Christof Geiss, Bernard Leclerc, Jan Schröer
Let L be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories sub(Q) for Q an injective L-module, and we introduce a mutation operation between complete rigid modules in sub(Q). This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.
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Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are homeomorphic to the lagrangian quiver varieties of Nakajima which have been well studied in the context of geometric representation theory. We then refine this result by finding quiver grassm…
We describe a categorification of the cluster algebra structure of multi-homogeneous coordinate rings of partial flag varieties of arbitrary Dynkin type using Cohen-Macaulay modules over orders. This completes the categorification of Geiss-Leclerc-Schröer by adding the missing coefficients. To achieve this, for an order $A$ and an idempotent $e \in A$, we introduce a subcategory $\operatorname{CM}\nolimits_e A$ of $\operatorname{CM}\nolimits A$ and study its properties. In particular, under some mild assumptions, we construct an equivalence of exact categories $(\operatorname{CM}\nolimits_e A)…
We apply tilting theory over preprojective algebras $Lambda$ to a study of moduli space of $Lambda$-modules. We define the categories of semistable modules and give an equivalence, so-called reflection functors, between them by using tilting modules over $Lambda$. Moreover we prove that the equivalence induces an isomorphism of algebraic varieties between moduli spaces. In particular, we study in the case when the moduli spaces related to the Kleinian singularity. We generalize a result of Crawley-Boevey which is known another proof of the McKay correspondence of Ito-Nakamura type.
We give a geometric construction of the Verma modules of a symmetric Kac-Moody Lie algebra in terms of constructible functions on the varieties of nilpotent finite-dimensional modules of the corresponding preprojective algebra.
We study preprojective algebras of graphs and their relationship to module categories over representations of quantum SL(2). As an application, ADE quiver varieties of Nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by Lusztig.
The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig's nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver.
We introduce for each quiver $Q$ and each algebraic oriented cohomology theory $A$, the cohomological Hall algebra (CoHA) of $Q$, as the $A$-homology of the moduli of representations of the preprojective algebra of $Q$. This generalizes the $K$-theoretic Hall algebra of commuting varieties defined by Schiffmann-Vasserot. When $A$ is the Morava $K$-theory, we show evidence that this algebra is a candidate for Lusztig's reformulated conjecture on modular representations of algebraic groups. We construct an action of the preprojective CoHA on the $A$-homology of Nakajima quiver varieties. We comp…
We study the multiplicative properties of the dual of Lusztig's semicanonical basis.The elements of this basis are naturally indexed by theirreducible components of Lusztig's nilpotent varieties, whichcan be interpreted as varieties of modules over preprojective algebras.We prove that the product of two dual semicanonical basis vectorsis again a dual semicanonical basis vector provided the closure ofthe direct sum of thecorresponding two irreducible components is again an irreducible component.It follows that the semicanonical basis and the canonical basiscoincide if and only if we are in Dynk…
Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A.
We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a "twisted case"; a consequence of our results is that our construction also includes the flag varieties for Kac-Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t…
We introduce the notion of flag Bott-Samelson variety as a generalization of Bott-Samelson variety and flag variety. Using a birational morphism from an appropriate Bott-Samelson variety to a flag Bott-Samelson variety, we compute Newton-Okounkov bodies of flag Bott-Samelson varieties as generalized string polytopes, which are applied to give polyhedral expressions for irreducible decompositions of tensor products of $G$-modules. Furthermore, we show that flag Bott-Samelson varieties are degenerated into flag Bott manifolds with higher rank torus actions, and find the Duistermaat-Heckman measu…
We describe the effect of Feigin's flat degeneration of the type $\textrm{A}$ flag variety on its Schubert varieties. In particular, we study when they stay irreducible and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some Schubert varieties with Richardson varieties in higher rank partial flag varieties.
We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank. We deduce that the corresponding degenerate flag varieties are isomorphic to Schubert varieties in any characteristic.
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures concerning their dimensions, and generalizes the superset method.
