PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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Distinct simple roots have nonpositive inner product

Statement

Let $Φ \subseteq E$ be a reduced crystallographic root system with positive system $Φ^{+}$ and simple roots $Δ$ (Positive systems and simple roots). If $α, β \in Δ$ are distinct, then $(α, β) \leq 0$ ; moreover $α - β$ is not a root.

Facts & Assumptions

Given: Distinct simple roots $α, β \in Δ$ of a reduced crystallographic root system $Φ$ with positive system $Φ^{+}$ .

[L1]

A simple root is a positive root that is not a sum of two positive roots; $Φ^{+}$ and $Φ^{-} = - Φ^{+}$ partition $Φ$ (Positive systems and simple roots).

[L2]

If $γ, δ \in Φ$ are nonproportional with $(γ, δ) > 0$ then $γ - δ \in Φ$ (Rank-two root-system classification).

Proof

technique · direct

1.1L1L2algebra

Assume $(α, β) > 0$ . Then $α - β \in Φ$ by [L2], and since $Φ$ is the disjoint union of its positive and negative roots, $α - β$ is either positive or negative.

2.1L1step 1.1algebra

The two alternatives of step 1.1 are impossible: if $α - β$ is positive, then $α = (α - β) + β$ exhibits the simple root $α$ as a sum of two positive roots; if $α - β$ is negative, then $β = (β - α) + α$ exhibits the simple root $β$ as a sum of two positive roots.

3.1L1step 1.1step 2.1algebra∎

Hence $(α, β) \leq 0$ . If $α - β$ were a root, the same dichotomy would apply verbatim and contradict simplicity, so $α - β \notin Φ$ .

Depends on

Used by

I₂(5) admits no crystallographic scaling and no reduced crystallographic root system with that base pairing Counterexample
The invariants and the coinvariant Hilbert series of I₂(m): an explicit computation and the noncrystallographic contrast Example
Simple roots are pairwise orthogonal False statement
Highest-root dominance and the fundamental alcove Lemma
Existence and uniqueness of the highest root Proposition
Properties of finite-type Cartan matrices Proposition

Dependency tree · two levels

8 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

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter II (standard reference, not scraped)