Schur Roots Research Articles

A proof of Lee-Lee's conjecture about geometry of rigid modules

This paper proves Lee-Lee's conjecture that establishes a coincidence between the set of associated roots of non-self-intersecting curves in an n-punctured disc and the set of real Schur roots of acyclic (valued) quivers with n vertices.

C-Vectors and Non-Self-Crossing Curves for Acyclic Quivers of Finite Type

Let $Q$ be an acyclic quiver and $k$ be an algebraically closed field. The indecomposable exceptional modules of the path algebra $kQ$ have been widely studied. The real Schur roots of the root system associated to $Q$ are the dimension vectors of the indecomposable exceptional modules. It has been shown in [N\'ajera Ch\'avez A., Int. Math. Res. Not. 2015 (2015), 1590-1600] that for acyclic quivers, the set of positive $c$-vectors and the set of real Schur roots coincide. To give a diagrammatic description of $c$-vectors, K-H. Lee and K. Lee conjectured that for acyclic quivers, the set of $c$-vectors and the set of roots corresponding to non-self-crossing admissible curves are equivalent as sets [Exp. Math., to appear, arXiv:1703.09113]. In [Adv. Math. 340 (2018), 855-882], A. Felikson and P. Tumarkin proved this conjecture for 2-complete quivers. In this paper, we prove a revised version of Lee-Lee conjecture for acyclic quivers of type $A$, $D$, and $E_{6}$ and $E_7$.

Picture groups and maximal green sequences

<p style='text-indent:20px;'>We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite "vertically and laterally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type).</p><p style='text-indent:20px;'>Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.</p>

Essential Dimension and Genericity for Quiver Representations

We study the essential dimension of representations of a fixed quiver with given dimension vector. We also consider the question of when the genericity property holds, i.e., when essential dimension and generic essential dimension agree. We classify the quivers satisfying the genericity property for every dimension vector and show that for every wild quiver the genericity property holds for infinitely many of its Schur roots. We also construct a large class of examples, where the genericity property fails. Our results are particularly detailed in the case of Kronecker quivers.

Rigid modules and Schur roots

Let $C$ be a symmetrizable generalized Cartan matrix with symmetrizer $D$ and orientation $\Omega$. In previous work we associated an algebra $H$ to this data, such that the locally free $H$-modules behave in many aspects like representations of a hereditary algebra $\tilde{H}$ of the corresponding type. We define a Noetherian algebra $\hat{H}$ over a power series ring, which provides a direct link between the representation theory of $H$ and of $\tilde{H}$. We define and study a reduction and a localization functor relating the module categories of these three types of algebras. These are used to show that there are natural bijections between the sets of isoclasses of tilting modules over $H$, $\hat{H}$ and $\tilde{H}$. We show that the indecomposable rigid locally free modules over $H$ and $\hat{H}$ are parametrized, via their rank vector, by the real Schur roots associated to $(C,\Omega)$. Moreover, the left finite bricks of $H$, in the sense of Asai, are parametrized, via their dimension vector, by the real Schur roots associated to the dual datum $(C^T,\Omega)$.

Acyclic cluster algebras, reflection groups, and curves on a punctured disc

We establish a bijective correspondence between certain non-self-intersecting curves in an n-punctured disc and positive c-vectors of acyclic cluster algebras whose quivers have multiple arrows between every pair of vertices. As a corollary, we obtain a proof of Lee–Lee conjecture [15] on the combinatorial description of real Schur roots for acyclic quivers with multiple arrows, and give a combinatorial characterization of seeds in terms of curves in an n-punctured disc.

ISOTROPIC SCHUR ROOTS

In this paper, we study the isotropic Schur roots of an acyclic quiver Q with n vertices. We study the perpendicular category $$ \mathcal{A}(d) $$ of a dimension vector d and give a complete description of it when d is an isotropic Schur δ. This is done by using exceptional sequences and by defining a subcategory ℛ(Q, δ) attached to the pair (Q, δ). The latter category is always equivalent to the category of representations of a connected acyclic quiver Q ℛ of tame type, having a unique isotropic Schur root, say δ ℛ. The understanding of the simple objects in $$ \mathcal{A}\left(\delta \right) $$ allows us to get a finite set of generators for the ring of semiinvariants SI(Q, δ) of Q of dimension vector δ. The relations among these generators come from the representation theory of the category ℛ(Q, δ) and from a beautiful description of the cone of dimension vectors of $$ \mathcal{A}\left(\delta \right) $$ . Indeed, we show that SI(Q, δ) is isomorphic to the ring of semi-invariants SI(Q ℛ, δ ℛ) to which we adjoin variables. In particular, using a result of Skowronski and Weyman, the ring SI(Q, δ) is a polynomial ring or a hypersurface. Finally, we provide an algorithm for finding all isotropic Schur roots of Q. This is done by an action of the braid group B n − 1 on some exceptional sequences. This action admits finitely many orbits, each such orbit corresponding to an isotropic Schur root of a tame full subquiver of Q.

Accumulation points of real Schur roots

Let k be an algebraically closed field and Q be an acyclic quiver with n vertices. Consider the category rep(Q) of finite dimensional representations of Q over k. The exceptional representations of Q, that is, the indecomposable objects of rep(Q) without self-extensions, correspond to the so-called real Schur roots of the usual root system attached to Q. These roots are special elements of the Grothendieck group Zn of rep(Q). When we identify the dimension vectors of the representations (that is, the non-negative vectors of Zn) up to positive multiple, we see that the real Schur roots can accumulate in some directions of Rn⊃Zn. This paper is devoted to the study of these accumulation points. After giving new properties of the canonical decomposition of dimension vectors, we show how to use this decomposition to describe the rational accumulation points. Finally, we study the irrational accumulation points and we give a complete description of them in case Q is of weakly hyperbolic type.

