arXiv subjects

Peter Jorgensen

Publications and source records attributed to Peter Jorgensen.

At least 55 records · Page 3Linked to original sources

arXiv2014

Generalized frieze pattern determinants and higher angulations of polygons

Frieze patterns (in the sense of Conway and Coxeter) are in close connection to triangulations of polygons. Broline, Crowe and Isaacs have assigned a symmetric matrix to each polygon triangulation and computed the determinant. In this paper we consider d-angulations of polygons and generalize the combinatorial algorithm for computing the entries in the associated symmetric matrices; we compute their determinants and the Smith normal forms. It turns out that both are independent of the particular d-angulation, the determinant is a power of d-1, and the elementary divisors only take values d-1 and 1. We also show that in the generalized frieze patterns obtained in our setting every adjacent 2x2-determinant is 0 or 1, and we give a combinatorial criterion for when they are 1, which in the case d=3 gives back the Conway-Coxeter condition on frieze patterns.

math.CO↗

arXiv2014

Cluster tilting vs. weak cluster tilting in Dynkin type A infinity

This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of C which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 <= \ell <= d-1, we show a weakly d-cluster tilting subcategory T_{\ell} which has an indecomposable object with precisely \ell mutations. The category C is the algebraic triangulated category generated by a (d+1)-spherical object and can be thought of as a higher cluster category of Dynkin type A infinity.

math.RT↗

arXiv2014

Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object

The (usual) Caldero-Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps "reachable" indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalises the idea that the cluster category is a "categorification" of the cluster algebra. The definition of the Caldero-Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster tilting object in the category. We study a modified version of the Caldero-Chapoton map which only requires the category to have a Serre functor, and only depends on a rigid object in the category. It is well-known that the usual Caldero-Chapoton map gives rise to so-called friezes, for instance Conway-Coxeter friezes. We show that the modified Caldero-Chapoton map gives rise to what we call generalised friezes, and that for cluster categories of Dynkin type A, it recovers the generalised friezes introduced by combinatorial means by Bessenrodt and us.

math.RT↗

arXiv2013

Classification of co-slicings and co-t-structures for the Kronecker algebra

In this paper we introduce the notion of a 'generalised' co-slicing of a triangulated category. This generalises the theory of co-stability conditions in a manner analogous to the way in which Gorodentsev, Kuleshov and Rudakov's t-stabilities generalise Bridgeland's theory of stability conditions. As an application of this notion, we use a complete classification of 'generalised' co-slicings in the bounded derived category of the Kronecker algebra, $D^b(KQ)$, to obtain a classification of co-t-structures in $D^b(Q)$. This is then used to compute the co-stability manifold of $D^b(KQ)$.

math.CT↗

arXiv2013

Ptolemy diagrams and torsion pairs in the cluster categories of Dynkin type D

We give a complete classification of torsion pairs in the cluster category of Dynkin type D_n, via a bijection to new combinatorial objects called Ptolemy diagrams of type D. For the latter we give along the way different combinatorial descriptions. One of these allows us to count the number of torsion pairs in the cluster category of type D_n by providing their generating function explicitly.

math.RT↗

arXiv2013

SL_2-tilings and triangulations of the strip

SL_2-tilings were introduced by Assem, Reutenauer, and Smith in connection with frieses and their applications to cluster algebras. An SL_2-tiling is a bi-infinite matrix of positive integers such that each adjacent 2 x 2-submatrix has determinant 1. We construct a large class of new SL_2-tilings which contains the previously known ones. More precisely, we show that there is a bijection between our class of SL_2-tilings and certain combinatorial objects, namely triangulations of the strip.

math.CO↗

arXiv2012

Torsion pairs in cluster tubes

We give a complete classification of torsion pairs in the cluster categories associated to tubes of finite rank. The classification is in terms of combinatorial 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.

math.RT↗

arXiv2011

Realising higher cluster categories of Dynkin type as stable module categories

We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class A_n, D_n, E_6, E_7 or E_8 are triangulated equivalent to u-cluster categories of the corresponding Dynkin type. The proof relies on the 'Morita' theorem for u-cluster categories by Keller and Reiten, along with the recent computation of Calabi-Yau dimensions of stable module categories by Dugas.

math.RT↗

arXiv2011

The co-stability manifold of a triangulated category

Stability conditions on triangulated categories were introduced by Bridgeland as a 'continuous' generalisation of t-structures. The set of locally-finite stability conditions on a triangulated category is a manifold which has been studied intensively. However, there are mainstream triangulated categories whose stability manifold is the empty set. One example is the compact derived category of the dual numbers over an algebraically closed field. This is one of the motivations in this paper for introducing co-stability conditions as a 'continuous' generalisation of co-t-structures. Our main result is that the set of nice co-stability conditions on a triangulated category is a manifold. In particular, we show that the co-stability manifold of the compact derived category of the dual numbers is the complex numbers.

math.RT↗

arXiv2011

Sparseness of t-structures and negative Calabi-Yau dimension in triangulated categories generated by a spherical object

