Published in Selecta Mathematica
We consider a cluster variety associated to a triangulated surface without punctures. The algebra of regular functions on this cluster variety possesses a canonical vector space basis parametrized by certain measured laminations on the surface. To each lamination, we associate a graded vector space, and we prove that the graded dimension of this ve...
Published in Selecta Mathematica
For fixed positive integers n and k, the Kneser graph KGn,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$KG_{n,k}$$\end{document} has vertices labeled by k-element su...
Published in Forum Mathematicum
For an interval finite quiver Q, we introduce a class of flat representations. We classify the indecomposable projective objects in the category rep ( Q ) {\mathrm{rep}(Q)} of pointwise finite dimensional representations. We show that an object in rep ( Q ) {\mathrm{rep}(Q)} is projective if and only if it is a direct sum of countably generated...
Published in Forum Mathematicum
Inspired by recent work of Carlson, Friedlander and Pevtsova concerning modules for p-elementary abelian groups E r {E_{r}} of rank r over a field of characteristic p > 0 {p>0} , we introduce the notions of modules with constant d-radical rank and modules with constant d-socle rank for the generalized Kronecker algebra 𝒦 r = k Γ r {\mathcal{K}_{r...
Published in Acta Mathematica Sinica, English Series
An algebra A is called derived-unique provided that any algebra which is derived equivalent to A is necessarily Morita equivalent to A. We classify derived-unique gentle algebras with at most one cycle using the combinatorial invariant introduced by Avella-Alaminos and Geiss.
Published in Algebras and Representation Theory
In this paper we study a class of algebras having n-dimensional pyramid shaped quiver with n-cubic cells, which we called n-cubic pyramid algebras. This class of algebras includes the quadratic dual of the basic n-Auslander absolutely n-complete algebras introduced by Iyama. We show that the projective resolutions of the simples of n-cubic pyramid ...
Published in Science China Mathematics
Bongartz (2013) and Ringel (2011) proved that there is no gaps in the sequence of lengths of indecomposable modules for the finite-dimensional algebras over algebraically closed fields. The present paper mainly studies this "no gaps" theorem as to cohomological length for the bounded derived category Db(A) of a gentle algebra A: if there is an inde...
Published in Algebras and Representation Theory
We investigate a general method that allows one to construct new integer sequences extending existing ones. We apply this method to the classic Somos-4 and Somos-5, and the Gale-Robinson sequences, as well as to more general class of sequences introduced by Fordy and Marsh, and produce a great number of new sequences. The method is based on the not...
Published in Japan Journal of Industrial and Applied Mathematics
The theory of persistence modules on the commutative ladders CLn(τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CL_n(\tau )$$\end{document} provides an extension of ...
Published in Mathematische Annalen
Consider a finite-dimensional algebra A and any of its moduli spaces M(A,d)θss\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {M}}}(A,\mathbf {d})^{ss}_{\the...
