Coxeter elements, the oriented Euler form, the skew form, and the periodic word

Let $S$ be finite and let $m$ be a Coxeter matrix on $S$; let $W$ be the presented group with length function $\mathrm{\ell }$ and reduced expressions (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), and let $V={\mathbb{R}}^S$ carry the Coxeter form $B$, with $B(e_s,e_s)=1$, $B(e_s,e_t)=-\cos (\pi \mathrm{/}m(s,t))$ for $s\mathrm{\ne }t$ finite and $B(e_s,e_t)=-1$ when $m(s,t)=\mathrm{\infty }$, together with the canonical reflection representation $\rho$ and simple roots $e_s$ (The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone). Write $n:=\mathrm{\mid }S\mathrm{\mid }$ and $S(w)$ for the support of $w$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (1)).

(1) Coxeter elements and Coxeter words. A word $s_1\cdots s_n$ in the alphabet $S$ is a Coxeter word when $S=\{s_1,\dots ,s_n\}$. An element $c\in W$ is a Coxeter element of $(W,S)$ when it is the value of a Coxeter word; a reduced Coxeter word for $c$ is any reduced expression of $c$. That every Coxeter word is reduced, that every reduced expression of a Coxeter element is again a Coxeter word, and how two Coxeter words for the same element are related, is proved in Coxeter words are commutation-connected; the Euler and skew forms depend only on the Coxeter element ↗; none of this is asserted here. Throughout the page, $c=s_1\cdots s_n$ denotes a Coxeter element together with a chosen reduced Coxeter word.

(2) The Cartan form and the oriented Euler form. Put $K:=2B$; then $K$ is symmetric bilinear with $K(e_s,e_s)=2$, $K(e_s,e_t)=-2\cos (\pi \mathrm{/}m(s,t))$ for finite $m(s,t)$ and $K(e_s,e_t)=-2$ when $m(s,t)=\mathrm{\infty }$. The oriented Euler form of the ordered word $(s_1,\dots ,s_n)$ is the bilinear form $E_c$ on $V$ with $E_c(e_{s_i},e_{s_j}):=\{\begin{array}{ll}K(e_{s_i},e_{s_j})& \text{if }i>j,\\ 1& \text{if }i=j,\\ 0& \text{if }i<j,\end{array}$ extended bilinearly. The skew form is ${\omega }_c:=E_c-E_c^{\mathsf{T}}$, that is, ${\omega }_c(\beta ,{\beta }^{\mathrm{\prime }})=E_c(\beta ,{\beta }^{\mathrm{\prime }})-E_c({\beta }^{\mathrm{\prime }},\beta )$; equivalently ${\omega }_c(e_{s_i},e_{s_j})=K(e_{s_i},e_{s_j})$ for $i>j$, $0$ for $i=j$, and $-K(e_{s_i},e_{s_j})$ for $i<j$. This normalization is used throughout: $E_c+E_c^{\mathsf{T}}=K=2B$, and the sign of ${\omega }_c$ on the roots of a rank-two subsystem is the orientation of that subsystem induced by $c$. That $E_c$ and ${\omega }_c$ depend only on $c$ and not on the chosen reduced Coxeter word is proved in Coxeter words are commutation-connected; the Euler and skew forms depend only on the Coxeter element ↗; the forms are not asserted here to be independent of the word.

(3) The periodic word and admissible position sets. Fix a reduced Coxeter word $s_1\cdots s_n$ for $c$ and form the half-infinite periodic word $c^{\mathrm{\infty }}:=s_1\cdots s_n\mathrm{\mid }{s}_1\cdots s_n\mathrm{\mid }\cdots ,$ where the symbols $\mathrm{\mid }$ are inert dividers after every block of $n$ letters and are ignored when subwords are evaluated. A position set for $c^{\mathrm{\infty }}$ is a finite strictly increasing sequence of positions $i_1<\cdots <i_k$; its value is $s_{i_1}\cdots s_{i_k}\in W$, and it is admissible for $w\in W$ when its value is $w$ and $k=\mathrm{\ell }(w)$, equivalently when its letters form a reduced expression of $w$. The $c^{\mathrm{\infty }}$-sorting word of $w$ is the lexicographically earliest admissible position set for $w$: least first position, then least second position, and so on. The block sequence of an admissible position set is the sequence $T_1,T_2,\dots$ in which $T_j\subseteq S$ is the set of letters of the subword occurring between the $(j-1)$-st and the $j$-th divider; it is read up to the last nonempty set.

(4) Well-definedness. The existence and uniqueness of the lexicographically earliest admissible position set for every $w\in W$, and the independence of its block sequence from the chosen reduced Coxeter word for $c$, are proved in The greedy scan computes the c-sorting word; commutation, conjugation and rank-two alignment ↗ (the greedy scan and its minimality) and Coxeter words are commutation-connected; the Euler and skew forms depend only on the Coxeter element ↗ (transport between Coxeter words); these are the recorded justifiers of this definition. Sortability of elements is defined later on this page (c-sortable elements, forced and unforced skips, skip roots, and the chamber cone).

(5) Abstentions. Nothing about finiteness of $W$, positivity or nondegeneracy of $B$, the sign of ${\omega }_c$ on roots, skip roots, cones or sortable elements is asserted here beyond the displayed formulas. No Choice is used.