Maitreyee C. Kulkarni
Buan, Iyama, Reiten and Smith proved that the superpotential of a quiver corresponding to an element of Coxeter group is rigid. In this paper, we extend this result to the Berenstein-Fomin-Zelevinsky quivers corresponding to double Bruhat cells of symmetric Kac-Moody algebras. We also realize these quivers as dimers models on cylinder over corresponding graphs.
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Noncommutative Double Bruhat cells and their factorizations
本文研究了非对易双Bruhat胞。我们的主要结果是通过非交换系数矩阵的拟子式在胞腔中的显式正矩阵分解。
Extending the Bruhat order and the length function from the Weyl group to the Weyl monoid
对于可对称化的Kac-Moody代数，容许表示范畴是半单李代数的有限维表示范畴的类似。通过广义Tannaka-Krein重构与这个范畴和限制性半群范畴相关联的幺半群包含Kac-Moody群作为开稠密单位群，并且具有与约化代数幺半群相似的性质。特别是有Bruhat和Birkhoff分解，外尔群被外尔幺半群，[M 1]取代。我们确定了Bruhat和Birkhoff胞腔的封闭关系，从而给出了Bruhat序从外尔群到外尔么半群的推广.我们证明了Bruhat和Birkhoff胞是不可约的，并且在它们的闭包中是主开的。我们给出了Bruhat和Birkhoff盒的乘积分解。我们定义了与扩展Bruhat阶相容的扩展长度函数。我们展示了一个推广的一些Tits公理孪生BN对。
On intersections of conjugacy classes and Bruhat cells
对于连通复半单李群$G$和$G$的相反Borel子群的固定对$(B, B^-)$，我们确定了$G$中的共轭类$C$与双陪集$BwB^-$的交何时非空，其中$w$在$G$的Weyl群$W$中.这个问题来自泊松几何，我们的答案是在$W$和一个对合$\mc \in W$相关的$C$的Bruhat秩序。研究了元素$\mc$的性质。对于$G = SL(n+1, \Cset)$，我们对每个共轭类$C$明确地描述了$\mc$，对于$w \in W$是对合的情况，我们也给出了$C \cap (BwB)$何时非空的明确答案.
Let $G$ be any connected and simply connected complex semisimple Lie group, equipped with a standard holomorphic multiplicative Poisson structure. We show that the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalized minors on every double Bruhat cell of $G$ are complete in the sense that all the integral curves of their Hamiltonian vector fields are defined on ${\mathbb{C}}$. It follows that all the Kogan-Zelevinsky integrable systems on $G$ have complete Hamiltonian flows, generalizing the result of Gekhtman and Yakimov for the case of $SL(n, {\mathbb{C}})$. We in fact construct…
The number of connected components in the double Bruhat cells for nonsimply-laced groups
我们计算了级数$B_n$和$C_n$以及例外群$F_4$的一般实双Bruhat胞腔中连通分量的个数.
Connected components of real double Bruhat cells
半单群中的双Bruhat胞腔是对应于两个相反的博雷尔子群的两个Bruhat分解中胞腔的交集。他们形成了一个几何框架的研究全积极的半单群，他们也密切相关的辛叶相应的泊松-李群。术语“细胞”可能会引起误解，因为它们的拓扑结构可能非常重要。作为理解这种拓扑结构的第一步，我们列举了真正的双Bruhat细胞的连接组件。这个结果推广了（从简单的情形推广到一般情形），并证明了在与B.Shapiro-M.Shapiro-A.Vainshtein的联合工作中提出的猜想;它也推广了B.Shapiro-M.Shapiro-A.Vainshtein和K. Rietsch早期的工作。
Double Bruhat cells and total positivity
我们研究了半单复李群中相对Bruhat胞的交叉点，以及相关的完全非负簇。
On cluster deep loci in double Bruhat cells
该研究针对A型双Bruhat胞上的约化词Berenstein-Fomin-Zelevinsky丛图谱深轨迹展开，推导其结构定理与递推公式，完成低秩算例并探讨其一般性质。
We apply the theory of fundamental strata of Bremer and Sage to find cohomologically rigid $G$-connections on the projective line, generalising the work of Frenkel and Gross. In this theory, one studies the leading term of a formal connection with respect to the Moy-Prasad filtration associated to a point in the Bruhat-Tits building. If the leading term is regular semisimple with centraliser a (not necessarily split) maximal torus $S$, then we have an $S$-toral connection. In this language, the irregular singularity of the Frenkel-Gross connection gives rise to the homogenous toral connection …
In this short note we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements $w$, $w'$ in the Weyl group $W(\mathfrak g)$, the corresponding real Bruhat cell $X_w$ intersects with the dual Bruhat cell $Y_{w'}$ iff $w\prec w'$ in the Bruhat order on $W(\mathfrak g)$. Here $\mathfrak g$ is a normal real form of a semisimple complex Lie algebra $\mathfrak g_\mathbb C$. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations…
Double Bruhat Cells in Kac-Moody Groups and Integrable Systems
我们构造了一族可积的哈密顿系统，推广了相对论周期Toda晶格，并将其作为一种特殊情况恢复。这些系统的相空间是对应于仿射外尔群中Coxeter元对的双Bruhat胞。在这个过程中，我们推广了各种结果的双重Bruhat细胞在简单的代数群的设置Kac穆迪群。我们还将Poisson-Lie理论中的一些基本结果推广到ind-代数群的情形，这是我们目前对可积系统的应用所不能及的.
