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Bruhat covers are right multiplication by positive-root reflections

Statement

Let $x, y \in W$ with $y ⊲ x$ a cover. Then there is a unique positive root $β \in Φ^{+}$ with $x = y s_{β}$ , and $ℓ (x) = ℓ (y) + 1$ . Conversely, if $β \in Φ^{+}$ and $ℓ (y s_{β}) = ℓ (y) + 1$ , then $y s_{β} ⊳ y$ is a cover. In the notation of the Bruhat graph the label $β$ of the arrow $x \to y$ is characterised by $s_{β} = y^{- 1} x$ , and it is also the unique positive root $γ$ with $y = x s_{γ}$ and $ℓ (x s_{γ}) = ℓ (x) - 1$ . The proof uses the standard sign criterion $ℓ (w s_{γ}) < ℓ (w) ⟺ w γ < 0$ together with the reflection-chain description of Bruhat order.

Facts & Assumptions

Given: The finite reduced crystallographic root system $Φ$ with positive system $Φ^{+}$ , the Weyl group $W$ with simple reflections, and the Bruhat order.

[F1]

$u \leq v$ in Bruhat order is equivalent to the existence of a saturated reflection chain $u = w_{0}, w_{1}, \dots, w_{k} = v$ with $w_{j + 1} = t_{j} w_{j}$ for root reflections $t_{j}$ and $ℓ (w_{j + 1}) = ℓ (w_{j}) + 1$ ; every such chain has exactly $ℓ (v) - ℓ (u)$ steps, so $u \leq v$ with $ℓ (v) = ℓ (u) + 1$ means $v = t u$ for a root reflection $t$ (Bruhat order on a finite Weyl group).

[F2]

For a positive root $γ \in Φ^{+}$ and its reflection $s_{γ}$ : $ℓ (s_{γ} w) < ℓ (w)$ if and only if $w^{- 1} γ < 0$ , and $ℓ (w s_{γ}) < ℓ (w)$ exactly when $w γ < 0$ ; only multiplication by a simple reflection is guaranteed to change length by one. Also $ℓ (w^{- 1}) = ℓ (w)$ (Finite Weyl strong exchange and deletion).

[F3]

Root reflections are the maps $s_{β}$ for roots $β$ ; $s_{β} = s_{- β}$ , and $s_{β} = s_{γ}$ forces $γ = \pm β$ because the only scalar multiples of a root in $Φ$ are $\pm$ itself; $W$ permutes the root set (Root reflections and the Weyl group action, Finite Weyl root system, lattice and chamber conventions).

Proof

1.1F1F2F3algebra

Let $y ⊲ x$ . By [F1] with a one-step chain, $x = t y$ for a root reflection $t$ ; write $t = s_{γ}$ with $γ$ a root and replace $γ$ by $- γ$ if necessary so that $γ \in Φ^{+}$ . Since $x = s_{γ} y$ and $ℓ (x) = ℓ (y) + 1 > ℓ (y)$ , the criterion in [F2] applied to $w = y$ forbids $y^{- 1} γ < 0$ ; hence $β : = y^{- 1} γ \in Φ^{+}$ . Conjugation gives $x = y (y^{- 1} s_{γ} y) = y s_{y^{- 1} γ} = y s_{β}$ with $β = y^{- 1} γ \in Φ^{+}$ .

2.1F3step 1.1

Suppose $x = y s_{β} = y s_{β^{'}}$ with $β, β^{'} \in Φ^{+}$ . Then $s_{β} = s_{β^{'}}$ , so $β^{'} = \pm β$ by [F3], and positivity forces $β^{'} = β$ . Thus the positive root in step 1.1 is unique, and multiplying $x = y s_{β}$ on the left by $y^{- 1}$ gives $y^{- 1} x = s_{β}$ , so the label is determined by the group elements.

2.2F1F3step 1.1

Conversely let $β \in Φ^{+}$ and suppose $ℓ (y s_{β}) = ℓ (y) + 1$ . Put $t : = y s_{β} y^{- 1} = s_{y β}$ by conjugation, so $y s_{β} = t y$ is a one-step saturated reflection chain; by [F1], $y \leq y s_{β}$ . If $z$ satisfied $y < z < y s_{β}$ , then by [F1] any saturated chain from $y$ to $y s_{β}$ through $z$ would have more than one step, so $ℓ (y s_{β}) - ℓ (y) \geq 2$ , contradicting the hypothesis. Hence $y s_{β}$ covers $y$ .

3.1step 1.1step 2.1step 2.2∎

Finally, if $γ \in Φ^{+}$ satisfies $y = x s_{γ}$ and $ℓ (x s_{γ}) = ℓ (x) - 1$ , then $x = y s_{γ}$ by multiplying on the right by $s_{γ}$ , and $ℓ (y s_{γ}) = ℓ (x) = ℓ (y) + 1$ ; step 1.1 applied to the cover $y ⊲ x$ (whose existence is the hypothesis $y = x s_{γ}$ ) gives $γ = β$ . Step 2.1 supplied the uniqueness of $β$ from the pair $(x, y)$ alone, so the two characterisations of the label coincide.

Depends on

Used by

Dependency tree · two levels

7 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

Fan Zhou, The classical and the functorial BGG resolutions (Columbia thesis 2021), Part I Sec. 3.1, p. 9 (standard reference, not scraped)
N. Hemelsoet and R. Voorhaar, A computer algorithm for the BGG resolution, arXiv:1911.00871, Sec. 2.1-2.2, pp. 3-5 (standard reference, not scraped)