C-sortable words as green mutation sequences

Yu Qiu Institutt for Matematiske Fag, NTNU, 7491 Trondheim, Norway

Combinatorics mathscidoc:2206.06001

Proceedings of the London Mathematical Society, 111, (5), 1052-1070, 2015.11
Let Q be an acyclic quiver and 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 D(Q), starting from the standard heart H_Q = modkQ and ending at another heart H_s in D(Q). Then we show that s is a green mutation sequence if and only if every heart in this simple tilting sequence is greater than or equal to HQ[−1]; it is maximal if and only if H_s = 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{w} induced by a c-sortable word w is a green mutation sequence. As a consequence, we obtain a bijection between c-sortable words and finite torsion classes in H_Q. As byproducts, the interpretations of inversions, descents and cover reflections of a c-sortable word w are given in terms of the combinatorics of green mutations.
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  title={C-sortable words as green mutation sequences},
  author={Yu Qiu},
  booktitle={Proceedings of the London Mathematical Society},
Yu Qiu. C-sortable words as green mutation sequences. 2015. Vol. 111. In Proceedings of the London Mathematical Society. pp.1052-1070. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626164524096205464.
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