We study the Jordan-Hoelder series for nearby cycles on certain Shimura varieties and Rapoport-Zink local models, and on finite-dimensional pieces of Beilinson's deformation of the affine Grassmannian to the affine flag variety (and their p-adic analogues). We give a formula for the multiplicities of irreducible constituents in terms of certain cohomology groups, and we also provide an algorithm to compute multiplicities, in terms of the affine Hecke algebra.
We study algebraic, combinatorial and geometric aspects of weighted PBW-type degenerations of (partial) flag varieties in type $A$. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.
For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X.
Recently E. Feigin introduced the $\mathbb G_a^N$-degenerations of semisimple algebraic groups and their associated degenerate flag varieties. It has been shown by Feigin, Finkelberg, and Littelmann that the degenerate flag varieties in types $A_n$ and $C_n$ are Frobenius split. In this paper we construct an associated degeneration of homogeneous coordinate rings of classical flag varieties in all types and show that these rings are Frobenius split in most types. It follows that the degenerate flag varieties of types $A_n$, $C_n$ and $G_2$ are Frobenius split. In particular we obtain an altern…
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of GL_2. At first we determine all such varieties up to isomorphy. After this we investigate the representations of the sigma-stabilizer of an element b of the group on the etale cohomology of the affine Deligne-Lusztig variety X_w(b). We describe such representations as inductions from compact subgroups and in terms of noncuspidal representations.
Let $G$ be a simple algebraic group of type $B_2$ over an algebraically closed field of odd characteristic. We prove that the flag variety $G/B$ is D-affine. This extends an earlier result of H.H.Andersen and M.Kaneda.
Our goal is to classify all generically transitive actions of commutative unipotent groups on flag varieties up to conjugation. We establish relationship between this problem and classification of multiplications with certain properties on Lie algebra representations. Then we classify multiplications with the desired properties and solve the initial classification problem.
Let $\Fl^a_\la$ be the PBW degeneration of the flag varieties of type $A_{n-1}$. These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$. We prove that $\Fl^a_\la$ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\la$ of $\Fl^a_\la$. The varieties $R_\la$ can be viewed as towers of successive $\bP^1$-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties $R_\la$ are Frobenius split. This gives us Frobenius splitting for the…
Tropical symplectic flag varieties: a Lie-theoretic approach
本文用李理论方法研究辛旗簇的热带化，显式刻画了一个极大素锥，并证明其内点对应的格罗布纳退化为FFLV多胞体相关的环面簇，关键工具是双有理序列。
This paper introduces a class of smooth projective varieties that generalise and share many properties with partial flag varieties of type A. The quiver flag variety M_\vartheta(Q,r) of a finite acyclic quiver Q (with a unique source) and a dimension vector r is a fine moduli space of stable representations of Q. Quiver flag varieties are Mori Dream Spaces, they are obtained via a tower of Grassmann bundles, and their bounded derived category of coherent sheaves is generated by a tilting bundle. We define the multigraded linear series of a weakly exceptional sequence of locally free sheaves E …
Multiplicative preprojective algebras are 2-Calabi-Yau
本研究证明含无向圈箭图的乘法预投射代数是2-Calabi-Yau代数，分析了不同箭图情形下的中心与结构性质，证明其dg版本的形式性，还得出乘法箭图簇的局部同构与辛奇点性质，并推测相关结论可推广至所有非Dynkin箭图。
We review the definition of quiver varieties and their relation to representation theory of Kac-Moody Lie algebras. Target readers are ring and representation theorists. We emphasize important roles of first extension groups of the preprojective algebra associated with a quiver.
Positivity in Weighted Flag Varieties
本文从李理论视角研究加权旗流形的环面等变上同调，证明其正性性质并推广了加权格拉斯曼流形及 A 型情形下的 Chevalley 公式、不动点限制与 GKM 描述等结果。
We study modules with 1-dimensional socle for preprojective algebras for type A quivers. In particular, we classify such modules, determine all homomorphisms between them, and then explain how they may be used to describe the components of Lusztig quiver varieties.
