c-sortable elements, forced and unforced skips, skip roots, and the chamber cone

Let $(W,S)$ be a Coxeter system of finite type with $S$ finite, with root system $\mathrm{\Phi }={\mathrm{\Phi }}_{+}\bigsqcup {\mathrm{\Phi }}_{-}$ and the identification $V\cong V^{\ast }$ by $B$ of The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset, and let $c=s_1\cdots s_n$ be a reduced Coxeter word with periodic word $c^{\mathrm{\infty }}$ as in Coxeter elements, the oriented Euler form, the skew form, and the periodic word; the block sequence of the $c^{\mathrm{\infty }}$-sorting word is well defined by The greedy scan computes the c-sorting word; commutation, conjugation and rank-two alignment (2).

(1) c-sortable elements. An element $v\in W$ is $c$-sortable when the block sequence $(T_1,T_2,\dots )$ of its $c^{\mathrm{\infty }}$-sorting word is weakly decreasing under inclusion: $T_1\supseteq T_2\supseteq \cdots$ (with the sequence read up to its last nonempty set). By The greedy scan computes the c-sorting word; commutation, conjugation and rank-two alignment (2) this condition is independent of the chosen reduced Coxeter word for $c$.

(2) Skips and forcedness. For any $v\in W$, fix a $c$-sorting word $a_1\cdots a_k$ of $v$ (so $a_1,\dots ,a_k$ are the letters of the sorting word in order and $k=\mathrm{\ell }(v)$), and let $r\in S$. The leftmost unselected occurrence of $r$ is the least position of $c^{\mathrm{\infty }}$ carrying $r$ which is not among the selected positions; it exists because the sorting word is finite and infinitely many occurrences of $r$ follow it. If $i$ is the number of selected letters preceding that position, the sorting word is said to skip $r$ in the $(i+1)$-st position, with associated reflection $t:=a_1\cdots a_{i}r{a}_i\cdots a_1$. The skip is forced when the word $a_1\cdots a_{i}r$ is not reduced, and unforced otherwise; write $t\in fsc(v)$ in the forced case and $t\in uf{s}_c(v)$ in the unforced case, and set ${\mathcal{A}}_c(v):=\{-{\beta }_t:t\in fsc(v)\}$ and ${\mathcal{B}}_c(v):=\{{\beta }_t:t\in uf{s}_c(v)\}$. By the definition of the sorting word, the leftmost unselected occurrence of $r$ is determined by $v$ and the chosen reduced Coxeter word for $c$; the reflection $t$ is determined by the selected prefix preceding it. Word independence for sortable $v$ is established by the justifier in (3).

(3) Skip roots. For $r\in S$ the skip root is $C_c^r(v):=\rho (a_1\cdots a_i)e_r=\pm {\beta }_t,$ with the sign rule $C_c^r(v)=-{\beta }_t⟺r\text{ is a forced skip of }v,C_c^r(v)=+{\beta }_t⟺r\text{ is an unforced skip of }v,$ where ${\beta }_t$ is the positive root of $t$ and $t$ is the reflection attached to the leftmost unselected occurrence of $r$ as in (2). The sign rule holds for every $v$: the root-length criterion of The root-length criterion and faithfulness of the canonical reflection representation (1) identifies the sign of $\rho (a_1\cdots a_i)e_r$ with whether the reduced prefix followed by $r$ is reduced.

When $v$ is $c$-sortable, the raw skip roots equivalently satisfy the following recursion of Reading--Speyer section 5: with $s$ initial in $c$, $C_c^r(v)=e_s$ if $v\mathrm{̸}{\ge }_{R}s$ and $r=s$; $C_c^r(v)=C_{sc}^r(v)$ if $v\mathrm{̸}{\ge }_{R}s$ and $r\mathrm{\ne }s$; and $C_c^r(v)=\rho (s)C_{scs}^r(sv)$ if $v{\ge }_{R}s$. For $c$-sortable $v$, agreement of the raw formula with this recursion, termination by induction on the pair (rank, length), and independence of the chosen reduced Coxeter word for $c$ are proved in Skip roots form a basis, negative skips are cover roots, and the cover decomposition of sortable elements ↗ (1),(2). The recursive description and these justifier assertions apply only in that sortable case; the raw formula and sign rule above remain defined and valid for every $v\in W$.

(4) The cone. For $c$-sortable $v$ put ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}_c(v):=\{x\in V:B(x,C_c^r(v))\ge 0\text{ for every }r\in S\},$ the intersection of the closed half-spaces with inward normals the skip roots, under the identification $V\cong V^{\ast }$ by $B$. Nothing beyond this definition is asserted here; that ${\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}_c(v)$ is a full-dimensional simplicial cone, that its walls are the root hyperplanes of all its skip roots, and that it is a union of chambers is proved in the later items of this page.

(5) Abstentions. Nothing about the projection ${\pi }_c$, greatest sortable elements, chamber unions, monotonicity, the traditional Cambrian congruence or noncrossing partitions is asserted here, and no finiteness of $W$ beyond the finite-type hypothesis of this page is used. No Choice is used.