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Root characterization of the inverse two-generator element

PDF p. 19: Remark 4.17

Statement

Let $(W, S)$ be the Coxeter system indexed by a set $Γ_{0}$ , possibly infinite, with its standard geometric representation and positive and negative roots $Φ^{+}$ and $Φ^{-}$ . For $L \subseteq Γ_{0}$ , let $W_{L}$ be the subgroup generated by the simple generators indexed by $L$ , let $Φ_{L}$ be its associated parabolic root set, and put $Φ_{L}^{\pm} = Φ_{L} \cap Φ^{\pm}$ . An associate of $J \subseteq Γ_{0}$ is a subset whose standard parabolic subgroup is conjugate to $W_{J}$ .

Let $I$ be an associate of $J$ , let distinct $a, b \in Γ_{0} ∖ I$ satisfy $∣ Φ_{I \cup {a, b}} ∖ Φ_{I} ∣ < \infty$ , and put $T = I \cup {a, b}$ . Let $v_{a, b, I}$ be the two-generator element constructed in the proof of the minimal common left multiple lemma. Then $v_{a, b, I}^{- 1}$ is the unique $w \in W_{T}$ satisfying $w (Φ_{T}^{+} ∖ Φ_{I}^{+}) \subseteq Φ_{T}^{-}$ and $w (Φ_{I}^{+}) \subseteq Φ_{T}^{+}$ .

Statement references

Coxeter-system, roots, length and weak-order conventions

Standard parabolic subgroups and parabolic roots

Associates of a parabolic subset

Standard expressions and their lengths

Two-generator minimal common left multiple

Facts & Assumptions

[F1]

The construction in the minimal common left multiple lemma takes $v_{a, b, I} = q$ , where $q = r^{- 1}$ and $r = w_{D} w_{D \cap I}$ , with $D$ the union of the components of the diagram induced on $T$ that meet ${a, b}$ . The relative-longest lemma makes $D$ spherical and gives $Φ^{+} \cap q Φ^{-} = Φ_{T}^{+} ∖ Φ_{I}^{+}$ . Here $w_{L}$ denotes the longest element of a finite standard parabolic subgroup $W_{L}$ . Two-generator minimal common left multiple Finite relative longest element without spherical ambient rank

[F2]

In a finite-rank Coxeter system, every root is positive or negative, with the two signs disjoint, and every nonidentity element sends at least one simple root to a negative root. Root sign coherence and the action of simple reflections on positive roots The root-length criterion and faithfulness of the canonical reflection representation

[F3]

Every group element has a finite word in the simple generators. For each finite $H \subseteq Γ_{0}$ , the canonical map from the Coxeter group presented by the restricted matrix on $H$ into $W$ is injective with image $W_{H}$ , by part 1 of Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank. Thus $W_{H}$ , with the generators indexed by $H$ , is a finite-rank Coxeter system. The geometric reflection formulas in Coxeter-system, roots, length and weak-order conventions restrict to its standard representation on $V_{H} = span {α_{h} : h \in H}$ . Its roots identify with $Φ_{H} = {x α_{h} : x \in W_{H}, h \in H}$ , and their positive and negative signs agree with the ambient signs because their coordinates outside $H$ are zero.

Proof

Given: The Coxeter system, roots, subsets and finite relative-root hypothesis in the Statement.

Proof technique: direct.

Goal: Prove the two root-sign conditions for $r = v_{a, b, I}^{- 1}$ and show that they determine it in $W_{T}$ .

1.1F1F2F3given

By [F1], $r$ lies in $W_{D} \subseteq W_{T}$ , and for a positive root $α$ the condition $r α \in Φ^{-}$ is equivalent to $α \in Φ^{+} \cap q Φ^{-} = Φ_{T}^{+} ∖ Φ_{I}^{+}$ . Indeed $q = r^{- 1}$ . The action of $W_{T}$ preserves $Φ_{T}$ , directly from its definition as the $W_{T}$ -orbit of the simple roots indexed by $T$ . Consequently $r$ sends the stated difference into $Φ_{T}^{-}$ and sends $Φ_{I}^{+}$ into $Φ_{T}^{+}$ ; for the latter, write a root under consideration as $x α_{i}$ and choose a finite set of indices containing $i$ and the letters in words for $x$ and $r$ . By [F3], both $x α_{i}$ and $r x α_{i}$ are roots of that finite-rank Coxeter system with their ambient signs, so [F2] gives root-sign dichotomy.

2.1F2F3step 1.1given∎

Let $w \in W_{T}$ satisfy both displayed conditions. Choose a finite $H \subseteq T$ containing the indices in words for both $w$ and $r$ . By [F3], $W_{H}$ is the Coxeter group for the restricted matrix, its roots are $Φ_{H} \subseteq Φ_{T}$ , and the actions of $w$ , $r$ and $w r^{- 1}$ on them are the finite-rank geometric actions with the same signs. For every positive root $α$ of the finite subsystem on $H$ , the two conditions for $w$ and step 1.1 show that $w α$ and $r α$ have the same sign: $α$ belongs either to $Φ_{I}^{+}$ or to its complement in $Φ_{T}^{+}$ . These images remain roots of the subsystem on $H$ . Thus $w r^{- 1}$ sends every positive root of that subsystem to a positive root: write such a root as either $r α$ or $- r α$ , with $α$ positive, and use the matching signs. By [F2] applied to this finite-rank Coxeter system, $w r^{- 1} = 1$ in $W_{H}$ , hence also in $W$ , so $w = r$ . This proves uniqueness and the assertion.

Source notes

Remark 4.17 attributes the characterization to the discussion following Proposition 2.1 of the article's reference [5]. The proof above uses the completed relative-longest supplier and a finite-subsystem argument; it does not claim to reproduce that discussion.

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 19, Remark 4.17