The weak parabolic projection, its adjoints, and the cover-join lemmas

Finite setup and notation. By [F1], $W$ is finite, and each $W_J$ is finite. Fix $J\subseteq S$ and $w\in W$. Write $w=w_{J}d$ for its unique length-additive factorization, with $d$ minimal in $W_{J}w$ by [F6]; $w_J$, $W_J$, ${\mathrm{\Phi }}_{J,+}$, $N$, $w_0$ and $w_0(J)$ have the meanings fixed in the Statement and [F7]-[F9],[F15].

The minimal representative above the parabolic factor. Let $L:=\mathrm{\ell }(w_0)$ and $L_J:=\mathrm{\ell }(w_0(J))$. By [F5], $q:=w_0(J)w_0$ has length $L-L_J$. For each $s\in J$, [F15] gives $\mathrm{\ell }(s{w}_0(J))=L_J-1$ and [F5] gives $\mathrm{\ell }(sq)=\mathrm{\ell }(s{w}_0(J)w_0)=L-(L_J-1)=\mathrm{\ell }(q)+1$. Thus $q$ has no left descent in $J$ and is the minimal representative of $W_J{w}_0$ by [F6]. Since $w_0(J{)}^2=1$ by [F15], $w_0=w_0(J)q$. Therefore $x:=zq=z{w}_0(J)w_0$ is the length-additive parabolic factorization with prefix $z$, and $\mathrm{\ell }(x)=\mathrm{\ell }(z)+L-L_J$. Also $N(w_0(J{)}^{-1})\subseteq {\mathrm{\Phi }}_{J,+}$ by [F8],[F16], and its cardinality is $L_J=\mathrm{\mid }{\mathrm{\Phi }}_{J,+}\mathrm{\mid }$ by [F16],[F23]; consequently $w_0(J)$ sends every positive subsystem root to a negative subsystem root.

Deleting a cover root. For $\alpha \in \mathrm{cov}(w)$ choose $s\in S$ with $t_{\alpha }w=ws$ and $\mathrm{\ell }(ws)=\mathrm{\ell }(w)-1$, and write a reduced word $w=qs$. Then $t_{\alpha }=ws{w}^{-1}=qs{q}^{-1}=t_{\rho (q)e_s}$ by [F25]; hence the unique positive root for this reflection is $\rho (q)e_s$ by [F10]. It is the last prefix root of $N(w^{-1})$; the prefix formula for $N((t_{\alpha }w{)}^{-1})=N(q^{-1})$ therefore gives $N((t_{\alpha }w{)}^{-1})=N(w^{-1})\setminus \{\alpha \}$. Conversely every cover predecessor $v{⋖}_{R}w$ is $v=ws$ for a generator $s$ by [F13], and its conjugate reflection $ws{w}^{-1}=t_{\alpha }$ has $\alpha \in N(w^{-1})$ by the prefix formula [F16]; hence $v=t_{\alpha }w$ with $\alpha \in \mathrm{cov}(w)$. Thus cover predecessors correspond exactly to deleting their cover roots.

Prefix inversion set. Choose a reduced expression $w_J=s_1\cdots s_k$ with letters in $J$ (available by [F12]) and a reduced expression $d=s_{k+1}\cdots s_n$. Their concatenation is reduced by [F6]. The prefix roots for indices $i\le k$ in the formula [F16] are the prefix roots of $N(w_J^{-1})$ and lie in ${\mathrm{\Phi }}_{J,+}$, because their reflections are words in $J$ and [F8] identifies the roots of $W_J$. If a suffix index $i>k$ had prefix root ${\alpha }_i\in {\mathrm{\Phi }}_{J,+}$, then $t_{{\alpha }_i}\in W_J$ by [F22]. Write $w=p{s}_{i}r$ with $p=s_1\cdots s_{i-1}$ and $r=s_{i+1}\cdots s_n$. By [F16], ${\alpha }_i=\rho (p)e_{s_i}$, and [F25] gives $t_{{\alpha }_i}=p{s}_i{p}^{-1}$; hence $t_{{\alpha }_i}w=pr=w_J{d}^{\mathrm{\prime }}$, where $d^{\mathrm{\prime }}$ is represented by the suffix word with $s_i$ deleted and has length at most $\mathrm{\ell }(d)-1$. Since $t_{{\alpha }_i}\in W_J$, $t_{{\alpha }_i}w\in W_{J}w=W_{J}d$, so $d^{\mathrm{\prime }}=w_J^{-1}{t}_{{\alpha }_i}w\in W_{J}d$. This contradicts the minimality of $d$. Thus no suffix root lies in ${\mathrm{\Phi }}_{J,+}$, while all prefix roots do; the inversion formula gives $N(w^{-1})\cap {\mathrm{\Phi }}_{J,+}=N(w_J^{-1})$.

