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The cyclic sieving phenomenon: a survey
"... The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. It was partly motivated as an extension of the q = −1 phenomenon introduced earlier by Stembridge. We give a survey of the current literature on cyclic sieving. The techniques used to prove that the phenomenon h ..."
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The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. It was partly motivated as an extension of the q = −1 phenomenon introduced earlier by Stembridge. We give a survey of the current literature on cyclic sieving. The techniques used to prove that the phenomenon holds are discussed, including ideas from representation theory such as Springer’s Theorem on regular elements in complex reflection groups.
FROM TRIANGULATED CATEGORIES TO MODULE CATEGORIES VIA LOCALISATION II: CALCULUS OF FRACTIONS
, 2011
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Cotorsion pairs in the cluster category of a marked surface
- Department of Mathematics, University of Leicester, University Road, Leicester
"... Dedicated to Professor Idun Reiten on the occasion of her seventieth birthday. We study extension spaces, cotorsion pairs and their mutations in the cluster category of a marked surface without punctures. Under the one-to-one correspondence between the curves, valued closed curves in the marked surf ..."
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Dedicated to Professor Idun Reiten on the occasion of her seventieth birthday. We study extension spaces, cotorsion pairs and their mutations in the cluster category of a marked surface without punctures. Under the one-to-one correspondence between the curves, valued closed curves in the marked surface and the indecomposable objects in the associated cluster category, we prove that the dimension of extension space of two indecom-posable objects in the cluster categories equals to the intersection number of the correspond-ing curves. By using this result, we prove that there are no non-trivial t−structures in the cluster categories when the surface is connected. Based on this result, we give a classifi-cation of cotorsion pairs in these categories. Moreover we define the notion of paintings of a marked surface without punctures and their rotations. They are a geometric model of cotorsion pairs and of their mutations respectively.
Cyclic sieving for torsion pairs in the cluster category of dynkin type an
"... Abstract. Recently, a combinatorial model for torsion pairs in the cluster category of Dynkin type An was introduced, and used to derive an explicit formula for their number. In this article we determine the number of torsion pairs that are invariant under b-fold application of Auslander-Reiten tran ..."
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Abstract. Recently, a combinatorial model for torsion pairs in the cluster category of Dynkin type An was introduced, and used to derive an explicit formula for their number. In this article we determine the number of torsion pairs that are invariant under b-fold application of Auslander-Reiten translation. It turns out that the set of torsion pairs together with Auslander-Reiten translation, and a natural q-analogue of the formula for the number of all torsion pairs exhibits the cyclic sieving phenomenon.
Torsion pairs in cluster tubes
"... Abstract. We give a complete classification of torsion pairs in the cluster cat-egories associated to tubes of finite rank. The classification is in terms of com-binatorial objects called Ptolemy diagrams which already appeared in our earlier work on torsion pairs in cluster categories of Dynkin typ ..."
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Abstract. We give a complete classification of torsion pairs in the cluster cat-egories associated to tubes of finite rank. The classification is in terms of com-binatorial objects called Ptolemy diagrams which already appeared in our earlier work on torsion pairs in cluster categories of Dynkin type A. As a consequence of our classification we establish closed formulae enumerating the torsion pairs in cluster tubes, and obtain that the torsion pairs in cluster tubes exhibit a cyclic sieving phenomenon. Dedicated to Idun Reiten on the occasion of her 70th birthday. 1.
TORSION PAIRS IN A TRIANGULATED CATEGORY GENERATED BY A SPHERICAL OBJECT
"... Abstract. We extend Ng’s characterisation of torsion pairs in the 2-Calabi-Yau triangulated category generated by a 2-spherical object to the characterisation of tor-sion pairs in the w-Calabi-Yau triangulated category, Tw, generated by a w-spherical object for any w ∈ Z. Inspired by the combinatori ..."
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Abstract. We extend Ng’s characterisation of torsion pairs in the 2-Calabi-Yau triangulated category generated by a 2-spherical object to the characterisation of tor-sion pairs in the w-Calabi-Yau triangulated category, Tw, generated by a w-spherical object for any w ∈ Z. Inspired by the combinatorics of Tw, we also characterise the torsion pairs in a certain w-Calabi-Yau orbit category of the bounded derived cate-gory of the path algebra of Dynkin type A. Contents 1. Torsion pairs, extension closure and functorial finiteness 3 2. Triangulated categories generated by w-spherical objects 4 3. Extensions with indecomposable outer terms in Tw for w 6 = 1 6 4. The combinatorial model and contravariant-finiteness 11
