On regularity in codimension one of irreducible components of module varieties

Orbit closures of directing modules are regular in codimension one

We show that the orbit closure of a directing module is regular in codimension one. In particular, this result gives information about a distinguished irreducible component of a module variety.

Irreducible components of affine Deligne-Lusztig varieties

By extending the method of semi-modules developed by de Jong, Oort, Viehmann and Hamacher, we introduce a stratification for the affne Deligne-Lusztig variety (in the affne Grassmannian) attached to attached to a minuscule cocharacter and a basic element. As an application, we complete the proof of a conjecture on the irreducible components of affne Deligne-Lusztig varieties, due to Miaofen Chen and Xinwen Zhu.

Irregular and singular loci of commuting varieties

We prove that the singular locus of the commuting variety of a noncommutative reductive Lie algebra is contained in the irregular locus and we compute the codimension of the latter. We prove that one of the irreducible components of the irregular locus has codimension 4. This yields the lower bound of the codimension of the singular locus, in particular, implies that it is at least 2. We also prove that the commuting variety is rational.

Irreducible components of two-row Springer fibers and Nakajima quiver varieties

We give an explicit description of the irreducible components of two-row Springer fibers in type A as closed subvarieties in certain Nakajima quiver varieties in terms of quiver representations. By taking invariants under a variety automorphism, we obtain an explicit algebraic description of the irreducible components of two-row Springer fibers of classical type. As a consequence, we discover relations on isotropic flags that describe the irreducible components.

Algebras with irreducible module varieties III: Birkhoff varieties

We study a family of affine varieties arising from a version of an old problem due to Birkhoff asking for the classification of embeddings of finite abelian p-groups. We show that all of these varieties are irreducible and have a dense orbit.

Irreducible components of module varieties: projective equations and rationality

We expand the existing arsenal of methods for exploring the irreducible components of the varieties $Rep(A,\bold d)$ which parametrize the representations with dimension vector $\bold d$ of a finite dimensional algebra $A$. To do so, we move back and forth between $Rep(A,\bold d)$ and a projective variety, $GRASS(A,\bold d)$, parametrizing the same set of isomorphism classes of modules. In particular, we show the irreducible components to be accessible in a highly compressed format within the projective setting. Our results include necessary and sufficient conditions for unirationality, smooth…

Codimension one connectedness of the graph of associated varieties

Let $ π$ be an irreducible Harish-Chandra $ (\mathfrak{g}, K) $-module, and denote its associated variety by $ AV(π) $. If $ AV(π) $ is reducible, then each irreducible component must contain codimension one boundary component. Thus we are interested in the codimension one adjacency of nilpotent orbits for a symmetric pair $ (G, K) $. We define the notion of orbit graph and associated graph for $ π$, and study its structure for classical symmetric pairs; number of vertices, edges, connected components, etc. As a result, we prove that the orbit graph is connecetd for even nilpotent orbits. Fina…

Minimum codimension of eigenspaces in irreducible representations of simple linear algebraic groups

Let $k$ be an algebraically closed field of characteristic $p \geq 0$, let $G$ be a simple simply connected classical linear algebraic group of rank $\ell$ and let $T$ be a maximal torus in $G$ with rational character group $X(T)$. For a nonzero $p$-restricted dominant weight $λ\in X(T)$, let $V$ be the associated irreducible $kG$-module. Define $ν_{G}(V)$ to be the minimum codimension of eigenspaces corresponding to non-central elements of $G$ on $V$. In this paper, we calculate $ν_{G}(V)$ for $G$ of type $A_{\ell}$, $\ell \geq 16$, and $dim(V) \leq \frac{\ell^{3}}{2}$; for $G$ of type $B_{\e…

Algebras with irreducible module varieties I

We call a finite-dimensional K-algebra A geometrically irreducible if for all d, all connected components of the affine scheme of d-dimensional A-modules are irreducible. We show that the geometrically irreducible algebras without loops (this includes all algebras of finite global dimension) are the hereditary algebras. We also show that truncated polynomial rings are the only geometrically irreducible local algebras. Finally, we formulate a conjectural classification of all geometrically irreducible algebras.