We determine the set of supports for the flat family of linear degenerations of flag varieties in terms of Motzkin combinatorics.
On the Classification of Schubert Varieties in Partial Flag Varieties
本文在部分旗流形中推广了Schubert流形的分类，通过根系折叠得到等价对，并确定了Schubert三重流形等价类的上界。
Independence of the Grothendieck classes of twisted symplectic flag varieties
本文利用非交换动机理论，证明扭曲辛旗簇的格罗滕迪克类在线性及齐次二次意义下，在代数簇的格罗滕迪克环中具有独立性。
We show that in type A or C any degenerate flag variety is in fact isomorphic to a Schubert variety in an appropriate partial flag manifold.
In 2016, Leclerc constructed conjectural cluster structures on open Richardson varieties using representations of preprojective algebras. A variant with more explicit seeds was obtained by Ménard in his thesis. We show that Ménard's seeds do yield *upper* cluster algebra structures on open Richardson varieties and discuss the problems that remain in order to prove that they are cluster algebra structures.
Let $n$ be a maximal nilpotent subalgebra of a complex symmetric Kac-Moody Lie algebra. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of nilpotent modules over a preprojective algebra of the same type as $n$. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important role in our work on the relationship between…
Following the historical track in pursuing $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation theory, such as canonical bases, Newton-Okounkov bodies, PBW-filtrations and cluster algebras come to push the subject forward.
Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open $G_{\mathbb{R}}$--orbits in flag varieties $G/P$. We investigate Hodge--theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, we relate the Griffiths--Yukawa coupling to the variety of lines on $G/P$ (under a minimal homogeneous embedding), construct a large class of polarized $G_{\mathbb{R}}$--orbits in $G/P$, and compute the associated Hodge--theoretic boundary components. An emphasis is placed through…
Cohomology of Flag Superschemes and Syzygies of Compositional Varieties
研究部分旗超概形的凝聚上同调，计算四类新无限族旗超概形的结构层上同调，引入复合簇并计算其Tor群，证明旗超概形上同调与普通旗簇上同调环及复合簇Tor群的张量积同构。
We will use the combinatorics of the $G$-stable pieces to describe the closure relation of the partition of partial flag varieties in \cite[section 4]{L3}.
A partial flag variety ${\mathcal {P}}_K$ of a Kac-Moody group $G$ has a natural stratification into projected Richardson varieties. When $G$ is a connected reductive group, a Bruhat atlas for ${\mathcal {P}}_K$ was constructed by He, Knutson and Lu: ${\mathcal {P}}_K$ is locally modeled with Schubert varieties in some Kac-Moody flag variety as stratified spaces. The existence of Bruaht atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties…
Dynkin abelianisations of flag varieties
本文引入Dynkin锥参数化经典李代数的阿贝尔化，将Cerulli Irelli和Lanini关于A、C型旗簇PBW退化为Schubert簇的结果推广到所有源自Dynkin锥的度向量退化情形，并计算了所有经典类型中退化单模的定义关系。
We introduce some new Frobenius subcategories of the module category of a preprojective algebra of Dynkin type, and we show that they have a cluster structure in the sense of Buan-Iyama-Reiten-Scott. These categorical cluster structures yield cluster algebra structures in the coordinate rings of intersections of opposed Schubert cells.
Newton--Okounkov bodies of partial flag varieties via cluster algebras
利用单胞上的簇结构构造部分旗簇中Schubert变体的Newton-Okounkov多面体，证明在无限型簇代数下存在无穷多互不等价的多面体，并应用于构造无穷多不同的单调拉格朗日环面。
The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More precisely, we generalize the construction from our previous paper relating the combinatorics and geometry of quiver Grassmanians to that of the totally nonnegative Grassmannians. The varieties we are interested in serve as realizations of local models of Shimura varieties. We exploit quiver representation techniques to study the quiver Grassmannians of interest and, …
Semiorthogonal decompositions for families of twisted flag varieties
该研究将Samokhin-van der Kallen的半正交分解推广至拟分裂半单群，证明光滑扭曲广义旗簇族的导出范畴可半正交分解，实现了Panin关于扭曲旗簇Quillen K-理论计算的范畴化。
We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by Gross-Hacking-Keel-Kontsevich: for the flag variety the cone is the tropicalization of their superpotential while for Schubert varieties a restriction of the superpotential is necessary. We prove…
We further develop the general theory of the "mixed modular derived category" introduced by the authors in a previous paper in this series. We then use it to study positivity and Q-Koszulity phenomena on flag varieties.
