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 ${\mathcal{H}}_Q=\mathrm{mod}\mathbf{k}Q$ 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 $\tilde{\mathbf{w}}$ 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.

Last version, to appear in PLMS

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Cited by in corpus (7)

• Stability conditions and quantum dilogarithm identities for Dynkin quivers
• Decorated marked surfaces: spherical twists versus braid twists
• Contractible stability spaces and faithful braid group actions
• A survey on maximal green sequences
• On Maximal Green Sequences
• Horizontal and vertical mutation fans
• Lengths of maximal green sequences for tame path algebras