Algebras with irreducible module varieties II: Two vertex case

We call a finite-dimensional K-algebra A geometrically irreducible if for all d all connected components of the affine scheme of d-dimensional A-modules are irreducible. We prove that a geometrically irreducible algebra with exactly two simple modules has to be of a very special form, which we describe. Based on this result we prove that every minimal geometrically irreducible algebra without shortcuts in its Gabriel quiver has at most two simple modules.

一类有限维代数表示簇的不可约分量

Irreducible components of varieties of representations of a class of finite dimensional algebras

针对一类三次截断且带额外关系的非交换多项式代数，本文通过证明固定根分层或底座分层子簇的不可约性并比较两类覆盖，确定了其表示簇的不可约分量。

Twisted orbital integrals and irreducible components of affine Deligne-Lusztig varieties

We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and $q$-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the $σ$-centralizer group to the Mirkovic-Vilonen basis of a certain weight space of a representation of the Langlands dual group.

A binary operation on irreducible components of Lusztig's nilpotent varieties {I}: definition and properties

We define a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. We study commutativity, cancellativity and associativity of this operation. We focus on rigid irreducible components and discuss inductive ways to construct them.

Parametrizations of canonical bases and irreducible components of nilpotent varieties

It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis for quantum groups. For each reduced expression of a Weyl group element, Geiß, Leclerc and Schröer has recently given a parametrization of irreducible components of nilpotent varieties in studying cluster algebras. In this paper we show that their parametrization coincides with Lusztig's parametrization of the canonical basis.

Cycles on Shimura varieties via geometric Satake

We construct (cohomological) correspondences between mod $p$ fibers of different Shimura varieties and describe the fibers of these correspondences by studying irreducible components of affine Deligne-Lusztig varieties. In particular, in the case of correspondences from a Shimura set to a Shimura variety, we obtain a description of the basic Newton stratum of the latter, and show that the irreducible components of the basic Newton stratum generate all the Tate classes in the middle cohomology of the Shimura variety, under a certain genericity condition. Along the way, we also determine the set…

On the varieties of representations and characters of a family of one-relator subgroups. Their irreducible components

Let us consider the group $G = < x,y \mid x^m = y^n>$ with $m$ and $n$ nonzero integers. In this paper, we study the variety of epresentations $R(G)$ and the character variety $X(G)$ in $SL(2,\C)$ of the group $G$,obtaining by elementary methods an explicit primary decomposition of the ideal corresponding to $X(G)$ in the coordinates $X=t_x$, $Y=t_y$ and $Z=t_{xy}$. As an easy consequence, a formula for computing the number of irreducible components of $X(G)$ as a function of $m$ and $n$ is given. We provide a combinatorial description of $X(G)$ and we prove that in most cases it is possible t…

普遍τ-正则模品种的不可约分量

Generically $τ$-regular irreducible components of module varieties

本文研究了有限维代数模品种中普遍τ-正则不可约分量的性质，证明了模的τ-正则性等价于其最小投射呈现为最大秩，并分类了三角代数的此类分量，同时讨论了τ-正则与τ-负则分量的对应关系。

A binary operation on irreducible components of Lusztig's nilpotent varieties {II}: applications and conjectures for representations of $GL_n$ over a non-archimedean local field

In the first part of the paper we defined and studied a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. For type $A$ we conjecture, following Geiss and Schröer, that this operation is compatible with taking the socle of parabolic induction of representations of general linear groups over a local non-archimedean field, at least when one of the irreducible components is rigid. We verify this conjecture in special cases.

Bad irreducible subgroups and singular locus for character varieties in $PSL(p,\mathbb{C})$

We describe the centralizer of irreducible representations from a finitely generated group $Γ$ to $PSL(p,\mathbb{C})$ where $p$ is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of representations whose centralizer strictly contains the center of the ambient group) of the irreducible part of the character variety $χ^i(Γ,PSL(p,\mathbb{C}))$. When $Γ$ is a free group of rank $l\geq 2$ or the fundamental group of a closed Riemann surface of genus $g\geq 2$, we give a complete description of this locus and prove that this locus is exactly the set of…