Let k be an algebraically closed field and let T be the k-linear algebraic triangulated category generated by a w-spherical object for an integer w. For certain values of w this category is classical. For instance, if w = 0 then it is the compact derived category of the dual numbers over k. As main results of the paper we show that for w \leq 0, the category T has no non-trivial t-structures, but does have one family of non-trivial co-t-structures, whereas for w \geq 1 the opposite statement holds. Moreover, without any claim to originality, we observe that for w \leq -1, the category T is a candidate to have negative Calabi-Yau dimension since Σ^w is the unique power of the suspension functor which is a Serre functor.

math.RT↗

arXiv2011

Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A_n

We give a complete classification of torsion pairs in the cluster category of Dynkin type A_n. Along the way we give a new combinatorial description of Ptolemy diagrams, an infinite version of which was introduced by Ng. This allows us to count the number of torsion pairs in the cluster category of type A_n. We also count torsion pairs up to Auslander-Reiten translation.

math.RT↗

arXiv2010

Duality for cochain DG algebras

This paper develops a duality theory for connected cochain DG algebras, with particular emphasis on the non-commutative aspects. One of the main items is a dualizing DG module which induces a duality between the derived categories of DG left-modules and DG right-modules with finitely generated cohomology. As an application, it is proved that if the canonical module $A / A^{\geq 1}$ has a semi-free resolution where the cohomological degree of the generators is bounded above, then the same is true for each DG module with finitely generated cohomology.

math.RA↗

arXiv2010

A Caldero-Chapoton map for infinite clusters

We construct a Caldero-Chapoton map on a triangulated category with a cluster tilting subcategory which may have infinitely many indecomposable objects. The map is not necessarily defined on all objects of the triangulated category, but we show that it is a (weak) cluster map in the sense of Buan-Iyama-Reiten-Scott. As a corollary, it induces a surjection from the set of exceptional objects which can be reached from the cluster tilting subcategory to the set of cluster variables of an associated cluster algebra. Along the way, we study the interaction between Calabi-Yau reduction, cluster structures, and the Caldero-Chapoton map. We apply our results to the cluster category D of Dynkin type A infinity which has a rich supply of cluster tilting subcategories with infinitely many indecomposable objects. We show an example of a cluster map which cannot be extended to all of D. The case of D also permits us to illuminate results by Assem-Reutenauer-Smith on SL_2-tilings of the plane.

math.RT↗

arXiv2010

Cluster categories, selfinjective algebras, and stable Calabi-Yau dimensions: type A

The preprints arXiv:math/0610728 and arXiv:math/0612451 are withdrawn due to a problem with Theorem 2.2 in arXiv:math/0610728. The theorem claims that for certain triangulated categories with finitely many indecomposable objects, the Calabi-Yau dimension can be computed combinatorially, by finding the smallest d for which the Serre functor and the d'th power of the suspension functor have the same action on the Auslander-Reiten quiver. This is false, and we are grateful to Alex Dugas for pointing out a counterexample; see Section 5 of his paper arXiv:math/0808.1311 for more details. Unfortunately, we are not presently able to come up with a corrected version of the theorem, and this means that we cannot compute the Calabi-Yau dimensions of concrete stable module categories. Since these dimensions are necessary for identifying the categories with higher cluster categories, we presently have no means to achieve such identifications.

math.RT↗

arXiv2010

Cluster categories, selfinjective algebras, and stable Calabi-Yau dimensions: types D and E

math.RT↗

arXiv2010

Symmetric Auslander and Bass categories

We define the symmetric Auslander category A^s(R) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right tails of totally acyclic complexes of projective modules. The symmetric Auslander category contains A(R), the ordinary Auslander category. It is well known that A(R) is intimately related to Gorenstein projective modules, and our main result is that A^s(R) is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories: \underline{GMor}(R) --> A^s(R) / K^b(Prj R). Here \underline{GMor}(R) is the stable category of Gorenstein projective objects in the abelian category Mor(R) of homomorphisms of R-modules, and K^b(Prj R) is the homotopy category of bounded complexes of projective R-modules. This result is set in the wider context of a theory for A^s(R) and B^s(R), the symmetric Bass category which is defined dually.

math.AC↗

arXiv2009

Cotorsion pairs induced by duality pairs

We introduce the notion of a duality pair and demonstrate how the left half of such a pair is often covering and preenveloping. As an application, we generalize a result by Enochs et al. on Auslander and Bass classes, and we prove that the class of Gorenstein injective modules introduced by Enochs and Jenda is covering when the ground ring has a dualizing complex.

math.AC↗

arXiv2009

On a triangulated category which behaves like a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon

This paper investigates a certain 2-Calabi-Yau triangulated category D whose Auslander-Reiten quiver is ZA_{\infty}. We show that the cluster tilting subcategories of D form a so-called cluster structure, and we classify these subcategories in terms of what one may call `triangulations of the infinity-gon'. This is reminiscent of the cluster category C of type A_n which is a 2-Calabi-Yau triangulated category whose Auslander-Reiten quiver is a quotient of ZA_n. The cluster tilting subcategories of C form a cluster structure and they are classified in terms of triangulations of the (n+3)-gon. The category D behaves like a `cluster category of type A_{\infty}'.

math.RT↗