−α2.\omega_1,\qquad \omega_1-\alpha_1,\qquad \omega_1-\alpha_1-\alpha_2.

Geometrically they are the vertices of an equilateral triangle centered at the origin in the two-dimensional real span of the roots. The S3S_3 permutes the three vertices.

Exterior power and dual

The second exterior power has weights

ε1+ε2=−ε3,ε1+ε3=−ε2,ε2+ε3=−ε1.\varepsilon_1+\varepsilon_2=-\varepsilon_3, \quad \varepsilon_1+\varepsilon_3=-\varepsilon_2, \quad \varepsilon_2+\varepsilon_3=-\varepsilon_1.

The determinant volume form identifies

Λ2V≅V∗,\Lambda^2V\cong V^*,

whose highest weight is ω2=−ε3\omega_2=-\varepsilon_3. This realizes the nontrivial symmetry of the A2A_2 as the exchange of the defining representation with its .

Group-level form

The same weight calculation describes the defining representation of SL3(C)SL_3(\mathbb C). It does not descend to PSL3(C)PSL_3(\mathbb C), because the center acts by nontrivial scalar matrices. Infinitesimal weight data alone do not record this obstruction.

References
  1. William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, §§12–15. Publisher record.
  2. Brian C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015, Chapters 7 and 10. Publisher record.