Advances in Mathematics最新文献

英文中文

BPS Lie algebras and the less perverse filtration on the preprojective CoHA

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-20 DOI: 10.1016/j.aim.2025.110114

Ben Davison

The affinization morphism for the stack

M (Π_{Q})

of representations of a preprojective algebra

Π_{Q}

is a local model for the morphism from the stack of objects in a general 2-Calabi–Yau category to the good moduli space. We show that the derived direct image of the dualizing complex along this morphism is pure, and admits a decomposition in the sense of the Beilinson–Bernstein–Deligne–Gabber decomposition theorem.

We introduce a new perverse filtration on the Borel–Moore homology of

M (Π_{Q})

, using this decomposition. We show that the zeroth piece of the resulting filtration on the cohomological Hall algebra built out of the Borel–Moore homology of

M (Π_{Q})

is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra

g_{Π_{Q}}

. This Lie algebra is defined via the Kontsevich–Soibelman theory of critical cohomological Hall algebras for 3-Calabi–Yau categories. We then lift this Lie algebra to a Lie algebra object in the category of perverse sheaves on the coarse moduli space of

Π_{Q}

-modules, and use this algebra structure to prove results about the summands appearing in the above decomposition theorem. In particular, we prove that the intersection cohomology of singular spaces of semistable

Π_{Q}

-modules provide “cuspidal cohomology” – a conjecturally complete canonical subspace of generators for

g_{Π_{Q}}

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引用次数: 0

Bumpless pipe dreams meet puzzles

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-16 DOI: 10.1016/j.aim.2025.110113

Neil J.Y. Fan , Peter L. Guo , Rui Xiong

Knutson and Zinn-Justin recently found a puzzle rule for the expansion of the product

G_{u} (x, t) \cdot G_{v} (x, t)

of two double Grothendieck polynomials indexed by permutations with separated descents. We establish its triple Schubert calculus version in the sense of Knutson and Tao, namely, a formula for expanding

G_{u} (x, y) \cdot G_{v} (x, t)

in different secondary variables. Our rule is formulated in terms of pipe puzzles, incorporating the structures of both bumpless pipe dreams and classical puzzles. As direct applications, we recover the separated-descent puzzle formula by Knutson and Zinn-Justin (by setting

y = t

) and the bumpless pipe dream model of double Grothendieck polynomials by Weigandt (by setting

v = id

and

x = t

). Moreover, we utilize the formula to partially confirm a positivity conjecture of Kirillov about applying a skew operator to a Schubert polynomial.

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引用次数: 0

Expression of concern “Notes on Plücker's relations in geometric algebra” [Adv. Math. 363 (2020) 106959]

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2025-01-01

引用次数: 0

Order-2 Delaunay triangulations optimize angles Order-2 Delaunay三角剖分优化角度

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-29 DOI: 10.1016/j.aim.2024.110055

Herbert Edelsbrunner , Alexey Garber , Morteza Saghafian

The local angle property of the (order-1) Delaunay triangulations of a generic set in

R^{2}

asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in

R^{2}

, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in

R^{2}

, the order-2 Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-1, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-2 Delaunay triangulations more attractive to applications as well.

R2中一般集合的（order-1） Delaunay三角剖分的局域角性质证明了一个公共边对的两个角的和小于π。本文将这一性质推广到更高阶，并将两个经典性质从order-1推广到order-2:(1)在R2中一般点集的完全level-2超三角剖分中，order-2 Delaunay三角剖分按字典顺序最大化排序角向量；(2)在R2中一般点集的最大level-2超三角剖分中，阶-2 Delaunay三角剖分是唯一具有局域角性质的。我们还利用建立式(2)的方法给出了（order-1） Delaunay三角剖分的角向量最优性的一个新的简短证明。对于阶-1，这两个性质在Delaunay三角剖分的许多应用中都很有用，我们期望它们的推广将使阶-2 Delaunay三角剖分在应用中也更有吸引力。

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引用次数: 0

Quasilinear tropical compactifications 拟线性热带紧化

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-29 DOI: 10.1016/j.aim.2024.110037

Nolan Schock

The prototypical examples of tropical compactifications are compactifications of complements of hyperplane arrangements, which posses a number of remarkable properties not satisfied by more general tropical compactifications of closed subvarieties of tori. We introduce a broader class of tropical compactifications, which we call quasilinear (tropical) compactifications, and which continue to satisfy the desirable properties of compactifications of complements of hyperplane arrangements. In particular, we show any quasilinear compactification is schön, and its intersection theory is described entirely by the intersection theory of the corresponding tropical fan. As applications, we prove the quasilinearity of the moduli spaces of 6 lines in

P^{2}

and marked cubic surfaces, obtaining results on the geometry of the stable pair compactifications of these spaces.

