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Root-permuting elements are minimum coset representatives

Statement

Let $(W, S)$ be an arbitrary-rank Coxeter system in its standard real root representation, with simple roots $Π = {α_{i} : i \in Γ_{0}}$ and length $ℓ$ . Fix $J \subseteq Γ_{0}$ , put $A_{J} = {α_{j} : j \in J}$ and $W_{J} = ⟨ s_{j} : j \in J ⟩$ , and write $M = {Norm}_{W} (W_{J})$ and $N_{J} = {n \in W : n A_{J} = A_{J}}$ . These are the conventions of Root-permuting normaliser complement; $N_{J}$ is a subgroup of $M$ .

Every $n \in N_{J}$ is the unique minimum-length element of $n W_{J}$ .

External input

Dyer–Gimenez, Proposition 2.10(b),(c), states that a reflection-subgroup coset $x U$ has a unique minimum-length element and that an element $y$ of it is that minimum exactly when $y Π_{U} \subseteq Φ^{+}$ . Proposition 2.11(c) identifies $Π_{W_{J}} = A_{J}$ for each standard parabolic subgroup. These assertions hold without finite rank by §1.4.

Facts & Assumptions

[F1]

By Root-permuting normaliser complement, $N_{J} = {n \in W : n A_{J} = A_{J}}$ is a subgroup of $M = {Norm}_{W} (W_{J})$ .

[F2]

The canonical simple-system identification and minimum-coset positivity criterion are the imported assertions in the External input section above.

[F3]

The arbitrary-rank conventions in Coxeter-system, roots, length and weak-order conventions define $ℓ (w)$ as the least number of simple generators representing $w$ , $Φ = {w α_{i} : w \in W, i \in Γ_{0}}$ , and $Φ^{+} = Φ \cap V_{+}$ , where $V_{+}$ consists of finite nonnegative linear combinations of the simple-root basis. In particular $A_{J} \subseteq Φ^{+}$ .

Proof

Given: The data in the statement. Goal: Prove its conclusion.

1.1F1F2F3

By [F1], $n A_{J} = A_{J}$ , and [F3] gives $A_{J} \subseteq Φ^{+}$ . The imported identification $Π_{W_{J}} = A_{J}$ in [F2] therefore gives $n Π_{W_{J}} \subseteq Φ^{+}$ .

2.1F2step 1.1∎

Apply the imported positivity criterion in [F2] to the reflection subgroup $W_{J}$ and the element $n \in n W_{J}$ . By step 1.1 its positivity condition holds, so $n$ is the unique minimum-length element of $n W_{J}$ , as claimed.

Source notes

The displayed external input is imported from the exact independent source locations recorded above. The argument applies those assertions to a standard parabolic subgroup; it does not reconstruct the source proofs or use the article under study as a prerequisite.

Depends on

Used by · 2

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink–Howlett groupoids · §1.4, PDF p.4; Proposition 2.10(b),(c) and Proposition 2.11(c), PDF p.23

Dyer–Gimenez, Proposition 2.10(b),(c), states that a reflection-subgroup coset $xU$ has a unique minimum-length element and that an element $y$ of it is that minimum exactly when $y\Pi_U\subseteq\Phi^+$. Proposition 2.11(c) identifies $\Pi_{W_J}=A_J$ for each standard parabolic subgroup. These assertions hold without finite rank by §1.4.

Hypotheses: Arbitrary-rank Coxeter system with standard based root system; arbitrary subset of simple roots; W_J is a reflection subgroup. No sphericity or finite-rank assumption.

Reading limits: Checked the retrieved v1 PDF text of §§1.3–1.4, §1.7, the §2.8 introduction, and Proposition 2.10(b),(c) and Proposition 2.11(c), PDF pp.3–4 and 22–23, for the root conventions, arbitrary-rank scope and exact imported criterion. The proof/reference paragraphs adjacent to these propositions were inspected; their cited older works were not read, and no reconstruction of their proofs is claimed.

Sources

Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink–Howlett groupoids · §1.4, PDF p.4; Proposition 2.10(b),(c) and Proposition 2.11(c), PDF p.23