Inversion set of the lift. Let $x=z{w}_0(J)w_0$. If $\alpha \in {\mathrm{\Phi }}_{J,+}$, then $\beta :=\rho (z^{-1})\alpha$ is a root of the subsystem. By step 1.2, $\rho (w_0(J))$ reverses the sign of subsystem roots, while $\rho (w_0)$ reverses the sign of every root; therefore $\rho (x^{-1})\alpha \in {\mathrm{\Phi }}_{-}$ exactly when $\rho (z^{-1})\alpha \in {\mathrm{\Phi }}_{-}$, so $\alpha \in N(x^{-1})$ exactly when $\alpha \in N(z^{-1})$. If $\alpha \in {\mathrm{\Phi }}_{+}\setminus {\mathrm{\Phi }}_{J,+}$, choose reduced words for $z^{-1}$ and $w_0(J)$; their letters lie in $J$ by [F12]. At every letter $r\in J$, the current root remains outside $V_J$: since $\rho (r)$ is invertible and preserves $V_J$ by [F8], it cannot send a vector outside $V_J$ into $V_J$. Thus the current root is never $e_r$, and [F17] keeps it positive. Then $\rho (w_0)$ sends the resulting positive root to a negative root by [F5], so every such $\alpha$ belongs to $N(x^{-1})$. This proves $N(x^{-1})=N(z^{-1})\cup ({\mathrm{\Phi }}_{+}\setminus {\mathrm{\Phi }}_{J,+})$.

Greatest prefix and order-preserving projection. If $v\in W_J$, [F12] says every reduced word for $v$ uses only letters of $J$, so every prefix root of a reduced word for $v$ lies in ${\mathrm{\Phi }}_{J,+}$ by [F8] and [F16]. Thus, when $v{\le }_{R}w$, the criterion [F3] and step 2.1 give $N(v^{-1})\subseteq N(w_J^{-1})$, so $v{\le }_R{w}_J$; conversely $w_J{\le }_{R}w$ by the factorization [F6] and the definition [F24]. This proves the greatest-element claim and $v{\le }_{R}w⟺v{\le }_R{w}_J$ for $v\in W_J$. If $x{\le }_{R}y$, intersect $N(x^{-1})\subseteq N(y^{-1})$ with ${\mathrm{\Phi }}_{J,+}$ and apply step 2.1 to both prefixes; [F3] gives $x_J{\le }_R{y}_J$.

Cover-join clause (4)(i). Let $J=S\setminus \{s\}$ and assume the hypotheses of (i). By [F16], $N(s^{-1})=\{e_s\}$, so $e_s\in N(w^{-1})$ implies $s{\le }_{R}w$ by [F3]; also $w_J{\le }_{R}w$ by the length-additive factorization [F6] and the definition [F24]. The element $w$ is therefore a common upper bound. Suppose a strict common upper bound $u{<}_{R}w$ existed. By [F20], choose a cover predecessor $v{⋖}_{R}w$ with $u{\le }_{R}v$; by step 1.3 it deletes some $\alpha \in \mathrm{cov}(w)$. If $\alpha =e_s$, then $e_s\mathrm{\notin }N(v^{-1})$, so $s\mathrm{̸}{\le }_{R}v$ by [F3]. If $\alpha \mathrm{\ne }{e}_s$, then $t_{\alpha }\in W_J$ by hypothesis and $\alpha \in {\mathrm{\Phi }}_{J,+}$ by [F22]; steps 2.1 and 1.3 give $N(v_J^{-1})=N(w_J^{-1})\setminus \{\alpha \}$, so $w_J\mathrm{̸}{\le }_{R}v$ by [F3]. Both cases contradict $u{\le }_{R}v$ and $s,w_J{\le }_{R}u$. Hence no strict common upper bound lies below $w$. The join $s\vee w_J$ exists by [F4], and by its least-upper-bound definition [F2] is at most $w$; it equals $w$.