热带紧化的典型例子是超平面排列补的紧化，它具有环面闭合亚种的更一般的热带紧化所不满足的许多显著性质。我们引入了一类更广泛的热带紧化，我们称之为拟线性（热带）紧化，它继续满足超平面排列补紧化的理想性质。特别地，我们证明了任何拟线性紧化都是schön，其相交理论完全由相应热带扇的相交理论来描述。作为应用，我们证明了P2和标记三次曲面上6条直线的模空间的拟线性性，得到了这些空间的稳定对紧化的几何结果。

引用次数: 0

The centre of the modular affine vertex algebra 模仿射顶点代数的中心

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-29 DOI: 10.1016/j.aim.2024.110052

Tomoyuki Arakawa , Lewis Topley , Juan J. Villarreal

The Feigin–Frenkel theorem states that, over the complex numbers, the centre of the universal affine vertex algebra at the critical level is an infinite rank polynomial algebra. The first author and W. Wang observed that in positive characteristics, the universal affine vertex algebra contains a large central subalgebra known as the p-centre. They conjectured that at the critical level the centre should be generated by the Feigin–Frenkel centre and the p-centre. In this paper we prove the conjecture for classical simple Lie algebras for p larger than the Coxeter number, and for exceptional Lie algebras in large characteristics. Finally, we give an example which shows that at non-critical level the center is larger than the p-centre.

Feigin-Frenkel定理指出，在复数上，普遍仿射顶点代数在临界水平上的中心是一个无限秩多项式代数。第一作者和W. Wang观察到，在正特征中，普遍仿射顶点代数包含一个大的中心子代数，称为p中心。他们推测，在临界水平上，中心应该由Feigin-Frenkel中心和p-中心产生。本文证明了p大于Coxeter数的经典简单李代数的猜想，以及具有大特征的例外李代数的猜想。最后，我们给出了一个例子，表明在非临界水平上，中心大于p中心。

引用次数: 0

Uniqueness up to inner automorphism of regular exact Borel subalgebras 正则精确Borel子代数的内自同构唯一性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-29 DOI: 10.1016/j.aim.2024.110049

Anna Rodriguez Rasmussen

In [18], Külshammer, König and Ovsienko proved that for any quasi-hereditary algebra

(A, \leq_{A})

there exists a Morita equivalent quasi-hereditary algebra

(R, \leq_{R})

containing a basic exact Borel subalgebra B. The Borel subalgebra B constructed in [18] is in fact a regular exact Borel subalgebra as defined in [7]. Later, Conde [9] showed that given a quasi-hereditary algebra

(R, \leq_{R})

with a basic regular exact Borel subalgebra B and a Morita equivalent quasi-hereditary algebra

(R^{'}, \leq_{R^{'}})

with a basic regular exact Borel subalgebra

B^{'}

, the algebras R and

R^{'}

are isomorphic, and Külshammer and Miemietz [20] showed that there is even an isomorphism

φ : R \to R^{'}

such that

φ (B) = B^{'}

In this article, we show that if

R = R^{'}

, then φ can be chosen to be an inner automorphism. Moreover, instead of just proving this for regular exact Borel subalgebras of quasi-hereditary algebras, we generalize this to an appropriate class of subalgebras of arbitrary finite-dimensional algebras. As an application, we show that if

(A, \leq_{A})

is a finite-dimensional algebra and G is a finite group acting on A via automorphisms, then under some natural compatibility conditions, there is a Morita equivalent quasi-hereditary algebra

(R, \leq_{R})

with a basic regular exact Borel subalgebra B such that

g (B) = B

for every

g \in G

在[18]中，k lshammer， König和Ovsienko证明了对于任意拟遗传代数（A,≤A）存在一个包含基本精确Borel子代数B的Morita等价拟遗传代数（R,≤R）。在[18]中构造的Borel子代数B实际上是[7]中定义的正则精确Borel子代数。随后，Conde[9]证明了给定一个具有基本正则精确Borel子代数B的拟遗传代数（R,≤R）和一个具有基本正则精确Borel子代数B ‘的Morita等价拟遗传代数（R ’，≤R ‘），代数R和R ’是同构的，并且k lshammer和Miemietz[20]证明了甚至存在一个同构φ:R→R ‘使得φ(B)=B ’。在本文中，我们证明了如果R=R '，那么φ可以被选为一个内自同构。此外，我们不仅在拟遗传代数的正则精确Borel子代数上证明了这一点，而且将其推广到任意有限维代数的一类适当的子代数上。作为一个应用，我们证明了如果（A,≤A）是有限维代数，G是通过自同构作用于A的有限群，那么在某些自然相容条件下，存在一个Morita等价拟遗传代数（R,≤R），它具有一个基本正则精确Borel子代数B，使得G (B)对每一个G∈G都=B。

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引用次数: 0

Bordism invariance of orientations and real APS index theory 方向的Bordism不变性与实APS指标理论

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-29 DOI: 10.1016/j.aim.2024.110048

Markus Upmeier

We show that orientations and Floer gradings for elliptic differential operators can be propagated through bordisms. This is based on a new perspective on APS indices for elliptic boundary value problems over the real numbers. Several applications to moduli spaces of this new bordism-theoretic point of view will be given in the sequel.

