Related papers: Acyclic cluster algebras, reflection groups, and c…
Le-diagrams are combinatorial objects that parametrize cells of the totally nonnegative Grassmannian, called positroid cells, and each Le-diagram gives rise to a cluster algebra which is believed to be isomorphic to the coordinate ring of…
We study alternating strand diagrams on the disk with an orbifold point. These are quotients by rotation of Postnikov diagrams on the disk, and we call them orbifold diagrams. We associate a quiver with potential to each orbifold diagram,…
We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…
The deep locus of a cluster variety is defined to be the set of its points that do not belong to any cluster torus. We show that, if the cluster variety has a seed whose mutable part is a tree without multiple edges, then the deep locus can…
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…
We introduce a notion of {\em cyclic Schur-positivity} for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux…
We consider frieze sequences corresponding to sequences of cluster mutations for affine D and E type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient…
Haile, Han and Kuo have studied certain non-commutative algebras associated to a binary quartic or ternary cubic form. We extend their construction to pairs of quadratic forms in four variables, and conjecture a further generalisation to…
We study cluster algebras that are associated to unpunctured surfaces with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…
We give a bijection between ordered $m$-clusters and (complete) $m$-exceptional sequences, a concept that we introduce for this purpose. This holds for all hereditary artin algebras. This extends the bijection in the $m = 1$ case shown in…
In this paper we study division algebras over the function fields of curves over $\Q_p$. The first and main tool is to view these fields as function fields over nonsingular $S$ which are projective of relative dimension 1 over the $p$ adic…
We study non-commutative projective lines over not necessarily algebraic bimodules. In particular, we give a complete description of their categories of coherent sheaves and show they are derived equivalent to certain bimodule species. This…
In this paper we study the combinatorial consequences of the relationship between rational Cherednik algebras of type $G(l,1,n)$, cyclic quiver varieties and Hilbert schemes. We classify and explicitly construct $\mathbb{C}^*$-fixed points…
We show that a proper algebraic n-dimensional scheme Y admits nontrivial vector bundles of rank n, even if Y is non-projective, provided that there is a modification containing a projective Cartier divisor that intersects the exceptional…
Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M. In particular, we obtain some sufficient conditions for smoothness, polynomial…
We start with observing that the only connected finite dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the radical…
This article discusses deviations from the Cohen-Lenstra heuristics when roots of unity are present. In particular, we propose an explanation for the discrepancy between the observed number of cyclic cubic fields whose 2-class group is…
It was recently shown by Gross, Hacking, and Keel that, in the absence of frozen indices, a cluster A-variety with generic coefficients is the universal torsor of the corresponding cluster X-variety with corresponding coefficients. We…
After recalling several constructions of the moduli space of curves of genus zero by different people we give our alternative construction of the moduli space. This gives a simple description of the intersection ring of this space. We give…
The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use…