On the zero set of semi-invariants for regular modules over tame canonical algebras

驯服箭图与仿射基II：非对称情形

Tame quivers and affine bases II: nonsimply-laced cases

本文用Ringel-Hall代数方法将仿射典范基推广到一般对称化情形，证明所得bar不变基与Lusztig带符号典范基一致，并在$\tilde{B}_n,\tilde{C}_n$型中直接给出典范基。

n-Representation infinite algebras

From the viewpoint of higher dimensional Auslander-Reiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call n-representation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: n-preprojective, n-preinjective and n-regular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext^1-orthogonal families of modules. Moreover we give general constructions of n-representation infi…

Canonical $β$-extensions

We compare the level zero part of the type of a representation of GL(n) over a non-archimedean local field with the tame part of its Langlands parameter restricted to inertia. By normalizing this comparison, we construct canonical $β$-extensions of maximal simple characters.

On regularity in codimension one of irreducible components of module varieties

Let A be a tame quasi-tilted algebra and d the dimension vector of an indecomposable A-module. In the paper we prove that each irreducible component of the variety of A-modules of dimension vector d is regular in codimension one.

Infinite dimensional representations of canonical algebras

The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to the more general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with canonical algebras. The investigation is centered around the generic and the Prüfer modules, and how other modules are determined by these modules.

Hecke algebras for tame supercuspidal types

Let $F$ be a non-archimedean local field of residue characteristic $p\neq 2$. Let $G$ be a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. Yu constructed types which are called tame supercuspidal types and conjectured that Hecke algebras associated with these types are isomorphic to Hecke algebras associated with depth-zero types of some twisted Levi subgroups of $G$. In this paper, we prove this conjecture. We also prove that the Hecke algebra associated with a regular supercuspidal type is isomorphic to the group algebra of a certain abelian group.

tame 余弦代数中原始元素的研究

Primitive elements in Ringel-Hall algebras of tame hereditary algebras

本文研究了tame hereditary代数的Ringel-Hall代数中的原始元素，扩展并改进了Hennecart的结果，并构造了显式基。

Constructing local L-packets for tame unitary groups

We generalize the work of DeBacker and Reeder to the case of unitary groups split by a tame extension. The approach is broadly similar and the restrictions on the parameter the same, but many of the details of the arguments differ. Let $G$ be a unitary group defined over a local field $K$ and splitting over a tame extension $E/K$. Given a Langlands parameter $φ: \mathcal{W}_K \rightarrow {^L G}$ that is tame, discrete and regular, we give a natural construction of an $L$-packet $Π_φ$ associated to $φ$, consisting of representations of pure inner forms of $G(K)$ and parametrized by the characte…

Geometry of regular modules over canonical algebras

We classify canonical algebras such that for every dimension vector of a regular module the corresponding module variety is normal (respectively, a complete intersection). We also prove that for the dimension vectors of regular modules normality is equivalent to irreducibility.

Canonical tilting modules over shod algebras are regular in codimension one

We show that for a class of modules over shod algebras, including the canonical tilting modules, the closures of the corresponding orbits in module varieties are regular in codimension one.

Spectra of Bethe subalgebras of $Y(\mathfrak{gl}_n)$ in tame representations

We study the eigenproblem for Bethe subalgebras of the Yangian $Y(\mathfrak{gl}_n)$ in tame representations, i.e. in finite dimensional representations which admit Gelfand-Tsetlin bases. Namely, we prove that for any tensor product of skew modules $V=\otimes_{i=1}^k V_{λ_i \setminus μ_i}(z_i)$ over the Yangian $Y(\mathfrak{gl}_n)$ with generic $z_i$'s, the family of Bethe subalgebras $B(X)$ with $X$ being a regular element of the maximal torus of $GL_n$ (or, more generally, with $X \in \overline{M_{0,n+2}}$) acts with a cyclic vector on $V$. Moreover, for $X$ in the real form of $\overline{M_{…

The cluster category of a canonical algebra

We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan-Iyama-Reiten-Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of X is non-negative, or equivalently, if A is of tame (domestic or tu…

A parameterization of the canonical bases of affine modified quantized enveloping algebras

For symmetrizable Kac-Moody Lie algebra $\textbf{g}$, Lusztig introduced the modified quantized enveloping algebra $\dot{\textbf{U}}(\textbf{g})$ and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra $\textbf{g}$ we define a set which depend only on the root category and prove that there is a bijection between the set and the canonical basis of $\dot{\textbf{U}}(\textbf{g})$, where the root category is the $T^2$-orbit category of the derived category of Dynkin or tame quiver. Our method bases on one theorem of Lin, Xiao and Zhang in [9], which gave th…

Maximal Cohen-Macaulay modules over non-isolated surface singularities and matrix problems

In this article we develop a new method to deal with maximal Cohen-Macaulay modules over non-isolated surface singularities. In particular, we give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen-Macaulay modules. Next, we prove that the degenerate cusp singularities have tame Cohen-Macaulay representation type. Our approach is illustrated on the case of $k\llbracket x,y,z\rrbracket/(xyz)$ as well as several other rings. This study of maximal Cohen-Macaulay modules over non-isolated singularities leads to a new …

Proof of the tree module property for exceptional representations of tame quivers