我们证明了椭圆型微分算子的定向和花分级可以通过边界传播。这是基于对实数上椭圆边值问题的APS指标的一种新的认识。在后续部分将给出这一新的边界理论观点在模空间中的几个应用。

引用次数: 0

Dual linear programming bounds for sphere packing via discrete reductions 通过离散还原实现球体包装的双重线性规划边界

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-27 DOI: 10.1016/j.aim.2024.110043

Rupert Li

The Cohn-Elkies linear program for sphere packing, which was used to solve the 8 and 24 dimensional cases, is conjectured to not be sharp in any other dimension

d > 2

. By mapping feasible points of this infinite-dimensional linear program into a finite-dimensional problem via discrete reduction, we provide a general method to obtain dual bounds on the Cohn-Elkies linear program. This reduces the number of variables to be finite, enabling computer optimization techniques to be applied. Using this method, we prove that the Cohn-Elkies bound cannot come close to the best packing densities known in dimensions

3 \leq d \leq 13

except for the solved case

d = 8

. In particular, our dual bounds show the Cohn-Elkies bound is unable to solve the 3, 4, and 5 dimensional sphere packing problems.

用于求解 8 维和 24 维情况的球体包装的 Cohn-Elkies 线性程序，据猜测在任何其他维度 d>2 下都不尖锐。通过离散还原法将这个无限维线性程序的可行点映射为有限维问题，我们提供了一种获得 Cohn-Elkies 线性程序对偶约束的通用方法。这就将变量的数量减少到有限，使计算机优化技术得以应用。利用这种方法，我们证明了除了 d=8 的求解情况外，Cohn-Elkies 边界无法接近维数 3≤d≤13 的已知最佳堆积密度。特别是，我们的对偶边界表明，Cohn-Elkies 边界无法解决 3 维、4 维和 5 维球体堆积问题。

引用次数: 0

On the profinite homotopy type of log schemes 论对数方案的无穷同调类型

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-11-26 DOI: 10.1016/j.aim.2024.110018

David Carchedi , Sarah Scherotzke , Nicolò Sibilla , Mattia Talpo

We complete the program, initiated in [8], to compare the many different possible definitions of the underlying homotopy type of a log scheme. We show that, up to profinite completion, they all yield the same result, and thus arrive at an unambiguous definition of the profinite homotopy type of a log scheme. Specifically, in [8], we define this to be the profinite étale homotopy type of the infinite root stack, and show that, over

C

, this agrees up to profinite completion with the Kato-Nakayama space. Other possible candidates are the profinite shape of the Kummer étale site

X_{k \overset{´}{e} t}

, or of the representable étale site of

. Our main result is that all of these notions agree, and moreover the profinite étale homotopy type of the infinite root stack is not sensitive to whether or not it is viewed as a pro-system in stacks, or as an actual stack (by taking the limit of the pro-system). We furthermore show that in the log regular setting, all these notions also agree with the étale homotopy type of the classical locus

X^{triv}

(up to an appropriate completion). We deduce that, over an arbitrary locally Noetherian base, the étale homotopy type of

G_{m}^{N}

agrees with that of

B μ_{\infty}^{N}

up to completion.

我们完成了[8]中提出的计划，比较了对数方案底层同调类型的多种可能定义。我们证明，在无限完备性上，它们都得出了相同的结果，从而得出了对数方案的无限同调类型的明确定义。具体地说，在 [8] 中，我们将其定义为无限根堆栈的无限 étale 同调类型，并证明在 C 上，直到无限完备为止，这个定义与加藤中山空间一致。我们的主要结果是，所有这些概念都是一致的，而且无限根堆栈的无限étale同调类型对它是否被视为堆栈中的原系统或实际堆栈（通过取原系统的极限）并不敏感。我们进一步证明，在对数正则环境中，所有这些概念也与经典位置 Xtriv 的 étale 同调类型一致（直到适当的补全）。我们推导出，在任意局部诺特基上，GmN 的 étale 同调类型与 Bμ∞N 直至完备的同调类型一致。

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