Largest-lift adjunction. If $y_J=z$, step 2.1 gives $N(y^{-1})\cap {\mathrm{\Phi }}_{J,+}=N(z^{-1})$, so $N(y^{-1})\subseteq N(x^{-1})$ by step 2.2 and $y{\le }_{R}x$ by [F3]. Conversely, if $y{\le }_{R}x$, order preservation from step 3.1 gives $y_J{\le }_R{x}_J=z$. More generally, if $y_J{\le }_{R}z$, then $N(y^{-1})\cap {\mathrm{\Phi }}_{J,+}=N(y_J^{-1})\subseteq N(z^{-1})$, while every root outside ${\mathrm{\Phi }}_{J,+}$ is in $N(x^{-1})$; hence $N(y^{-1})\subseteq N(x^{-1})$ and $y{\le }_{R}x$. This proves the largest-lift claim and its stated equivalence.

Meet preservation. The finite weak orders on $W$ and $W_J$ are lattices by [F4] and [F14]. For $u\in W_J$, step 3.1 gives $u{\le }_{R}v$ exactly when $u{\le }_R{v}_J$. By the meet definition [F2], for every $u\in W_J$, $u{\le }_R{x}_J\wedge y_J$ exactly when $u{\le }_R{x}_J$ and $u{\le }_R{y}_J$, exactly when $u{\le }_{R}x$ and $u{\le }_{R}y$, exactly when $u{\le }_{R}x\wedge y$, exactly when $u{\le }_R(x\wedge y{)}_J$. Both candidate meets lie in $W_J$, so antisymmetry [F19] yields $(x\wedge y{)}_J=x_J\wedge y_J$.

Join preservation and surjectivity. Put $z:=x_J\vee y_J\in W_J$ and let $X:=z{w}_0(J)w_0$ be its largest lift from step 4.1. Since $x_J,y_J{\le }_{R}z$, the adjunction in step 4.1 gives $x,y{\le }_{R}X$, so $x\vee y{\le }_{R}X$ and order preservation gives $(x\vee y{)}_J{\le }_{R}z$. Conversely, order preservation applied to $x,y{\le }_{R}x\vee y$ gives $x_J,y_J{\le }_R(x\vee y{)}_J$, hence $z{\le }_R(x\vee y{)}_J$. The join definition [F2] makes $z$ the least upper bound of $x_J,y_J$; the two bounds and antisymmetry [F19] yield $(x\vee y{)}_J=z$. If $z\in W_J$, then its parabolic factorization is $z=z\cdot 1$, so $z_J=z$; the projection is onto $W_J$.

Cover-join clause (4)(ii): the simple root and other parabolic covers. Let $y\in W_J$ and put $z:=s\vee y$. By [F12], every reduced expression of $y$ uses letters in $J$, so [F16] and [F8] give $N(y^{-1})\subseteq {\mathrm{\Phi }}_{J,+}$. Since $J=S\setminus \{s\}$, the simple basis vector $e_s$ is not in $V_J$ and hence not in ${\mathrm{\Phi }}_{J,+}$. Thus $N(s^{-1})\cap {\mathrm{\Phi }}_{J,+}=\mathrm{\varnothing }$ by [F16], and step 2.1 gives $N(s_J^{-1})=\mathrm{\varnothing }=N(1)$. The inversion criterion [F3] and partial order [F19] imply $s_J=1$. Hence projection-join preservation in step 5.1 gives $z_J=y$. Further, $s\mathrm{̸}{\le }_{R}y$, because $N(s^{-1})=\{e_s\}$ by [F16] and $e_s\mathrm{\notin }{\mathrm{\Phi }}_{J,+}$; thus $y{<}_{R}z$. Choose a cover predecessor $v{⋖}_{R}z$ with $y{\le }_{R}v$ by [F20]. Since $s{\le }_{R}z$, [F16] and [F3] give $e_s\in N(z^{-1})$; if $e_s\mathrm{\notin }\mathrm{cov}(z)$, step 1.3 says no cover predecessor deletes it, so $e_s\in N(v^{-1})$ and $s{\le }_{R}v$, contradicting that $z$ is the join of $s$ and $y$. Hence $e_s\in \mathrm{cov}(z)$. If $\alpha \in \mathrm{cov}(z)\setminus \{e_s\}$ and $t_{\alpha }\mathrm{\notin }{W}_J$, then $\alpha \mathrm{\notin }{\mathrm{\Phi }}_{J,+}$ by [F22]. The predecessor $v=t_{\alpha }z$ deletes only $\alpha$ by step 1.3, so it retains $N(y^{-1})\subseteq N(z^{-1})\cap {\mathrm{\Phi }}_{J,+}$ and retains $e_s$; hence $y,s{\le }_{R}v{<}_{R}z$, again contradicting the join. Therefore every such $t_{\alpha }$ lies in $W_J$.

