The cone criterion, monotonicity of the projection, and the greatest sortable element below w

Chamber signs. For $x=\rho (w)y$ with $y\in C^{\circ }$ and $\beta \in {\mathrm{\Phi }}_{+}$, invariance gives $B(x,\beta )=B(y,\rho (w^{-1})\beta )$; its sign is negative exactly when $\beta \in N(w^{-1})$. Thus $wC\subseteq {\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}_c(v)$ exactly when every negative skip root $-{\beta }_t$ has $t\in I(w)$ and every positive skip root ${\beta }_t$ has $t\mathrm{\notin }I(w)$. Closure extends the interior signs to the entire chamber. We use this dictionary throughout, so root-hyperplane geometry introduces no dependence on the final cone theorem.

The identity case. For any $c$, ${\pi }_c(w)=1$ implies $w=1$. Prove this by rank induction: if an initial $s$ is below $w$, descent detection excludes value $1$; otherwise ${\pi }_c(w)={\pi }_{sc}(w_{J_s})$, so rank induction gives $w_{J_s}=1$. If $w\mathrm{\ne }1$, a first letter $r$ of a reduced word for $w$ is a left descent and differs from $s$, hence $r\in J_s$ and $r{\le }_R{w}_{J_s}$ by [F4], a contradiction. Conversely ${\pi }_c(1)=1$. The cone for $1$ is $C$, and $wC\subseteq C$ exactly when $w=1$ by disjoint chamber interiors. This is the base for the following inductions on (rank, length of the sortable element).

We prove monotonicity by induction on (rank, $\mathrm{\ell }(y)$), simultaneously for every Coxeter element and pair $x{\le }_{R}y$. The base $y=1$ is immediate. It suffices to handle covers. First establish the auxiliary consequence under these inductive hypotheses: for any simple $t{\le }_{R}y$, one has $t{\le }_R{\pi }_c(y)$. Choose initial $s$ of $c$. If $s=t$, descent detection proves this. If $s\mathrm{̸}{\le }_{R}y$, then $t\in J_s$ and $t{\le }_R{y}_{J_s}$; rank induction gives $t={\pi }_{sc}(t){\le }_R{\pi }_{sc}(y_{J_s})={\pi }_c(y)$, since every simple generator is sortable (its one selected occurrence is in the first block).

Cover case with neither $x$ nor $y$ above $s$. The prefix map preserves $x_{J_s}{\le }_R{y}_{J_s}$, and rank induction gives ${\pi }_{sc}(x_{J_s}){\le }_R{\pi }_{sc}(y_{J_s})$, the desired projections. No parabolic membership of $x,y$ is needed. If both are above $s$, left translation gives $sx{\le }_{R}sy$ with the upper length smaller; length induction followed by [F7] gives $s{\pi }_{scs}(sx){\le }_{R}s{\pi }_{scs}(sy)$.

Comparable criterion, neither element above initial $s$. Only the sortable $v$, not an arbitrary non-descent $w$, is asserted to belong to $W_{J_s}$ by [F12]. Its skip set is $\{e_s\}\cup C_{sc}(v)$. The $e_s$-inequality holds for $wC$ since $s\mathrm{̸}{\le }_{R}w$; all other inequalities involve subsystem roots and therefore depend only on $N(w_{J_s}^{-1})$ by [F4]. Consequently $wC\subseteq {\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}_c(v)$ is equivalent to $w_{J_s}{C}_{J_s}\subseteq {\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}_{sc}(v)$. Since $v{\le }_{R}w$ gives $v{\le }_R{w}_{J_s}$, rank induction identifies this with ${\pi }_{sc}(w_{J_s})=v$, the recursion for ${\pi }_c(w)$.

Comparable criterion, both elements above $s$. Then $sv{\le }_{R}sw$ by [F7], $sv$ is $scs$-sortable, and ${\pi }_c(w)=s{\pi }_{scs}(sw)$. Root transport gives ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}_c(v)=\rho (s){\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}_{scs}(sv)$, so inclusion of $wC$ is equivalent to inclusion of $(sw)C$ in the latter cone. Induction on the strictly smaller length of $sv$ proves the equivalence with ${\pi }_{scs}(sw)=sv$, hence ${\pi }_c(w)=v$.

Auxiliary consequence when $s{\le }_{R}y$ and $s\mathrm{\ne }t$. Put $z=s\vee t$, the rank-two longest element by [F9]; then $z{\le }_{R}y$, so $sz{\le }_{R}sy$ by [F7]. In the rank-two system $sz$ has an alternating reduced word of length $m(s,t)-1$ beginning with $t$, by [F11], so it is sortable for $ts$, the restriction of $scs$ (where $s$ is final). Parabolic restriction and the fixed-point property give ${\pi }_{scs}(sz)=sz$. Since $\mathrm{\ell }(sy)<\mathrm{\ell }(y)$, length induction gives $sz{\le }_R{\pi }_{scs}(sy)$. Both sides are not above $s$: the left because $s(sz)=z$ lengthens, the right because it is below $sy$, which is not above $s$. Apply [F7] to their left multiples to obtain $z{\le }_{R}s{\pi }_{scs}(sy)={\pi }_c(y)$, hence $t{\le }_R{\pi }_c(y)$. This proves the auxiliary consequence for all simple $t$ and all $c$ under the stated inductive hypotheses.

