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2012

C-sortable words as green mutation sequences

Qiu, Yu

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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.

C-sortable words as green mutation sequences · Around

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Then

BibliographyK. Nagao, Donaldson-Thomas theory and cluster algebras, Duke Math. J. 162 (2013), 1313-1367. ( arXiv:1002.4884v2 )
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