The geometric inversion set $N(w)$ of an element of a Coxeter group

Let $S$ be a finite set, $m$ a Coxeter matrix, $W$ the presented group with length function $\mathrm{\ell }$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), $V={\mathbb{R}}^S$ with Coxeter form $B$ and canonical reflection homomorphism $\rho :W\to \mathrm{G}\mathrm{L}(V)$ (The canonical reflection homomorphism, roots, reflections, and the positive cone), and let $\mathrm{\Phi }={\mathrm{\Phi }}_{+}\bigsqcup {\mathrm{\Phi }}_{-}$ be the signed root system (Root sign coherence and the action of simple reflections on positive roots).

(1) Definition. For $w\in W$ the inversion set of $w$ is $N(w):=\{\alpha \in {\mathrm{\Phi }}_{+}:\rho (w)\alpha \in {\mathrm{\Phi }}_{-}\}={\mathrm{\Phi }}_{+}\cap \rho (w{)}^{-1}{\mathrm{\Phi }}_{-}\mathrm{.}$ Its elements are the roots inverted by $w$. This is a subset of ${\mathrm{\Phi }}_{+}$, defined before any finiteness or independence assertion; by construction $\rho (w)N(w)\subseteq {\mathrm{\Phi }}_{-}$, and $N(w)$ is determined by the linear map $\rho (w)$.

(2) Elementary identities. $N(1)=\mathrm{\varnothing }$; for every $w\in W$ $N(w^{-1})=-\rho (w)N(w),$ and consequently $\mathrm{\mid }N(w^{-1})\mathrm{\mid }=\mathrm{\mid }N(w)\mathrm{\mid }$ whenever one of the two sets is finite. For every $u\in W$ and $s\in S$, with $sN(u):=\{\rho (s)\beta :\beta \in N(u)\}$, $\mathrm{\ell }(us)>\mathrm{\ell }(u)⟹N(us)=\{e_s\}\bigsqcup sN(u),\mathrm{\ell }(us)<\mathrm{\ell }(u)⟹N(us)=s(N(u)\setminus \{e_s\});$ in the first case $sN(u)\subseteq {\mathrm{\Phi }}_{+}\setminus \{e_s\}$ and $e_s\mathrm{\notin }N(u)$, so the union is disjoint.

The identities are elementary: $N(1)=\mathrm{\varnothing }$ because $\rho (1)={\mathrm{i}\mathrm{d}}_V$ would force $\alpha \in {\mathrm{\Phi }}_{+}\cap {\mathrm{\Phi }}_{-}$ for $\alpha \in N(1)$, and ${\mathrm{\Phi }}_{+},{\mathrm{\Phi }}_{-}$ are disjoint. For the second identity let $\alpha \in {\mathrm{\Phi }}_{+}$: then $\alpha \in N(w^{-1})$ means $\rho (w{)}^{-1}\alpha \in {\mathrm{\Phi }}_{-}$, and with $\beta :=-\rho (w{)}^{-1}\alpha$ one has $\beta \in {\mathrm{\Phi }}_{+}$ and $\rho (w)\beta =-\alpha \in {\mathrm{\Phi }}_{-}$, so $\beta \in N(w)$ and $\alpha =-\rho (w)\beta$; conversely $\alpha =-\rho (w)\beta$ for some $\beta \in N(w)$ gives $\alpha \in {\mathrm{\Phi }}_{+}$ because ${\mathrm{\Phi }}_{-}=-{\mathrm{\Phi }}_{+}$, and $\rho (w{)}^{-1}\alpha =-\beta \in {\mathrm{\Phi }}_{-}$, so $\alpha \in N(w^{-1})$. Since $\beta \mapsto -\rho (w)\beta$ is a bijection, the cardinality statement follows. For the recursion, let $\beta \in {\mathrm{\Phi }}_{+}$; then $\rho (us)\beta =\rho (u)\rho (s)\beta$, and $\rho (s)$ maps ${\mathrm{\Phi }}_{+}\setminus \{e_s\}$ bijectively onto itself while $\rho (s)e_s=-e_s$ (Root sign coherence and the action of simple reflections on positive roots (3)), so $\beta \mapsto \rho (s)\beta$ is a bijection of ${\mathrm{\Phi }}_{+}\setminus \{e_s\}$ that carries $N(us)\setminus \{e_s\}$ onto $N(u)\setminus \{e_s\}$. By the root-length criterion (The root-length criterion and faithfulness of the canonical reflection representation (1)) one has $e_s\in N(u)⟺\rho (u)e_s\in {\mathrm{\Phi }}_{-}⟺\mathrm{\ell }(us)<\mathrm{\ell }(u)$ and $e_s\in N(us)⟺\rho (u)\rho (s)e_s=-\rho (u)e_s\in {\mathrm{\Phi }}_{-}⟺\rho (u)e_s\in {\mathrm{\Phi }}_{+}⟺\mathrm{\ell }(us)>\mathrm{\ell }(u)$. If $\mathrm{\ell }(us)>\mathrm{\ell }(u)$ this gives $e_s\in N(us)$, $e_s\mathrm{\notin }N(u)$ and hence $N(us)=\{e_s\}\bigsqcup sN(u)$ with $sN(u)\subseteq {\mathrm{\Phi }}_{+}\setminus \{e_s\}$; if $\mathrm{\ell }(us)<\mathrm{\ell }(u)$ it gives $e_s\mathrm{\notin }N(us)$, $e_s\in N(u)$ and hence $N(us)=s(N(u)\setminus \{e_s\})$.