Double Bruhat cells in symmetric spaces
本文研究对称空间双 Bruhat 胞腔的闭包（大双 Bruhat 簇），建立其与 Richardson 几何的联系，证明正特征下的 Frobenius 分裂性质及特征零下的正则型与奇点性质。
Cluster algebras III: Upper bounds and double Bruhat cells
我们继续研究了在math.RT/0104151和math.RA/0208229中开始的簇代数。我们开发了一种新的方法的概念上的集群代数，定义为某些劳伦多项式环的交集的基础上。本文从数学RT/0104151出发，证明了在“无环性”假设下，一个簇代数与它的“上”对应物重合，并且是双生成的。在这种情况下，我们还描述了它的定义理想，并构造了一个标准的单项基。证明了半单复李群中任意双Bruhat胞元的坐标环都自然同构于由相关组合数据定义的上簇代数.
Q-Systems, Factorization Dynamics, and the Twist Automorphism
我们提供了一个具体的实现与Q-系统的集群结构的合并双Bruhat细胞在简单的代数群的集群代数。对于非单花边群，这提供了扭曲型Q-系统的簇代数公式。它也产生了一个统一的证明，这些Q-系统的离散可积性，通过识别它们与动力学的因子分解映射的双Bruhat细胞。在双Bruhat细胞本身，我们发现这些动力学是密切相关的Fomin-Zelevinsky扭曲映射。这导致了一个明确的公式表示扭曲的集群变量作为劳伦单项式在untwisted集群变量从相应的突变序列。这适用于Coxeter双Bruhat细胞在任何可对称化的Kac-Moody群，我们表明，在仿射型的类似的因子分解映射也是可积的。
Kontsevich and Soibelman defined the Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety can produce an example of such a category, whose corresponding Donaldson-Thomas invariants are encoded by a special formal automorphism of the cluster variety, known as the Donaldson-Thomas transformation. In this paper we prove a conjecture of Goncharov and Shen in the case of $\mathrm{GL}_n$, which describes the Donaldson-Thomas transformation of the double quotient of the double Bruhat cells $H \backslash \mathrm{GL}_n^{u,v}/H$ where $H$ is a maximal t…
Double Bruhat cells $G^{u,v}$ were studied by Fomin and Zelevinsky. They provide important examples of cluster algebras and cluster Poisson varieties. Cluster varieties produce examples of 3d Calabi-Yau categories with stability conditions, and their Donaldson-Thomas invariants, defined by Kontsevich and Soibelman, are encoded by a formal automorphism on the cluster variety known as the Donaldson-Thomas transformation. Goncharov and Shen conjectured in that for any semisimple Lie group $G$, the Donaldson-Thomas transformation of the cluster Poisson variety $H\backslash G^{u,v}/H$ is a slight m…
For a complex reductive Lie group G with Lie algebra g, Cartan subalgebra h and Weyl group W, we describe the category of perverse sheaves on h/W smooth w.r.t the natural stratification. The answer is given in terms of mixed Bruhat sheaves, which are certain mixed sheaf-cosheaf data on cells of a natural cell decomposition of h/W. Using the parabolic Bruhat decomposition, we relate mixed Bruhat sheaves with the properties of various procedures of parabolic induction and restriction that connect different Levi subgroups in G.
