C-sortable words as green mutation sequences

Abstract

Let $Q$ be an acyclic quiver and $\mathbf{s}$ be a sequence with elements in the vertex set $Q_0$. We describe an induced sequence of simple (backward) tilting in the bounded derived category $\mathcal{D}(Q)$, starting from the standard heart $\mathcal{H}_Q=\operatorname{mod}\mathbf{k}Q$ and ending at another heart $\mathcal{H}_\mathbf{s}$ in $\mathcal{D}(Q)$. Then we show that $\mathbf{s}$ is a green mutation sequence if and only if every heart in this simple tilting sequence is greater than or equal to $\mathcal{H}_Q[-1]$; it is maximal if and only if $\mathcal{H}_\mathbf{s}=\mathcal{H}_Q[-1]$. This provides a categorical way to understand green mutations. Further, fix a Coxeter element $c$ in the Coxeter group $W_Q$ of $Q$, which is admissible with respect to the orientation of $Q$. We prove that the sequence $\widetilde{\mathbf{w}}$ induced by a $c$-sortable word $\mathbf{w}$ is a green mutation sequence. As a consequence, we obtain a bijection between $c$-sortable words and finite torsion classes in $\mathcal{H}_Q$. As byproducts, the interpretations of inversions, descents and cover reflections of a $c$-sortable word $\mathbf{w}$ are given in terms of the combinatorics of green mutations.