ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-09-05

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Degree-two and degree-three Harish-Chandra generators for ${s l}_{3}$

Example

For ${s l}_{3}$ , the Weyl group is $S_{3}$ permuting $t_{1}, t_{2}, t_{3}$ . On the Cartan subalgebra, the invariant polynomials

$s_{2} = t_{1}^{2} + t_{2}^{2} + t_{3}^{2}, s_{3} = t_{1}^{3} + t_{2}^{3} + t_{3}^{3}$

generate $S (h)^{W}$ , so their inverse Harish-Chandra images give independent degree-two and degree-three generators of $Z (U ({s l}_{3}))$ .

Facts & Assumptions

Given: The Cartan subalgebra of diagonal traceless matrices $diag (t_{1}, t_{2}, t_{3})$ in ${s l}_{3}$ , with $t_{1} + t_{2} + t_{3} = 0$ .

Verification

technique · direct

1.1

The symmetric polynomials in $t_{1}, t_{2}, t_{3}$ are generated by the elementary symmetric functions, and the trace-zero relation $t_{1} + t_{2} + t_{3} = 0$ removes the degree-one generator. Thus the invariant ring is generated by the degree-two and degree-three symmetric polynomials $s_{2}$ and $s_{3}$ .

givenalgebra

2.1

By The center of the enveloping algebra is polynomial on rank-many generators, the center of $U ({s l}_{3})$ is polynomial on two homogeneous generators. The Harish-Chandra isomorphism identifies those generators with any algebraically independent pair generating $S (h)^{W}$ , so the preimages of $s_{2}$ and $s_{3}$ give the required quadratic and cubic central elements.

step 1.1∎

Depends on

The center of the enveloping algebra is polynomial on rank-many generators

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yiannis Sakellaridis, Verma Modules and the Category O (standard reference, not scraped)
Pavel Etingof, Representations of Lie Groups (standard reference, not scraped)

Degree-two and degree-three Harish-Chandra generators for sl3sl3​