arXiv subjects
Publications and source records attributed to Emine Yıldırım.
We provide multiple combinatorial expansion formulas - in terms of snake graphs, labelled posets, matrices, and $T$-walks - for elements in generalized cluster algebras associated to arcs on punctured orbifolds and illustrate their equivalence. This work generalizes and unifies existing work on combinatorial expansion formulas from surfaces and unpunctured orbifolds.
We analyse the growth coefficients of infinite frieze patterns arising from cluster algebras using cluster modular groups and cluster categories. For a fixed cluster category of affine type, we prove that the collection of infinite frieze patterns given by both the homogeneous and non-homogeneous stable tubes all have the same growth coefficients. We also derive and verify an explicit formula for the $k$-th growth coefficient, expressed directly in terms of data from homogeneous tubes, or, alternatively, from appropriate elements of the corresponding cluster algebra.
We introduce a deformed squared Markov equation given by $X^2 + Y^2 + Z^2 + (q+q^{-1})(XY+YZ+XZ) = 3(1 + q + q^{-1})XYZ$. Symmetric solutions of this new equation present a remarkable factorization property which allows us to talk about their square roots. These square roots give a natural $q$-deformation of the Markov numbers that has not previously occurred in the literature. We call them mirror Markov numbers. We prove a characterization of mirror Markov numbers and discover a mutation rule, mirror mutation, to generate them all. We also prove a geometric realization of the corresponding mirror mutation on a once-punctured sphere with three orbifold points. Our mirror deformation leads to deformations of Fibonacci and Pell branches for which we give precise formulas. Furthermore, the deformed squared Markov equation specializes to many other very well known generalized Markov equations. We also obtain the super Markov numbers from a specialization of the deformed squared Markov numbers, which we use to prove a conjecture of Musiker.
The homogeneous coordinate ring of the Grassmannian $\rm{Gr}(k,n)$ has a well-known cluster structure. There is a categorification of this cluster structure via a category of modules for a ring $A_{k,n}$ due to Jensen-King-Su, building on work of Geiss-Leclerc-Schröer, in which cluster variables correspond to indecomposable rigid modules. We give a combinatorial description of modules that correspond to rank $2$ cluster variables by using webs. This conjecturally gives the categorification of all rank $2$ cluster variables.
Since their introduction, torsion theories have played a key role in the study of abelian and pointed categories. In representation theory, torsion theories and lattices of torsion classes of mod$ A$, for $A$ a finite-dimensional algebra, have been widely studied. The more recent definition of pretorsion theories, that can be given for any category, has expanded the theory, giving many more instances of ``non-pointed torsion theories'' in unexpected settings. In this work, we introduce and study the lattice $\mathcal{L}_t(A)$ of pretorsion classes of mod$ A$. These lattices are in close connection with the lattices tors$ A$ of torsion classes of mod$ A$. We fully describe the completely join-irreducible elements of $\mathcal{L}_t(A)$. Moreover, we characterise and give a full classification of when $\mathcal{L}_t(A)$ is distributive and further describe when it can be identified with the \emph{distributive closure} of tors$ A$. Finally, we show how the lattices of pretorsion classes, together with their duals, can be used to build pretorsion theories in mod$ A$.
We study the representation category of thread quivers and their quotients. A thread quiver is a quiver in which some arrows have been replaced by totally ordered sets. Pointwise finite-dimensional (pwf) representations of such a thread quiver admit a Krull-Remak-Schmidt-Azumaya decomposition. We show that an indecomposable representation is induced from an indecomposable representation of a quiver obtained from the original quiver by replacing some of its arrows by a finite linear $\mathbb{A}_n$ quiver. We study injective and projective pwf indecomposable representations and we fully classify them when the quiver satisfies a mild condition. We give a characterization of the indecomposable pwf representations of certain categories whose representation theory has similar properties to finite type or tame type. We further construct new hereditary abelian categories, including a Serre subcategory of pwf representations that includes every indecomposable representation.
Let $P$ be a finite poset and $L$ the associated distributive lattice of order ideals of $P$. Let $ρ$ denote the rowmotion bijection of the order ideals of $P$ viewed as a permutation matrix and $C$ the Coxeter matrix for the incidence algebra $kL$ of $L$. Then we show the identity $(ρ^{-1} C)^2=id$, as was originally conjectured by Sam Hopkins. Recently it was noted that the rowmotion bijection is a special case of the much more general grade bijection $R$ that exists for any Auslander regular algebra. This motivates to study the interaction of the grade bijection and the Coxeter matrix for general Auslander regular algebras. For the class of higher Auslander algebras coming from $n$-representation finite algebras we show that $(R^{-1} C)^2=id$ if $n$ is even and $(R^{-1}C+id)^2=0$ when $n$ is odd.
In this article, we study infinite friezes arising from cluster categories of affine type $D$ and determine the growth coefficients for these friezes. We prove that for each affine type $D$, the friezes given by the tubes all have the same growth behavior.
We give two new combinatorial methods for computing cluster expansion formulas for arcs coming from possibly punctured surfaces. The first is by using $T$-walks, an extension of the $T$-path model for unpunctured surfaces to general surfaces. We also introduce a new way of generating $T$-paths. The second method is by using order ideals of labeled posets associated to arcs. We also use the theory of oriented posets to give a quick way to calculate the expressions using $2$ by $2$ matrices. The techniques introduced are applicable to different settings in cluster algebras and beyond.
A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply-laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the $F$-polynomials of the corresponding cluster algebras. In addition, we show that the toric variety associated to the g-vector fan has the property that its nef cone is simplicial.
We use Khovanov and Kuperberg's web growth rules to identify the minimal term in the invariant associated to an $\text{SL}_3$ web diagram, with respect to a particular term order.
For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for Hom($X,τY$). We use this to describe support $τ$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $τ$-tilting modules and functorially finite torsion classes. We give an explicit basis of Ext$^1(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.
Let $\mathcal{X}$ be a class of $R$-modules. In this paper, we investigate \;$\mathcal{X}$-(f.g.)injective ((f.g.)projective) and DG-$\mathcal{X}$-injective (projective) complexes which are generalizations of injective (projective) and DG-injective (projective) complexes. We prove that some known results can be extended to the class of \;$\mathcal{X}$-(f.g.)injective ((f.g.)projective) and DG-$\mathcal{X}$-injective (projective) complexes for this general settings.
In this paper we give a generalization of injective and projective complexes.
Let $R$ be any associative ring with unity and $\mathcal{X}$ be a class of $R$-modules of closed under direct sum (and summands) and with extension closed. We prove that every complex has an $C(\mathcal{X^{*}})$-cover ($C(\mathcal{X^{*}})$-envelope) if every module has an $\mathcal{X}$-cover ($\mathcal{X}$-envelope) where $C(\mathcal{X^{*}})$ is the class of complexes of modules in $\mathcal{X}$ such that it is closed under direct and inverse limit.