Cluster Ensembles and Kac-Moody Groups
我们研究了双Bruhat细胞上的两组坐标之间的关系，由Berenstein，Fomin和Zelevinsky引入的簇变量和由Fock和Goncharov的余权参数化定义的$\CX$ -坐标。在这些坐标系中，我们表明，广义室Anastomic的Fomin和Zelevinsky是一个非退化的版本的集群变量和$\CX$ -坐标定义的一个共同的交换矩阵之间的典型单项变换。我们证明了这一点的设置中的任意对称化的Kac-穆迪组，推广了许多以前的结果的双Bruhat细胞的半单代数群。特别地，我们在可对称化的Kac-Moody群中的任意双Bruhat胞的坐标环上构造了一个上簇代数结构，证明了Berenstein、Fomin和Zelevinsky的一个猜想。
Poisson structures on complex flag manifolds associated with real forms
对于复半单群G及其实形式G 0，我们在G的旗簇上定义一个Poisson结构，使得所有的Bruhat胞腔（对于G的博雷尔子群的适当选择）以及所有的G 0-轨道都是Poisson子流形.我们表明，每个交叉点的G 0-轨道与Bruhat细胞是一个经常泊松子流形，我们计算其辛叶的尺寸。
Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space
通过在GL_{m+n}（C）的极大环面T作用下的辛叶轨道，研究了矩形矩阵簇M_{m,n}（C）上的标准Poisson结构.这些轨道，有限的数量，被证明是光滑的不可约局部闭的M_{m,n}（C），同构于交叉的对偶Schubert细胞在全旗品种的GL_{m+n}（C）。得到了M_{m,n}（C）中辛叶T轨道的三种不同表示：（a）作为GL_{m+n}（C）中Bruhat胞腔在特定映射下的拉回;（b）作为矩形子矩阵的秩条件;（c）作为GL_m（C）和GL_n（C）中类似于双Bruhat胞腔的集合的矩阵积.在表示（a）中，叶的轨道由Weyl群的子集S_{m+n}参数化，使得Zorbiki闭包的包含对应于Bruhat阶。演示（b）允许明确的轨道计算。从表示（c）可以得出，直到Zebraki闭包，叶的每个轨道都是一个具有固定列-梯形式的轨道和一个具有固定行-梯形式的轨道的矩阵乘积。最后，得到了M_{m,n}（C）中广义双Bruhat胞腔（关于部分置换矩阵对）到辛叶T-轨道并的分解.
We prove the Berenstein-Zelevinsky conjecture that the quantized coordinate rings of the double Bruhat cells of all finite dimensional simple algebraic groups admit quantum cluster algebra structures with initial seeds as specified by [4]. We furthermore prove that the corresponding upper quantum cluster algebras coincide with the constructed quantum cluster algebras and exhibit a large number of explicit quantum seeds. Along the way a detailed study of the properties of quantum double Bruhat cells from the viewpoint of noncommutative UFDs is carried out and a quantum analog of the Fomin-Zelev…
Berenstein, Fomin and Zelevinsky defined functions on double Bruhat cells which they called generalized minors. By relating certain double Bruhat cells to configuration spaces of flags, we give formulas for these generalized minors as tensor invariants. This allows us to verify certain weight identities. The weights of the tensor invariants can then be used to construct the quiver for the cluster structure on the configuration space of three flags. We also show a converse statement--that the weights of tensor invariants can by computed from the structure of the quiver. The weight identities ar…
We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential function to provide a minimal set of inequalities via tropicalization and conjecture an equivalence.
Cluster algebras: notes for 2004 IMCC (Chonju, Korea, August 2004)
这是2004年国际数学会议（全北国立大学，2004年8月4-6日）上的两个演讲笔记的扩展版本。第一堂课讨论了簇代数的起源，重点是半单群中双Bruhat细胞的全正性和几何。第二讲介绍了簇代数，讨论了一些基本结果，未解决的问题和解答。
Cluster algebras and Grassmannians of type G2
本文证明了Geiss，Leclerc和Schröer的一个猜想，即在部分旗簇的多齐次坐标环上产生簇代数结构，对于$G_2$ .因此，我们通过证明双Bruhat胞腔$G^{e,w_0}$的坐标环是一个上簇代数，从而进一步明确了这一已知事实.
We show that the cohomology ring of a quiver Grassmannian asssociated with a rigid quiver representation has property (S): there is no odd cohomology and the cycle map is an isomorphism; moreover, its Chow ring admits explicit generators defined over any field. From this we deduce the polynomial point count property. By restricting the quiver to finite or affine type, we are able to show a much stronger assertion: namely, that a quiver Grassmannian associated to an indecomposable (not necessarily rigid) representation admits a cellular decomposition. As a corollary, we establish a cellular dec…
Cluster algebras of finite type via Coxeter elements and principal minors
给出了主系数位于任意非循环种子上的任意有限型簇代数的一致几何实现。该代数被实现为相同Cartan-Killing型的单连通半单代数群中某个既约双Bruhat胞的坐标环。在这种实现中，聚类变量表现为某些（广义）主子式。
It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of its bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland's representation theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all well-known cluster algeb…
We establish the relation of the potential function constructed by Gross-Hacking-Keel-Kontsevich's and Berenstein-Kazhdan's decoration function on the open double Bruhat cell in the base affine space $G/\mathcal{N}$ of a simple, simply connected, simply laced algebraic group $G$. As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on $G/\mathcal{N}$ arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations.
Total positivity in twisted product of flag varieties
本文证明 Kac-Moody 群旗簇扭积的全非负部分具有胞腔分解和显式参数化，并证明双旗簇、辫子簇及双 Bruhat 胞腔链环均为同胚于闭球的正则 CW 复形，解决了 Fomin-Zelevinsky 公开问题。
We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the Heisenberg double $\mathcal{H}_q$ of the quantum Borel subalgebra $U_{\geq0}$, which we relate to $\mathcal{O}_q[G]$ via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck-Springer resolution studied by Evens and Lu, and the quantum Beilinson-Bernstein theorem investigated by Backelin, Kremnitzer, …
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