Coxeter elements, the noncrossing interval [1,c], and the Kreweras map w ↦ w⁻¹c

Let $(W,S)$ be a Coxeter system with $S$ finite, Coxeter diagram $\mathrm{\Gamma }$, word length $\mathrm{\ell }$, and presented group $W$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Coxeter diagrams: edges, labels, components and finite type). Let $V={\mathbb{R}}^S$ have its Coxeter form $B$ and canonical reflection homomorphism $\rho :W\to \mathrm{G}\mathrm{L}(V)$; its reflection set is $T=\{ws{w}^{-1}:w\in W,s\in S\}$ (The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone). Extend the finite-type formulas of Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator (1),(2) to this possibly infinite group by defining ${\mathrm{\ell }}_T(w):=\min \{k\in \mathbb{N}:w=t_1\cdots t_k,t_i\in T\},u{\le }_{T}v⟺{\mathrm{\ell }}_T(v)={\mathrm{\ell }}_T(u)+{\mathrm{\ell }}_T(u^{-1}v)\mathrm{.}$ The minimum exists because $S\subseteq T$ generates $W$; the empty product is $1$.

(1) Coxeter elements. Put $n=\mathrm{\mid }S\mathrm{\mid }$ and choose a bijection $\sigma :\{1,\dots ,n\}\to S$. The product $c_{\sigma }=\sigma (1)\cdots \sigma (n)$ is the Coxeter element for that ordering; a Coxeter element of $(W,S)$ is any such product, with each simple reflection used exactly once. For $n=0$ the unique empty ordering has empty product $1$; for $n=1$ the product is the sole simple reflection. If $\mathrm{\Gamma }$ is connected and $W$ is of finite type, the bipartite construction of The bipartite Coxeter element, its ordered prefix roots, and the conditional vector map mu(a) = -2(c-1)^{-1}a (1) gives one particular Coxeter element. This definition makes no claim that distinct orderings give conjugate elements.

(2) The noncrossing interval. For an irreducible system and a Coxeter element $c$, define $\mathrm{NC}(W,c):=[1,c{]}_{{\le }_T}=\{w\in W:1{\le }_{T}w{\le }_{T}c\},$ with the order induced by ${\le }_T$. Since $T$ generates $W$, ${\mathrm{\ell }}_T$ is a word length: it is subadditive, vanishes only at $1$, and ${\mathrm{\ell }}_T(1)=0$. Thus $u{\le }_{T}u$; if $u{\le }_{T}v$ and $v{\le }_{T}u$, adding the two defining equalities gives ${\mathrm{\ell }}_T(u^{-1}v)={\mathrm{\ell }}_T(v^{-1}u)=0$, so $u=v$. If $u{\le }_{T}v{\le }_{T}w$, subadditivity gives ${\mathrm{\ell }}_T(w)\le {\mathrm{\ell }}_T(u)+{\mathrm{\ell }}_T(u^{-1}w)\le {\mathrm{\ell }}_T(u)+{\mathrm{\ell }}_T(u^{-1}v)+{\mathrm{\ell }}_T(v^{-1}w)={\mathrm{\ell }}_T(w),$ so equality holds throughout and $u{\le }_{T}w$. Hence ${\le }_T$ is a partial order, and $1$ is the least element of the interval. The chosen $c$ is part of the definition; independence up to isomorphism for finite type is proved in Finite noncrossing intervals are lattices, independently of the Coxeter element ↗ (4), not assumed here.

(3) Reducible systems. If the connected components of $\mathrm{\Gamma }$ have vertex sets $S_1,\dots ,S_k$, then $W\cong W_{S_1}\times \cdots \times W_{S_k}$ by Disconnected diagrams, direct products, and comparison of invariant forms (1), where $W_{S_i}=⟨S_i⟩$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups). A Coxeter element $c$ has coordinates $c_i$, each a Coxeter element for $(W_{S_i},S_i)$. Define $\mathrm{NC}(W,c):={\prod }_{i=1}^k\mathrm{NC}(W_{S_i},c_i)$ with componentwise order; for $k=0$ this is the one-element empty product. This agrees with the ambient interval $[1,c{]}_{{\le }_T}$: conjugates of a simple generator stay in its component, so $T$ is the disjoint union of the component reflection sets $T_i$ in their respective factors. Any reflection factorization of $(w_i)$ projects to one in each factor, giving ${\mathrm{\ell }}_T((w_i))\ge {\sum }_i{\mathrm{\ell }}_{T_i}(w_i)$; concatenating shortest factorizations in the factors gives the reverse inequality. Hence reflection length is the sum of the component lengths, and the absolute-order relation is componentwise. For $k=0$, $W=\{1\}$ and both the interval and product are singletons.

(4) The Kreweras map. Define $K:\mathrm{NC}(W,c)\to W,K(w):=w^{-1}c\mathrm{.}$ This is well-defined as a map to $W$ by the group operations. It is not defined here as a map into $\mathrm{NC}(W,c)$, and no bijectivity or order-reversal is asserted; those properties are proved in The Kreweras complement of [1,c], and the type-A model by noncrossing set partitions ↗ (1).

(5) Abstentions. This definition asserts no finiteness, lattice property, conjugacy of Coxeter elements, independence from $c$, or Kreweras-complement property beyond the definitions above. In the reducible case the product definition in (3) is justified locally as the ambient absolute interval; the lattice theorem remains a separate finite-type result. No form of the Axiom of Choice is used.

• The set of Coxeter elements need not be a union of $W$-conjugacy classes. Take the presentation with generators $s_1,s_2,s_3$ and only the relations $s_i^2=1$ (the free product $C_2\ast C_2\ast C_2$). On the set $X$ of words with no equal adjacent letters, let $s_i$ delete the first letter when it is $s_i$, and otherwise prepend $s_i$. Each operation is an involution in the permutation group $\mathrm{Sym}(X)$ (The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition, $\mathrm{Sym}(X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements), so the presentation's universal property gives a homomorphism to that group. A word with no equal adjacent letters sends the empty word to its own letter string, whereas a product of $k$ generators sends it to a string of length at most $k$. Therefore $s_2{s}_1{s}_2{s}_3{s}_2$ has word length five. It equals $s_2(s_1{s}_2{s}_3)s_2$, but cannot be a once-each product of three generators.