Rotation parameters and the two switchback completions

Let $(W,S)$ be a Coxeter system and let $T\subseteq S$ be finitary, meaning that $W_T=⟨T⟩$ is finite. All the following constructions take place in the finite standard parabolic Coxeter system $(W_T,T)$. For $J\subseteq T$, let $w_J$ be its parabolic longest element and let $\mathrm{\ell }$ be Coxeter length in $(W_T,T)$.

Use the following singular-expression conventions. Write $M(p)$ for the unique Bruhat-maximal element of a double coset $p$, using the Bruhat order of $(W_T,T)$. Let $\ast$ be the Coxeter monoid product on the underlying set $W_T$, with the Coxeter braid relations and idempotence $c\ast c=c$ for $c\in T$. For $p:J\to K$ and $q:K\to H$, where these arrows are respectively $(J,K)$- and $(K,H)$-double cosets, set $p\ast q=W_J(M(p)\ast M(q))W_H$. A singular expression $[J_0,\dots ,J_m]$ with successive subsets differing by one vertex denotes the composite of the double cosets $W_{J_{i-1}}e{W}_{J_i}$ in the displayed order. The notation $[J+c-d]$ records the path $J,J\cup \{c\},(J\cup \{c\})\setminus \{d\}$, with an addition required to add a missing vertex and a deletion required to remove a present one. An expression is reduced if each composition of a prefix with the next factor is reduced; a composition $p\ast q$ through $K$ is reduced when $\mathrm{\ell }((M(p)w_K^{-1})M(q))=\mathrm{\ell }(M(p)w_K^{-1})+\mathrm{\ell }(M(q))$. Brackets with larger inclusions or deletions abbreviate the corresponding monotone single-vertex chains. In particular $[I,T,\bar{I}]$ means an inclusion chain from $I$ to $T$ followed by a deletion chain from $T$ to $\bar{I}$. An atomic expression concatenates the three-subset expressions of its atomic factors.

Let $T=I\bigsqcup \{a,b\}$ with $a\mathrm{\ne }b$. Put $\bar{c}=w_{T}c{w}_T$, $\bar{J}=\{\bar{c}:c\in J\}$ for $J\subseteq T$, $s=a$, $t=\bar{b}$, $u_{-1}=b$, $u_0=a$, $I_j=T\setminus \{u_j\}$ and $u_{j+1}=w_{I_j}{u}_{j-1}{w}_{I_j}$. Let $d=d(T,a,\bar{b})$ be Ko's maximal index for which the rotation word is reduced. The first word $A$ consists, for $j=0,\dots ,d$, of the atomic steps $[L_j+u_{j-1}-u_{j+1}]$, where $L_j=T\setminus \{u_j,u_{j-1}\}$. The second word $B$ is obtained by reversing $A$ and applying $c\mapsto \bar{c}$ to every subset and vertex. Both are atomic reduced expressions from $I$ to $\bar{I}$. They express the same coset as the compressed expression $[I,T,\bar{I}]$. Their first added vertices are respectively $b$ and $a$.

The singular-expression construction in the Statement is the imported convention of Ko Sections 1.2–1.3 and 2.1–2.2: Section 2.1 specifies the unique maximal double-coset representatives with respect to the Bruhat order, Definition 2.1 specifies composition by the Coxeter monoid product of these representatives, (2.4)–(2.5) specify the subset and plus/minus notation, Definitions 2.2–2.3 specify reducedness, and Definition 3.9 specifies atomic expressions. The monoid presentation, identification with $W_T$, existence of the Bruhat-maximal elements and well-definedness of this composition are imported, not proved here. The compressed notation expands inclusions and deletions into monotone single-vertex chains; the intermediate subsets have no effect on the composite, since the longest element of the largest parabolic absorbs the generators of each smaller one in the Coxeter monoid.

Ko's Proposition 4.3, Proposition 4.5, Lemma 4.7, Remark 4.8 and Proposition 4.9 supply atomicity, the endpoint identities, reducedness and the switchback equality. These are imported in the finite system $(W_T,T)$, with $s=a$, $t=\bar{b}$. The cited Elias–Ko proofs underlying the switchback result were not independently reconstructed.