Comparable criterion, $v\mathrm{̸}{\ge }_{R}s$ and $w{\ge }_{R}s$. Descent detection makes ${\pi }_c(w)\mathrm{\ne }v$, while $e_s$ is a positive skip root of $v$ and interior points of $wC$ have negative pairing with it. Both sides fail. The fourth possibility $v{\ge }_{R}s$, $w\mathrm{̸}{\ge }_{R}s$ is excluded by $v{\le }_{R}w$. These cases prove (1) using only rank and sortable-length induction.

Mixed cover $x\mathrm{̸}{\ge }_{R}s$, $y{\ge }_{R}s$. Their inversion sets differ by one root, necessarily $e_s$ by [F8]; deleting its cover reflection gives $x=sy$. Put $u={\pi }_{scs}(x)$. It is below $x$ and not above $s$. The auxiliary consequence in steps 1.3 and 2.3, applied to $scs$ at the present upper element $y$, gives ${\pi }_{scs}(y){\ge }_{R}s$, so it differs from $u$. Comparable criterion (1), already proved independently, gives $xC\subseteq {\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}_{scs}(u)$ and $yC\subseteq ̸{\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}_{scs}(u)$, since $u{\le }_{R}x{\le }_{R}y$. The sign dictionary and the single new inversion $e_s$ show that the skip inequality which changes from satisfied on $xC$ to violated on $yC$ must have positive normal $e_s$. Thus $e_s\in C_{scs}(u)$, and transport gives $-e_s\in C_c(su)$; the negative-skip/cover dictionary makes $s$ a cover reflection of $su={\pi }_c(y)$. Therefore $u=s{\pi }_c(y){<}_R{\pi }_c(y)$. Length induction at $x$, together with parabolic restriction, gives ${\pi }_c(x)={\pi }_{sc}(x_{J_s})={\pi }_{scs}(x_{J_s}){\le }_{R}u$. Hence ${\pi }_c(x){\le }_R{\pi }_c(y)$. This proves (2). The separating wall here is $H_{e_s}$; no identification with $H_{\rho (x)e_s}$ is used.

Greatest sortable element. The projection is sortable and below $w$ by [F3]. If sortable $v{\le }_{R}w$, monotonicity gives $v={\pi }_c(v){\le }_R{\pi }_c(w)$. Antisymmetry proves uniqueness.

Full criterion: induct again on (rank, $\mathrm{\ell }(v)$), now with arbitrary $w$. The base $v=1$ is step 1.2. Cases where neither element is above initial $s$, or both are above it, use exactly the sign/prefix and conjugation computations in steps 2.1-2.2, with this full induction replacing the comparable induction; no comparison is needed. The case $v\mathrm{̸}{\ge }_{R}s$, $w{\ge }_{R}s$ is step 3.1. In the remaining case $v{\ge }_{R}s$, $w\mathrm{̸}{\ge }_{R}s$, descent detection excludes equality. Monotonicity gives ${\pi }_{scs}(sw){\ge }_R{\pi }_{scs}(s)=s$, since $sw{\ge }_{R}s$; but $sv\mathrm{̸}{\ge }_{R}s$. The projections therefore differ. Full induction on the shorter sortable element $sv$ gives $(sw)C\subseteq ̸{\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}_{scs}(sv)$, hence $wC\subseteq ̸{\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}_c(v)$ by conjugation. This proves (3) without applying the comparable criterion to an unverified comparable pair.

Parabolic compatibility. Since $w_J{\le }_{R}w$, monotonicity and restriction give ${\pi }_{c{\mathrm{\mid }}_J}(w_J)={\pi }_c(w_J){\le }_R{\pi }_c(w)$, hence it is below ${\pi }_c(w{)}_J$. Conversely ${\pi }_c(w){\le }_{R}w$ implies ${\pi }_c(w{)}_J{\le }_R{w}_J$. This prefix is $c{\mathrm{\mid }}_J$-sortable by [F10], so applying monotonicity of ${\pi }_{c{\mathrm{\mid }}_J}$ gives ${\pi }_c(w{)}_J{\le }_R{\pi }_{c{\mathrm{\mid }}_J}(w_J)$. Antisymmetry proves (4).

Clauses (1)-(4) have been proved in the order comparable criterion, monotonicity, greatest/full criterion, and parabolic compatibility. All choices use finite words, roots or chambers and no Choice is invoked.