The cover roots inside the parabolic. Since $z_J=y$ by step 5.1, write $z=yd$ with $d$ the minimal right-coset representative. If $\alpha \in \mathrm{cov}(z)\setminus \{e_s\}$, step 6.1 gives $t_{\alpha }\in W_J$, so $\alpha \in {\mathrm{\Phi }}_{J,+}$ by [F22]. Since $W_J{t}_{\alpha }z=W_{J}z$, $d$ remains the minimal representative of this coset by [F6], and $t_{\alpha }z=(t_{\alpha }y)d$ is the length-additive parabolic factorization. Thus its prefix is $t_{\alpha }y$; as $t_{\alpha }z$ covers $z$, lengths give $\mathrm{\ell }(t_{\alpha }y)=\mathrm{\ell }(y)-1$. Intersecting the deletion formula of step 1.3 with ${\mathrm{\Phi }}_{J,+}$ and using step 2.1 gives $N((t_{\alpha }y{)}^{-1})=N(y^{-1})\setminus \{\alpha \}$. By [F3], $t_{\alpha }y{\le }_{R}y$, and the length difference one makes it a cover by [F20]; hence $\alpha \in \mathrm{cov}(y)$. Conversely let $\alpha \in \mathrm{cov}(y)$. By [F12], a reduced word for $y$ uses only letters in $J$; its prefix roots have the form in [F16] and lie in ${\mathrm{\Phi }}_{J,+}$ by [F8], so $t_{\alpha }\in W_J$ by [F22]. Put $y^{\mathrm{\prime }}:=t_{\alpha }y$ and $z^{\mathrm{\prime }}:=s\vee y^{\mathrm{\prime }}$. Since $y^{\mathrm{\prime }}=yr$ for some $r\in J$ with $\mathrm{\ell }(y^{\mathrm{\prime }})=\mathrm{\ell }(y)-1$, [F24] gives $y^{\mathrm{\prime }}{<}_{R}y$; therefore $z=s\vee y$ is an upper bound of $s,y^{\mathrm{\prime }}$ and [F2] gives $z^{\mathrm{\prime }}=s\vee y^{\mathrm{\prime }}{\le }_{R}z$. Step 5.1 gives $z_J^{\mathrm{\prime }}=y^{\mathrm{\prime }}$ and $z_J=y$, so $z^{\mathrm{\prime }}\mathrm{\ne }z$; hence $z^{\mathrm{\prime }}{<}_{R}z$. Choose $v{⋖}_{R}z$ with $z^{\mathrm{\prime }}{\le }_{R}v$ by [F20], and let $\beta \in \mathrm{cov}(z)$ be its deleted root from step 1.3. Since $s,y^{\mathrm{\prime }}{\le }_{R}v$, if $\beta \mathrm{\notin }N(y^{-1})$ then $N(y^{-1})\subseteq N(v^{-1})$ and $y{\le }_{R}v$, contradicting the join. If $\beta \in N(y^{-1})$, then step 1.3 gives $N((y^{\mathrm{\prime }}{)}^{-1})=N(y^{-1})\setminus \{\alpha \}$; because $y^{\mathrm{\prime }}{\le }_{R}v$, the deleted root $\beta$ cannot lie in this latter set, so $\beta =\alpha$. Therefore $v=t_{\alpha }z$ and $\alpha \in \mathrm{cov}(z)$. This proves $\mathrm{cov}(z)=\mathrm{cov}(y)\cup \{e_s\}$.

Choice and conclusion. Steps 2.1 and 3.1 prove (1); steps 1.2, 2.2 and 4.1 prove (2); steps 4.2 and 5.1 prove (3); step 3.2 proves (4)(i); and steps 6.1 and 7.1 prove (4)(ii). Since $W$ is finite and each witness is selected from a finite interval or one fixed reduced expression at a time, no Axiom of Choice is used.