We give a classification of (co)torsion pairs in finite 2-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting. These finite 2-Calabi-Yau triangulated categories are divided into two main classes: one denoted by An,t called of type A, and the other denoted by Dn,t called of type D [BPR]. By using the geometric model of torsion pairs in cluster categories of type A, or type D in [HJR1, HJR3], we give a geometric description of torsion pairs in An,t or Dn,t respectively, via defining the periodic Ptolemy diagrams. This allows to count the number of (co)torsion pairs in these categories. Finally, we determine the hearts of (co)torsion pairs in all finite 2-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting via quivers and relations. Classification of torsion pairs (equivalently cotorsion pairs) has been studied by many people recently. Ng gave a classification of torsion pairs in the cluster category of type A ∞ by Ptolemy diagrams of an ∞-gon[Ng]. Holm, Jørgensen and Rubey gave a classification of torsion pairs in the cluster category of type A n via Ptolemy diagrams of a regular (n + 3)-gon [HJR1, Theorem A], they also did the same work for the cluster category of type D n by Ptolemy diagrams of a regular 2n-gon [HJR3, Theorem 1.1] and for cluster tubes [HJR2, Theorem 1.1]. Zhang, Zhou and Zhu gave a classification of torsion pairs in the cluster category of a marked surface [ZZZ, Theorem 4.5]. Zhou and Zhu gave a construction and a classification of torsion pairs in any 2-Calabi-Yau triangulated category with cluster tilting objects[ZZ2, Theorem 4.4]. Cluster categories associated with finite dimensional hereditary algebras [BMRRT] (see also [CCS] for type A) and the stable categories of the preprojective algebras Λ of Dynkin quivers [GLS] have been used for the categorification of cluster algebras. These categories are 2-Calabi-Yau triangulated categories with an important class of objects called cluster-tilting objects, which are the analogues of clusters in cluster algebras. The cluster-tilting objects are closely related to a class of objects called maximal rigid objets. Indeed, cluster-tilting objects are maximal rigid objects, but the converse is not true in general [BIKR, BMV, KZ]. For a 2-Calabi-Yau triangulated category, either all maximal rigid objects are cluster tilting, or none of them are [ZZ1, Theorem 2.6]. Triangulated categories with finitely many indecomposable objects (which we call finite triangulated categories) are a special class of locally finite triangulated categories. By Amiot [A] and Burban-Iyama-Keller-Reiten [BIKR] (see also [BPR]), finite 2-Calabi-Yau triangulated categories with non-zero maximal rigid objects have a classification which depends on wether the maximal rigid objects are cluster tilting or not. Standard finite 2-Calabi-Yau triangulated categories with non-zero maximal rigidobjects which are not cluster tilting are exactly the following orbit categories:, where n ≥ 1 and t > 1;• (Type D) D n...
Let [Formula: see text] be a maximal subgroup of an [Formula: see text]-group [Formula: see text] with odd index and let [Formula: see text] be primitive. Lewis proved in this situation that [Formula: see text] divides [Formula: see text], and Isaacs and Wilde further refined this result by showing that either [Formula: see text] or [Formula: see text]. In this paper, we present an independent and simpler proof for these remarkable results and thereby obtain more detailed information regarding the structure of the group [Formula: see text] and the primitive character [Formula: see text]. In particular, [Formula: see text] is strongly irreducible in the sense of Brauer.
Let [Formula: see text] be a quasi-primitive character with odd degree, and suppose that [Formula: see text] is a [Formula: see text]-solvable group. Wilde associated to [Formula: see text] a unique conjugacy class of subgroups [Formula: see text] satisfying [Formula: see text]. We construct in this situation a sequence of character pairs [Formula: see text], where [Formula: see text] is quasi-primitive and each [Formula: see text] is uniquely determined up to conjugacy in [Formula: see text], such that [Formula: see text] and [Formula: see text]. Furthermore, we have [Formula: see text] for each [Formula: see text], and in particular [Formula: see text]. We also prove that the subgroups [Formula: see text] and [Formula: see text] are conjugate in [Formula: see text], and thus present a new description for Wilde’s result.
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