ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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The weak and Bruhat orders on $S_{3}$

Example

On $S_{3}$ , the weak-order cover relations are

$012 ≺ 021, 012 ≺ 102, 021 ≺ 120, 102 ≺ 201, 120 ≺ 210, 201 ≺ 210,$

while the Bruhat-order cover relations are

$012 ≺ 021, 012 ≺ 102, 021 ≺ 120, 021 ≺ 201, 102 ≺ 120, 102 ≺ 201, 120 ≺ 210, 201 ≺ 210.$

So the Bruhat order is strictly finer than the weak order already on $S_{3}$ .

Facts & Assumptions

Given: The weak order by inversion inclusion and the Bruhat order by rank inequalities (The weak order on $S_{n}$ by inversion-set inclusion, The Bruhat order on $S_{n}$ by rank inequalities).

Verification

technique · direct

1.1

Computing inversion sets gives the six weak-order covers displayed above. In particular, $021$ and $201$ are incomparable in weak order because their inversion sets are ${(1, 2)}$ and ${(0, 1), (0, 2)}$ respectively.

given

2.1

Computing the rank inequalities shows that $021 < 201$ and $102 < 120$ in Bruhat order, producing the two extra cover relations listed above. Thus Bruhat order is strictly finer than weak order on $S_{3}$ .

step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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The weak and Bruhat orders on S3S3​