The bipartite Coxeter element, its ordered prefix roots, and the conditional vector map mu(a) = -2(c-1)^{-1}a

For the irreducible case, let $(W,S)$ be a finite-type Coxeter system with $S$ finite of cardinality $n\ge 1$, length function $\mathrm{\ell }$, Coxeter diagram $\mathrm{\Gamma }$ and standard parabolics $W_T=⟨s:s\in T⟩$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Coxeter diagrams: edges, labels, components and finite type, The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups); let $V={\mathbb{R}}^S$, let $B$ be the Coxeter form with $B(e_s,e_s)=1$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Descent of the reflection representation, unit root norms, and conjugation of reflections (3)), and let $\rho :W\to \mathrm{G}\mathrm{L}(V)$, $\mathrm{\Phi }=\{\rho (w)e_s\}$ and $T=\{ws{w}^{-1}\}$ be the canonical reflection representation, the root system and the reflection set (The canonical reflection homomorphism, roots, reflections, and the positive cone); assume $\mathrm{\Gamma }$ is connected, equivalently $(W,S)$ is irreducible (Coxeter diagrams: edges, labels, components and finite type). The form $B$ is positive definite (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite). Write ${\alpha }_s:=e_s$ for the simple roots and $r_a$ for the reflection with normal $a$. Clause (5) separately specifies the componentwise extension to reducible finite-type systems.

(1) The bipartition. $\mathrm{\Gamma }$ is connected and has no cycle, hence is a tree (Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (2)); a tree has a bipartition, i.e. there is a partition $S=J\bigsqcup K$ with $m(s,t)=2$ for all distinct $s,t$ in the same part (A bipartite graph and a proper two-colouring of its vertices, A finite graph is bipartite if and only if it has no odd cycle). Concretely, fix $s_0\in S$, let $J$ be the set of vertices at even distance from $s_0$ in $\mathrm{\Gamma }$ and $K$ the set at odd distance, and note that the pair $\{J,K\}$ is determined up to interchanging the two classes. Choose such a bipartition and order the simple reflections and their simple roots as ${\alpha }_1,\dots ,{\alpha }_n$, with corresponding simple reflections $s_1,\dots ,s_n$ and reflections $R_i:=r_{{\alpha }_i}$, so that $J=\{s_1,\dots ,s_r\}$ and $K=\{s_{r+1},\dots ,s_n\}$ for $r:=\mathrm{\mid }J\mathrm{\mid }$. For distinct $s_i,s_j$ in either class, $m(s_i,s_j)=2$; the Coxeter relators $s_i^2=s_j^2=(s_i{s}_j{)}^2=1$ therefore imply $s_i{s}_j=s_j{s}_i$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups). Thus the products $a:={\prod }_{i=1}^r{s}_i$ and $b:={\prod }_{i=r+1}^n{s}_i$ do not depend on the order of their factors, and

$c:=ab=s_1{s}_2\cdots s_n\in W,h:=\mathrm{ord}(c)\in \mathbb{N}$

are well defined ($W$ is finite, so $h\ge 1$). For $n=1$ one has $J=\{s_1\}$, $K=\mathrm{\varnothing }$, $a=s_1$, $b=1$, $c=s_1$ and $h=2$; empty products are $1$. For $n\ge 2$ both classes are nonempty because $\mathrm{\Gamma }$ is connected.

(2) Cyclic indexing. Read subscripts $i$ of $s_i$, ${\alpha }_i$, $R_i$ and ${\beta }_i$ cyclically modulo $n$: $s_{i+n}:=s_i$, ${\alpha }_{i+n}:={\alpha }_i$, $R_{i+n}:=R_i$, and ${\beta }_{i+n}:={\beta }_i$ for the dual family below. The cyclic indexing of the ${\beta }_i$ is part of the convention: it is what makes the vector ${\mu }_i$ below well defined for every $i\ge 1$ (see (3)).

