The sortable projection kernel and the c-Cambrian quotient

Let $(W,S)$ be a Coxeter system of finite type, let $c$ be a Coxeter element (Coxeter elements, the oriented Euler form, the skew form, and the periodic word) and let ${\pi }_c\text{\hspace{0.17em}⁣}:W\to W$ be the sortable projection of The recursive initial-letter sortable projection. By Skip bases, cover roots, greatest-sortable projections, and the chamber union of each cone (1), ${\pi }_c$ is well defined, independent of the initial-letter choices in its recursion, idempotent and order preserving, and ${\pi }_c(w)$ is the unique greatest $c$-sortable element below $w$ in the right weak order ${\le }_R$ (c-sortable elements, forced and unforced skips, skip roots, and the chamber cone (1), The right and left weak orders, intervals, covers, and meets and joins of subsets). The right weak order on the finite group $W$ is a lattice with meet $\wedge$ and join $\vee$ (Weak order is a meet-semilattice, finite Coxeter groups are lattices, and joins of simple reflections exist exactly for finite parabolics).

(1) Sortable equivalence and quotient order. Define the sortable equivalence ${\sim }_c$ on $W$ by

$x{\sim }_{c}y:⟺{\pi }_c(x)={\pi }_c(y),$

write $[x{]}_c:=\{y\in W:{\pi }_c(y)={\pi }_c(x)\}$ for the ${\sim }_c$-class of $x$ and $W\mathrm{/}{\sim }_c:=\{[x{]}_c:x\in W\}$, and let $p_c\text{\hspace{0.17em}⁣}:W\to W\mathrm{/}{\sim }_c$, $p_c(x):=[x{]}_c$, be the quotient map. The sortable quotient order on $W\mathrm{/}{\sim }_c$ is

$[x{]}_c{\le }_c[y{]}_c:⟺{\pi }_c(x){\le }_R{\pi }_c(y)\mathrm{.}$

This is independent of the chosen representatives because each class has a single ${\pi }_c$-image; it is the order induced by ${\le }_R$ on the image of ${\pi }_c$.

(2) Proposed quotient operations. On classes define

$[x{]}_c\vee [y{]}_c:=[x\vee y{]}_c,[x{]}_c\wedge [y{]}_c:=[x\wedge y{]}_c,$

the proposed quotient operations of Finite lattice congruences, interval endpoints and descending rooted-chain labels (1) specialized to the weak-order lattice $W$.

(3) Scope and abstentions. The set $W\mathrm{/}{\sim }_c$ with the order (1) and the operations (2) is the sortable quotient of the finite weak order; throughout this library c-Cambrian quotient (or Cambrian quotient) denotes this sortable-kernel construction and nothing else. The definition asserts only the displayed constructions: it does not assert that the proposed operations are independent of the chosen representatives (equivalently, that ${\sim }_c$ is a lattice congruence), that every class is an interval of ${\le }_R$, that ${\pi }_c$ preserves meets and joins, or that $p_c$ is a lattice homomorphism. Those statements are proved in Sortable elements form a sublattice and the c-Cambrian quotient is its lattice-homomorphic image and The upper endpoint of a c-Cambrian fiber, interval fibers and the explicit formula u_c(w) = pi_{c^{-1}}(ww0)w0 ↗; the well-definedness target of this definition recorded in its justification is the second of these. The quotient is not identified here with the separate least lattice congruence contracting the oriented rank-two cover pairs determined by the rank-two orientations induced by $c$ (Coxeter elements, the oriented Euler form, the skew form, and the periodic word (2)); that identification is not asserted. No claim about noncrossing partitions, cluster fans, associahedra or $W$-Catalan counting is made. No Choice is used.