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The corank-two atomic coset has relative longest minimum

Statement

In the finite setting $T = I ⊔ {a, b}$ of the rotation construction, both words have minimum representative $q = w_{I} w_{T}$ and target $\overset{ˉ}{I} = w_{T} I w_{T}$ . This $q$ is the finite-relative-longest morphism used to construct $v_{a, b, I}$ .

Facts & Assumptions

Given: The data and hypotheses in the Statement.

[F1]

The rotation words express $[I, T, \overset{ˉ}{I}]$ . Rotation parameters and the two switchback completions

[F2]

A core coset with maximum $M$ and source $I$ has minimum $w_{I} M$ . Minimal representatives identify core cosets with root-transport arrows

[F3]

The finite-relative-longest construction is $r = w_{D} w_{D \cap I}$ , $q = r^{- 1}$ , where $D$ is the union of components of $T$ meeting $a$ or $b$ . Common components of finite longest elements commute and cancel. Finite relative longest element without spherical ambient rank Finite parabolic longest elements and disconnected factors

[F4]

The two-generator lemma explicitly chooses $v_{a, b, I} = (w_{D} w_{D \cap I})^{- 1} = r^{- 1}$ , with $D$ as in [F3]. Its finite-root-difference hypothesis holds here because $W_{T}$ is finite. Two-generator minimal common left multiple

External input

Ko Definition 2.1 defines singular composition by star multiplication of maxima. In $W_{T}$ , the longest element satisfies $w_{T} * w_{T} = w_{T}$ , by the defining idempotence and the fact that every simple generator of $T$ is a descent of $w_{T}$ . This determines the maximum of $[I, T, \overset{ˉ}{I}]$ .

Proof

technique · direct

Goal: Establish the stated dictionary or identification.

1.1F1F2

The compressed expression $[I, T, \overset{ˉ}{I}]$ has maximum $w_{T}$ . Indeed its first inclusion has maximum $w_{T}$ , and its final deletion is represented by the coset with maximum $w_{T}$ again; multiplication in the singular Coxeter monoid leaves $w_{T} * w_{T} = w_{T}$ . It is the core coset $W_{I} w_{T} W_{\overset{ˉ}{I}}$ because $w_{T} \overset{ˉ}{I} w_{T} = I$ . Thus [F2] gives the minimum $w_{I} w_{T}$ , and [F1] gives the same minimum for each rotation word.

2.1F3F4step 1.1∎

Write $H = T ∖ D$ . Then $H \subseteq I$ consists of the components not meeting $a, b$ . The finite component cancellation in [F3] gives $w_{I} w_{T} = w_{D \cap I} w_{D} = r^{- 1}$ . This is precisely the construction of $q$ in [F3], including source $I$ and target $\overset{ˉ}{I}$ . By [F4], it is the element $v_{a, b, I}$ .

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

Hankyung Ko, An atomic Coxeter presentation, arXiv:2312.16666v3 · §2.2, Definition 2.1 and (2.4), PDF p. 9; Remark 4.8 (4.8), PDF p. 15

Composition of singular cosets is induced by star multiplication of their maximal elements. The switchback extension is a reduced expression of the compressed coset [I,T,bar I].

Hypotheses: A Coxeter system; all subsets indexing singular expressions are finitary. Section 4 is applied inside a finite standard parabolic (W_T,T).

Reading limits: For this adjudication, read the extracted text of §2.2 through Definition 2.1 and (2.4), PDF p. 9, and Remark 4.8 with (4.8), PDF p. 15, plus the finite-system convention at the start of §4. No visual inspection, full-paper reading, reconstruction of the cited background proofs, or inherited reading credit is claimed. The rotation-word equality is supplied by the existing rotation-words lemma; this item imports only the singular-composition definition for its local calculation.

Sources

Hankyung Ko, An atomic Coxeter presentation, arXiv:2312.16666v3 · §2.2, Definition 2.1 and (2.4), PDF p. 9; Remark 4.8 (4.8), PDF p. 15