(3) Prefix roots and dual vertices. Let $G=(B({\alpha }_j,{\alpha }_k){)}_{j,k=1}^n$ be the Gram matrix. It is invertible: for $x\mathrm{\ne }0$, the basis property gives ${\sum }_j{x}_j{\alpha }_j\mathrm{\ne }0$, so $x^{\mathsf{T}}Gx=B({\sum }_j{x}_j{\alpha }_j,{\sum }_j{x}_j{\alpha }_j)>0$. Set ${\beta }_i:={\sum }_{k=1}^n(G^{-1}{)}_{ki}{\alpha }_k$; symmetry of $G$ gives $B({\beta }_i,{\alpha }_j)={\sum }_k{G}_{jk}(G^{-1}{)}_{ki}={\delta }_{ji}$. These vectors are unique, since a vector orthogonal to every basis vector is orthogonal to itself and hence is zero by positive definiteness. Thus $({\beta }_1,\dots ,{\beta }_n)$ is the $B$-dual family of $({\alpha }_1,\dots ,{\alpha }_n)$. Define, for every integer $i\ge 1$,

${\rho }_i:=R_1{R}_2\cdots R_{i-1}{\alpha }_i,{\mu }_i:=R_1{R}_2\cdots R_{i-1}{\beta }_i,$

the empty product for $i=1$ being the identity. Put $c_V:=\rho (c)\in \mathrm{G}\mathrm{L}(V)$. The recursions ${\rho }_{i+n}=c_V{\rho }_i$ and ${\mu }_{i+n}=c_V{\mu }_i$ (valid for all $i\ge 1$) follow by separating the first $n$ factors and using cyclic indexing; they are also recorded in The Coxeter plane, ordered-root enumeration, and invertibility of rho(c) - id ↗.

(4) The conditional vector map $\mu$. For $v\in V$ put

$\mu (v):=-2(c_V-{\mathrm{i}\mathrm{d}}_V{)}^{-1}v,$

defined only if the linear map $c_V-{\mathrm{i}\mathrm{d}}_V\in \mathrm{G}\mathrm{L}(V)$ is invertible (Invertible linear maps, linear isomorphisms, and inverse linear maps). This definition asserts neither the invertibility of $c_V-{\mathrm{i}\mathrm{d}}_V$ nor the identity $\mu ({\rho }_i)={\mu }_i$; both are proved in The Coxeter plane, ordered-root enumeration, and invertibility of rho(c) - id ↗, the recorded justifier of this definition. Once defined, $\mu$ is a linear map on all of $V$, with $\mu ({\rho }_i)={\mu }_i$ for every $i\ge 1$ and $\mu (c_V^{k}v)=c_V^k\mu (v)$ for all $k\in \mathbb{Z}$ and $v\in V$, since $c_V$ commutes with $c_V-{\mathrm{i}\mathrm{d}}_V$.

(5) Reducible and empty systems. For a finite-type system with connected components $S_1,\dots ,S_m$, apply (1)--(4) to each irreducible factor $(W_i,S_i)$, where $W_i=W_{S_i}$ (Disconnected diagrams, direct products, and comparison of invariant forms). With component bipartitions $S_i=J_i\bigsqcup K_i$, let $a_i,b_i,c_i$ be the resulting group elements and $h_i=\mathrm{ord}(c_i)$. The product $c=c_1\cdots c_m$ is a product of the simple reflections in every component. Under the direct-product decomposition, $c^k=1$ exactly when $c_i^k=1$ for every $i$, so its order is $h=\mathrm{lcm}(h_1,\dots ,h_m)$. Choose a block order of the components and list the root and dual-vector families in that order. The operator $c_V=\rho (c)$ is the direct sum of $c_{V_i}=\rho (c_i)$; as in (4), the map $\mu$ is defined exactly when every $c_{V_i}-{\mathrm{i}\mathrm{d}}_{V_i}$ is invertible, and then is their direct sum. For $S=\mathrm{\varnothing }$ one has $m=0$, $W=\{1\}$, $c=1$, $h=1$ (the empty lcm is $1$), $V=\{0\}$ and empty root and dual-vector families; the unique endomorphism of $V$ is invertible, so $\mu$ is that unique map. No single number $nh\mathrm{/}2$ is claimed for reducible systems whose components have unequal Coxeter numbers.

(6) Abstentions. Each ${\rho }_i$ is a root by definition, since $R_1\cdots R_{i-1}=\rho (s_1\cdots s_{i-1})$ and ${\alpha }_i=e_{s_i}$. This item does not assert that the first $nh\mathrm{/}2$ roots enumerate the positive roots, any sign pattern for $(B({\mu }_i,{\rho }_j))$, or a spherical realization of the ordered root complex; those are proved by later items in this pair. No Choice is used.