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Opposition as conjugation by a longest Artin lift

Statement

For a spherical Artin system with standard generators $σ_{s}$ indexed by $S$ , let $A$ be its Artin group. For $K \subseteq S$ , write $w_{K}$ for the longest element of $W_{K}$ and $Δ_{K} = σ_{w_{K}}$ for its positive reduced-word lift in $A$ . Let $ι_{K} : K \to K$ be the opposition permutation specified by $w_{K} e_{i} = - e_{ι_{K} (i)}$ . For every $J \subseteq S$ , define $A_{J}^{+}$ to be the submonoid of $A$ generated by ${σ_{j} : j \in J}$ , including the empty product, and $A_{J}$ to be the subgroup generated by the same elements. Then $Δ_{K}^{- 1} σ_{i} Δ_{K} = σ_{ι_{K} (i)}$ for $i \in K$ . Consequently $Δ_{K}^{- 1} A_{I}^{+} Δ_{K} = A_{ι_{K} (I)}^{+}$ and $Δ_{K}^{- 1} A_{I} Δ_{K} = A_{ι_{K} (I)}$ whenever $I \subseteq K$ .

Facts & Assumptions

[F1]

In a spherical system, the positive Artin monoid given by the standard braid presentation embeds as the positive submonoid of its Artin group. Conjugation $x \mapsto Δ^{- 1} x Δ$ is its Garside automorphism. (Independent spherical Artin Garside input)

[F2]

The positive lift of an element of the Coxeter group is independent of its reduced expression, and the homomorphism $π : A \to W$ sends $σ_{s}$ to $s$ and $σ_{w}$ to $w$ . (Reduced-word lifts and the prefix-order convention)

[F3]

The longest element $w_{K}$ is an involution and induces the permutation $ι_{K}$ on the simple roots of the parabolic subsystem; the restricted canonical representation agrees with the ambient representation on their span. (Finite parabolic longest elements and disconnected factors, Statement and Proof, step 1.1)

[F4]

For any finite Coxeter system, its longest element $w_{0}$ satisfies $w_{0}^{2} = 1$ and $w_{0} s w_{0} = τ (s)$ , where $w_{0} e_{s} = - e_{τ (s)}$ . (The longest element as the opposition of the chamber, and longest elements of finite parabolics, Statement (1)(iv)–(v))

[F5]

For $K \subseteq S$ , the restricted Coxeter presentation identifies with $(W_{K}, K)$ and its intrinsic length equals the ambient length on $W_{K}$ . (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification, Statement, part 2)

Proof

Given: The spherical Artin system, its ambient Artin group and the subsets in the statement.

Goal: Identify longest-lift conjugation on generators and on the specified parabolic submonoids and subgroups.

1.1F1F5

By [F5], $(W_{K}, K)$ is the Coxeter system of the restricted matrix, with the same lengths as in $W$ . It is finite because $W$ is finite, and its longest element is $w_{K}$ . Let $A^{(K)}$ be the Artin group of this restricted matrix, with standard generators ${\hat{σ}}_{i}$ , positive monoid $M^{(K)}$ and longest lift ${\hat{Δ}}_{K}$ . Apply [F1] to this system. Its positive monoid has the standard braid presentation, whose relations preserve word length and have length at least two. Hence the atoms are precisely the standard generators: every positive word of length at least two factors into two nonidentity words, whereas a word of length one cannot do so. Any monoid automorphism permutes atoms, so conjugation by ${\hat{Δ}}_{K}$ permutes the ${\hat{σ}}_{i}$ .

2.1F2F3F4step 1.1

For $i \in K$ , project ${\hat{Δ}}_{K}^{- 1} {\hat{σ}}_{i} {\hat{Δ}}_{K}$ to $W_{K}$ using [F2]. Its image is $w_{K}^{- 1} i w_{K} = ι_{K} (i)$ by [F4] applied to $(W_{K}, K)$ ; its root-labelled permutation is the one in [F3], since the restricted representation agrees on the simple-root span. Different standard atoms have different images in $K$ . Thus the permuted atom in step 1.1 is exactly ${\hat{σ}}_{ι_{K} (i)}$ .

3.1F1F2F3F5step 2.1∎

The restricted braid relations also hold among the ambient generators, so the group presentations in [F1] give a homomorphism $A^{(K)} \to A$ sending ${\hat{σ}}_{i}$ to $σ_{i}$ . A reduced expression for $w_{K}$ in $K$ remains reduced in $S$ by [F5]. Therefore [F2] identifies the image of ${\hat{Δ}}_{K}$ with $Δ_{K} = σ_{w_{K}}$ , and step 2.1 gives the claimed generator identity in $A$ . Conjugation consequently maps all finite positive products with indices in $I$ onto all such products with indices in $ι_{K} (I)$ , because $ι_{K}$ is a permutation. This is exactly the stated equality of the defined submonoids. Applying the same argument to products of generators and their inverses proves the subgroup equality. If $K = \emptyset$ , both parabolics and the longest lift are trivial, so the assertions hold as well.

Depends on

Used by · 3

Dependency tree · 2 levels

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