This document serves as an arXiv entry point for the appendix to the paper [13] (the ancillary file e6_proof.pdf -- ``Proof of the tree module property for exceptional representations of the quiver $\widetilde{\mathbb{E}}_6$'') and the appendix to the paper [12] (the ancillary file d6_proof.pdf -- ``Proof of the tree module property for exceptional representations of the quiver $\widetilde{\mathbb{D}}_6$''). The ancillary files contain the computer generated part of the proofs of the main results in [13] respectively [12], giving a complete and general list of tree representations correspondin…

Stratifications of derived categories from tilting modules over tame hereditary algebras

In this paper, we consider the endomorphism algebras of infinitely generated tilting modules of the form $R_{\mathcal U}\oplus R_{\mathcal U}/R$ over tame hereditary $k$-algebras $R$ with $k$ an arbitrary field, where $R_{\mathcal{U}}$ is the universal localization of $R$ at an arbitrary set $\mathcal{U}$ of simple regular $R$-modules, and show that the derived module category of $\End_R(R_{\mathcal U}\oplus R_{\mathcal U}/R)$ is a recollement of the derived module category $\D{R}$ of $R$ and the derived module category $\D{{\mathbb A}_{\mathcal{U}}}$ of the adèle ring ${\mathbb A}_{\mathcal{U…

Ringel-Hall numbers and the Gabriel-Roiter submodules of simple homogeneous regular modules

Let k be an arbitrary field and Q an acyclic quiver of tame type. Consider the path algebra kQ and the category of finite dimensional right modules Mod kQ. In the first part of the paper we deduce that the Gabriel-Roiter inclusions in preprojective indecomposables and simple homogeneous regulars (with dimension the minimal radical vector) as well as their Gabriel-Roiter measures are field independent. Using this result we can prove in a more general setting a theorem by Bo Chen which states that the Gabriel-Roiter submodule of a simple homogeneous regular (with dimension the minimal radical ve…

The Gabriel-Roiter measure for $\widetilde{\mathbb{A}}_n$ II

Let $Q$ be a tame quiver of type $\widetilde{\mathbb{A}}_n$ and $\Rep(Q)$ the category of finite dimensional representations over an algebraically closed field. A representation is simply called a module. It will be shown that a regular string module has, up to isomorphism, at most two Gabriel-Roiter submodules. The quivers $Q$ with sink-source orientations will be characterized as those, whose central parts do not contain preinjective modules. It will also be shown that there are only finitely many (central) Gabriel-Roiter measures admitting no direct predecessors. This fact will be generaliz…

The closure of the set of periodic modules over a concealed canonical algebra is regular in codimension one

Let A be a concealed canonical algebra and d the dimension vector of an A-module which is periodic respect to the action of the Auslander-Reiten translation In the paper, we investigate the union of the closures of the orbits of the periodic A-modules of dimension vector d. We show that this set is closed and regular in codimension one.

Stratifying systems over hereditary algebras

This paper deals with stratifying systems over hereditary algebras. In the case of tame hereditary algebras we obtain a bound for the size of the stratifying systems composed only by regular modules and we conclude that stratifying systems can not be complete. For wild hereditary algebras with more than 2 vertices we show that there exists a complete stratifying system whose elements are regular modules. In the other case, we conclude that there are no stratifing system over them with regular modules. In one example we built all the stratifying systems, with a specific form, having maximal num…

半稳定挠类与Grothendieck群中的典范分解

Semistable torsion classes and canonical decompositions in Grothendieck groups

本文研究了推广函子有限挠类的半稳定挠类和态射挠类，建立了TF等价类与典范分解锥的联系，并给出了$\widetilde{\mathbb{A}}$型预投射代数TF等价类的显式描述。

仿射 GLS 代数局部自由表示的一般分类

A generic classification of locally free representations of affine GLS algebras

本文研究仿射 GLS 代数的局部自由表示，通过广义一点扩张构造稳定表示单参数族，并折叠路代数的 Kac 典范分解，建立一般分类并证明其 $E$-驯顺。

来自曲面Fukaya范畴的矩阵分解的典范形式

Canonical form of matrix factorizations from Fukaya category of surface

该研究针对pair-of-pants曲面与奇点$xyz=0$的同调镜像对称，证明高重数带型极大Cohen-Macaulay模对应闭测地线高秩局部系统，给出对应矩阵分解显式典范形式并解释相关代数运算的几何意义。

Depth zero supercuspidal L-packets for inner forms of GSp_4

We show that for any tame regular discrete series parameter of GSp_4 or its inner form GU_2(D), the L-packet attached by the local Langlands conjecture agrees with the L-packet of depth zero supercuspidal representations constructed by DeBacker and Reeder.

Regular irreducible represntations of classical groups over finite quotient rings

A parametrization of irreducible representations associated with a regular adjoint orbit of a classical group over finite quotient rings of the ring of integer of a non-dyadic non-archimedean local field is presented. The parametrization is given by means of (a subset of) the character group of the centralizer of a representative of the regular adjoint orbit. Our method is based upon Weil representations over finite fields. More explicit parametrization in terms of tamely ramified extensions of the base field is given for the general linear group, the special linear group, the symplectic group…