来自三角化曲面的分母向量与维数向量

铺砌上的相交向量及其在温和代数与丛代数中的应用

Intersection vectors over tilings with applications to gentle algebras and cluster algebras

本文证明铺砌上允许弧的多重集在温和条件下由相交向量唯一确定，并据此刻画温和代数中 $\tau$-刚性模维数向量的唯一性，同时建立 $\mathbb{A}\mathbb{B}\mathbb{C}$ 型丛代数的分母猜想。

On stellated spheres and a tightness criterion for combinatorial manifolds

We introduce the $k$-stellated spheres and consider the class ${\cal W}_k(d)$ of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its subclass ${\cal W}^{\ast}_k(d)$ consisting of the $(k+1)$-neighbourly members of ${\cal W}_k(d)$. We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of its Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-v…

Relation between $f$-vectors and $d$-vectors in cluster algebras of finite type or rank 2

We study $f$-vectors, which are the maximal degree vectors of $F$-polynomials in cluster algebra theory. For a cluster algebra is of finite type, we find that positive $f$-vectors correspond with $d$-vectors, which are exponent vectors of denominators of cluster variables. Furthermore, using this correspondence and properties of $d$-vectors, we prove that cluster variables in a cluster are uniquely determined by their $f$-vectors when the cluster algebra is of finite type or rank $2$.

Average Betti numbers of induced subcomplexes in triangulations of manifolds

We study a variation of Bagchi and Datta's $σ$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of $C$. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sph…

A Duality in Buchsbaum rings and triangulated manifolds

Let $Δ$ be a triangulated homology ball whose boundary complex is $\partialΔ$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $Δ$, $\mathbb F[Δ]$, is isomorphic to the Stanley--Reisner module of the pair $(Δ, \partialΔ)$, $\mathbb F[Δ,\partial Δ]$. This result implies that an Artinian reduction of $\mathbb F[Δ,\partial Δ]$ is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of $\mathbb F[Δ]$. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with …

Cluster algebras and triangulated orbifolds

We construct geometric realization for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on certain hyperbolic orbifolds. We also compute growth rate of these cluster algebras, provide positivity of Laurent expansions of cluster variables, and prove sign-coherence of c-vectors.

Newton polytopes of rank 3 cluster variables

We characterize the cluster variables of skew-symmetrizable cluster algebras of rank 3 by their Newton polytopes. The Newton polytope of the cluster variable $z$ is the convex hull of the set of all $\mathbf{p}\in\mathbb{Z}^3$ such that the Laurent monomial ${\bf x}^{\mathbf{p}}$ appears with nonzero coefficient in the Laurent expansion of $z$ in the cluster ${\bf x}$. We give an explicit construction of the Newton polytope in terms of the exchange matrix and the denominator vector of the cluster variable. Along the way, we give a new proof of the fact that denominator vectors of non-initial c…

一个仿射几乎正根模型

An affine almost positive roots model

本文将有限型簇代数的几乎正根模型推广到统一的有限/仿射型模型，定义了几乎正Schur根Φ_c和兼容度，给出了新的兼容度公式，并建立了簇扩张与g-向量扇的仿射同构。

Diagrammatic description of c-vectors and d-vectors of cluster algebras of finite type

We provide an explicit Dynkin diagrammatic description of the c-vectors and the d-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface realization of cluster algebras for types A_n and D_n, then we apply the folding method to D_{n+1} and A_{2n-1} to obtain types B_n and C_n. Exceptional types are done by direct inspection with the help of a computer algebra software. We also propose a conjecture on the root property of c-vectors for a general cluster algebra.

A categorification of Morelli's theorem

We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli's description of the K-theory of a smooth projective toric variety. Specifically, let $X$ be a proper toric variety of dimension $n$ and let $M_\bR = \mathrm{Lie}(T_\bR^\vee)\cong \bR^n$ be the Lie algebra of the compact dual (real) torus $T_\bR^\vee\cong U(1)^n$. Then there is a corresponding conical Lagrangian $Λ\subset T^*M_\bR$ and an equivalence of triangulated dg categories $\Perf_T(X) \cong \Sh_{…

离散 Laplace 算子——球面与双曲情形

Discrete Laplacians -- spherical and hyperbolic

本文定义球面与双曲三角曲面上的离散 Laplace 算子，保持 Delaunay 非负权、面积表示、共形特征函数及无穷小等距变形等几何结构。

Dimensions of multi-fan algebras

Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. This construction includes many important examples, such as cohomology of toric varieties and quasitoric manifolds, and Gorenstein algebras of triangulated homology manifolds, introduced by Novik and Swartz. In all these examples the dimensions of graded components of such duality algebras do not depend on the vector coloring. It was conjectured…

The Poincaré series of the algebra of rational functions which are regular outside hyperplanes

Let $Δ$ be a finite set of nonzero linear forms in several variables with coefficients in a field $\mathbf K$ of characteristic zero. Consider the $\mathbf K$-algebra $R(Δ)$ of rational functions on V which are regular outside $\bigcup_{α\inΔ} \kerα$. Then the ring $R(Δ)$ is naturally doubly filtered by the degrees of denominators and of numerators. In this paper we give an explicit combinatorial formula for the Poincaré series in two variables of the associated bigraded vector space $\bar{R}(Δ)$. This generalizes the main theorem of Terao, H.: Algebras generated by reciprocals of linear forms…