Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator

Let $(W,S)$ be a Coxeter system of finite type with $S$ finite and length function $\mathrm{\ell }$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Coxeter diagrams: edges, labels, components and finite type), with canonical reflection representation $\rho :W\to \mathrm{G}\mathrm{L}(V)$ on $V={\mathbb{R}}^S$, Coxeter form $B$, root system $\mathrm{\Phi }$ and reflection set $T=\{ws{w}^{-1}:w\in W,s\in S\}$ (The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone). Since $W$ is finite, $B$ is positive definite (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite), so $(V,B)$ is a real inner product space (Real and complex inner-product spaces and their induced length), and $\rho (w)$ preserves $B$ for every $w\in W$ (Descent of the reflection representation, unit root norms, and conjugation of reflections (2)). Write $\mathrm{O}(V)$ for the group of $B$-preserving invertible linear maps $V\to V$ (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces).

(1) Reflection length. For $w\in W$ put

${\mathrm{\ell }}_T(w):=\min \{k\in \mathbb{N}:\text{ there are }{t}_1,\dots ,t_k\in T\text{ with }w=t_1{t}_2\cdots t_k\},$

where the empty product ($k=0$) is the identity. The minimum exists because $S\subseteq T$ and $S$ generates $W$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), so the admitted $k$ form a nonempty subset of $\mathbb{N}$, which has a least element (The well-ordering principle).

(2) Absolute order. For $u,v\in W$ define $u{\le }_{T}v$ if and only if

${\mathrm{\ell }}_T(v)={\mathrm{\ell }}_T(u)+{\mathrm{\ell }}_T(u^{-1}v)\mathrm{.}$

(3) Moved and fixed spaces. For a linear map $A:V\to V$ (Linear map between vector spaces over the same field) define the moved space and the fixed space

$M(A):=\mathrm{im}(A-{\mathrm{i}\mathrm{d}}_V),F(A):=\ker (A-{\mathrm{i}\mathrm{d}}_V)$

(Kernel and image of a linear map). For $A,B\in \mathrm{O}(V)$ define the relation $B{\le }_{\mathrm{O}}A$ if and only if

$\dim M(A)=\dim M(B)+\dim M(B^{-1}A)\mathrm{.}$

(4) Conventions and abstentions. For $u\in W$ write $M(u):=M(\rho (u))$ and $F(u):=F(\rho (u))$. This definition asserts no property of ${\le }_T$ and ${\le }_{\mathrm{O}}$ beyond the displayed formulas: it asserts neither that either relation is a partial order, nor that ${\mathrm{\ell }}_T(w)=\dim M(w)$, nor that $B{\le }_{\mathrm{O}}A$ means that a shortest reflection factorization of $B$ is a prefix of one of $A$. Those properties are proved in Carter's reflection-length formula, the absolute order on a finite Coxeter group, and moved-space rigidity under a common upper bound ↗, the recorded justifier of this definition, and by the restriction and factorization lemmas of this page. No Choice is used: $S$, $W$, $\mathrm{\Phi }$ and $T$ are finite and every object is finite-dimensional or set-theoretic.