We show that the Borel-equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of dg modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves. This proves a conjecture of Soergel and Lunts in the case of flag varieties.
On the Smooth Locus in Flat Linear Degenerations of Flag Varieties
本文利用矩图技术给出旗簇平坦线性退化中光滑轨迹的组合刻画，推广了Feigin退化的相关结果，并提出用传递竞赛图判定固定点光滑性的新组合判据。
Let $\SF^a_λ$ be the degenerate symplectic flag variety. These are projective singular irreducible $\bG_a^M$ degenerations of the classical flag varieties for symplectic group $Sp_{2n}$. We give an explicit construction for the varieties $\SF^a_λ$ and construct their desingularizations, similar to the Bott-Samelson resolutions in the classical case. We prove that $\SF^a_\la$ are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel-Weil-Bott theo…
We continue, generalize and expand our study of linear degenerations of flag varieties from [G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, Math. Z. 287 (2017), no. 1-2, 615-654]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degener…
Following the program of algebraic Frobenius splitting begun by Kumar and Littelmann, we use representation-theoretic techniques to construct a Frobenius splitting of the cotangent bundle of the flag variety of a semisimple algebraic group over an algebraically closed field of positive characteristic. We also show that this splitting is the same as one of the splittings constructed by Kumar, Lauritzen, and Thomsen.
We study equivariant contact structures on complex projective varieties arising as partial flag varieties $G/P$, where $G$ is a connected, simply-connected complex simple group of type $ADE$ and $P$ is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these types. The result can be deduced from Boothby's classification of compact simply-connected complex contact manifolds with transitive action by contact automorphisms, but our proof is completely independent and relies on properties of $G$-equivariant vector bundles on $G/P$. A byprod…
We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T. Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T-orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schröder number and the Poincaré polynomial is given by a natural statistics counting the number of diagonal ste…
Seidel product formula in equivariant quantum $K$-theory of flag varieties
本文在旗种的等价量子K-理论中证明了Seidel乘积公式，并基于Kato的Peterson同构和Ikeda等人研究的nil-Hecke代数给出了证明。
Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proven that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits, and a cell…
For a connected semisimple algebraic group $G$, we consider some special infinite series of tensor products of simple $G$-modules whose $G$-fixed point spaces are at most one-dimensional. We prove that their existence is closely related to the existence of open $G$-orbits in multiple flag varieties and address the problem of classifying such series.
Let G be a reductive algebraic group over the complex number filed, and K = G^θ be the fixed points of an involutive automorphism θof G so that (G, K) is a symmetric pair. We take parabolic subgroups P and Q of G and K respectively and consider a product of partial flag varieties G/P and K/Q with diagonal K-action. The double flag variety G/P \times K/Q thus obtained is said to be of finite type if there are finitely many K-orbits on it. A triple flag variety G/P^1 \times G/P^2 \times G/P^3 is a special case of our double flag varieties, and there are many interesting works on the triple flag …
We prove a new family of total positivity criteria for partial flag varieties for simply-connected complex algebraic group in the simply laced case.
A short informal survey on the topics listed in the title. For the proceedings of the International Conference on Rings and Algebras X.
For a given flag variety, we characterize the primes $p$ for which there exists a weight $λ$ such that the Hard Lefschetz Theorem holds for multiplication by $λ$ on the cohomology of the flag variety with coefficients in an infinite field of characteristic $p$.
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