(3) Convention on reduced words. We keep the left action $\rho$ of $W$ on $V$. If $w=s_1\cdots s_n$ is a reduced expression, then the suffix roots $\rho (s_{i+1}\cdots s_n{)}^{-1}{e}_{s_i}$ $(1\le i\le n)$ are elements of $N(w)$, and the prefix roots $\rho (s_1\cdots s_{i-1})e_{s_i}$ are elements of $N(w^{-1})$. That these lists are exactly $N(w)$ and $N(w^{-1})$, that their elements are pairwise distinct positive roots, and that in particular $\mathrm{\mid }N(w)\mathrm{\mid }=\mathrm{\ell }(w)$, is proved in The inversion formula $\mathrm{\mid }N(w)\mathrm{\mid }=\mathrm{\ell }(w)$, the root-reflection dictionary and strong exchange ↗; no finiteness or independence beyond the identities of (2) is asserted here.

For the two membership statements put $w_{i-1}:=s_1\cdots s_{i-1}$ and ${\alpha }_i:=\rho (s_{i+1}\cdots s_n{)}^{-1}{e}_{s_i}$. Since $\mathrm{\ell }(w_{i-1}{s}_i)=i>\mathrm{\ell }(w_{i-1})=i-1$ by reducedness of the expression, the root-length criterion gives ${\gamma }_i:=\rho (w_{i-1})e_{s_i}\in {\mathrm{\Phi }}_{+}$ and also, applied to the reversed reduced expression of $w^{-1}=s_n\cdots s_1$, gives ${\alpha }_i\in {\mathrm{\Phi }}_{+}$. Moreover $(s_1\cdots s_n)(s_{i+1}\cdots s_n{)}^{-1}=s_1\cdots s_i$, so $\rho (w){\alpha }_i=\rho (s_1\cdots s_i)e_{s_i}=-\rho (w_{i-1})e_{s_i}=-{\gamma }_i\in {\mathrm{\Phi }}_{-}$, which shows ${\alpha }_i\in N(w)$; and $(s_n\cdots s_1)(s_1\cdots s_{i-1})=s_n\cdots s_i$, so $\rho (w^{-1}){\gamma }_i=\rho (s_n\cdots s_i)e_{s_i}=-\rho (s_n\cdots s_{i+1})e_{s_i}=-{\alpha }_i\in {\mathrm{\Phi }}_{-}$, which shows ${\gamma }_i\in N(w^{-1})$.