The authors study sortable elements in a general Coxeter group \(W\). An element \(c\in W\) is called a ‘Coxeter element’ if it is of the form \(s_1s_2\cdots s_n\) for some ordering of the set \(S:=\{s_1,s_2,\dots,s_n\}\) of simple reflections. For any word \(z\) in the alphabet \(S\), the ‘support’ \(\text{supp}(z)\subseteq S\) of \(z\) is the set of letters occurring in \(z\). An element \(w\in W\) is called \(c\)-sortable if \(w\) has a reduced word which is the concatenation \(z_1z_2\cdots z_k\) of words \(z_i\) subject to the following conditions: (1) Each \(z_i\) is a subword of \(s_1\cdots s_n\). (2) The supports of the \(z_i\)’s satisfy \(\text{supp}(z_1)\supseteq\text{supp}(z_2)\supseteq\cdots\supseteq\text{supp}(z_k)\). This word is called a \(c\)-sorting word for \(w\).
The authors show that the element \(w\in W\) is \(c\)-sortable if and only if \(w\) has a reduced word whose associated sequence of reflections is compatible with the skew-symmetric form \(\omega_c\).
They characterize \(c\)-sortability of \(w\) directly in terms of inversions of \(w\).
They define the ‘\(c\)-Cambrian cone’ associated to \(v\) for each \(c\)-sortable element \(v\).
They give a recursive definition of the projection \(\pi_{\downarrow}^c\) and show that \(\pi_{\downarrow}^c\) is order preserving and that each fiber \((\pi_{\downarrow}^c)^{-1}(v)\) is the intersection of the Tits cone with \(\text{Cone}_c(v)\).
They show that the \(c\)-sortable elements are a sub-semilattice of the weak order, that the map \(\pi_{\downarrow}^c\) respects the meet operation and the join operation.
They show that the Cambrian cones \(\text{Cone}_c(v)\) are the maximal cones of a fan inside the Tits cone.
The last section contains pictorial examples illustrating many of the results.