On the c-Vectors of an Acyclic Cluster Algebra

We prove that the set of c-vectors of the cluster algebra associated to an acyclic quiver Q coincides with the set of real Schur roots and their opposites in the root system associated to Q.

Tubular cluster algebras I: categorification

We present a categorification of four mutation finite cluster algebras by the cluster category of the category of coherent sheaves over a weighted projective line of tubular weight type. Each of these cluster algebras which we call tubular is associated to an elliptic root system. We show that via a cluster character the cluster variables are in bijection with the positive real Schur roots associated to the weighted projective line. In one of the four cases this is achieved by the approach to cluster algebras of Fomin–Shapiro–Thurston using a 2-sphere with 4 marked points whereas in the remaining cases it is done by the approach of Geiss–Leclerc–Schroer using preprojective algebras.

Cluster fans, stability conditions, and domains of semi-invariants

We show that the cone of finite stability conditions of a quiver Q without oriented cycles has a fan covering given by (the dual of) the cluster fan of Q. Along the way, we give new proofs of Schofield's results (1991) on perpendicular categories. From our results, we recover Igusa-Orr-Todorov-Weyman's theorem on cluster complexes and domains of semi-invariants for Dynkin quivers. For arbitrary quivers, we also give a description of the domains of semi-invariants labeled by real Schur roots in terms of quiver exceptional sets.

Linear free divisors and the global logarithmic comparison theorem

A complex hypersurface D in C-n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. By analogy with Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of C-n D. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields is reductive. For n at most 4, we show that the GLCT holds for all LFDs. We show that LFDs arising naturally as discriminants in quiver representation spaces (of real Schur roots) fulfill the GLCT. As a by-product we obtain a topological proof of a theorem of V. Kac on the number of irreducible components of such discriminants.

Generalized cluster complexes via quiver representations

We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Using d-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d-compatibility degree (���) on any pair of almost positive real Schur roots which generalizes previous definitions on the noncolored case and call two such roots compatible, provided that their d-compatibility degree is zero. Associated to the root system � corresponding to the valued quiver, using this compatibility relation, we define a simplicial complex which has colored almost positive real Schur roots as vertices and d-compatible subsets as simplices. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is the generalized cluster complex defined by Fomin and Reading.

THE COMPOSITION ALGEBRA AND THE COMPOSITION MONOID OF THE KRONECKER QUIVER

The Kronecker quiver is considered, and the relations for the specialisation at of the generic composition algebra are given, as well as those for Reineke's composition monoid. As a corollary, it is deduced that the composition monoid is a proper factor of the specialisation of the composition algebra. A normal form is also obtained for the varieties occurring in the composition monoid in terms of Schur roots.

On homomorphisms from a fixed representation to a general representation of a quiver

We study the dimension of the space of homomorphisms from a given representation X of a quiver to a general representation of dimension vector 3. We prove a theorem about this number, and derive two corollaries concerning its asymptotic behaviour as 3 increases. These results are related to work of A. Schofield on homological epimorphisms from the path algebra to a simple artinian ring. In this paper we prove a number of results about hom(X, 3), the dimension of the space of homomorphisms from a fixed representation X of a quiver to a general representation of dimension vector 3. Our basic result relates this number to the dimension of a subset of a Grassmannian of submodules of X. This result is in the spirit of A. Schofield's paper on general representations of quivers [S2], and generalizes one of his results. We derive two corollaries concerning the asymptotic behaviour of hom(X, r/3), of interest in themselves, and also because of their connection with another theorem of Schofield, that the homological epimorphisms from a path algebra to a simple artinian ring are in 1-1 correspondence with indivisible Schur roots. (A homological epimorphism is a ring homomorphism R -* S with S OR S -S and Tori(S, S) = 0 for i > 0. This correspondence was announced by Schofield in a lecture in March 1995 in Krippen, Germany, but actually dates from 1991.) In fact our second corollary is already known, proved by Schofield and used in the proof of the correspondence, but his proof of the corollary involves difficult results about semi-invariants of quivers, whereas the proof here is quite elementary. Let K be an algebraically closed field, Q a finite quiver with vertex set I, and let (-,-) be the Ringel form on Z'. If 3 E N' we write RePKQW() = 17 Hom(K8('), K:(j)) a:i-j for the configuration space of representations of Q of dimension vector 3, and if y E RePKQ(/) we write Ky for the corresponding left KQ-module. If X is a finitely generated left KQ-module and y E RePKQ (/3) then Hom(X, Ky) is a finite dimensional vector space. Moreover the function y F dim Hom(X, Ky) is an upper semicontinuous function on RePKQ(/3), and it follows that the minimum value of this function, denoted hom(X,/3), is also its general value. The rank of a homomorphism X -* Ky is the dimension vector of its image. For any Ol E N' the set of homomorphisms of rank at most Ol is a closed subset of Hom(X, Ky). It follows that there is a unique maximal rank Yx,y of homomorphisms from X to Ky, Received by the editors July 21, 1995. 1991 Mathematics Subject Classification. Primary 16G20; Secondary 14M15. ?1996 American Mathematical Society