Pub Date : 2024-07-18DOI: 10.4310/pamq.2024.v20.n4.a4
Lydia Bieri
We derive new results on radiation, angular momentum at future null infinity and peeling for a general class of spacetimes. For asymptotically-flat solutions of the Einstein vacuum equations with a term homogeneous of degree $-1$ in the initial data metric, that is it may include a non-isotropic mass term, we prove new detailed behavior of the radiation field and curvature components at future null infinity. In particular, the limit along the null hypersurface $C_u$ as $t to infty$ of the curvature component $rho =frac{1}{4}{R_{3434}}$ multiplied with $r^3$ tends to a function $P(u, theta, phi)$ on $mathbb{R} times S^2$. When taking the limit $u rightarrow + infty$ (which corresponds to the limit at spacelike infinity), this function tends to a function $P^+(theta, phi)$ on $S^2$. We prove that the latter limit does not have any $l=1$ modes. However, it has all the other modes, $l = 0, l geq 2$. Important derivatives of crucial curvature components do not decay in $u$, which is a special feature of these more general spacetimes We show that peeling of the Weyl curvature components at future null infinity stops at the order $r^{-3}$, that is $(r^{-4}|u|^{+1}$, for large data, and at order $r^{-frac{7}{2}}$ for small data. Despite this fact, we prove that angular momentum at future null infinity is well defined for these spacetimes, due to the good behavior of the $l=1$ modes involved.
我们推导出了关于辐射、未来空无穷远处的角动量以及一般空间的剥离的新结果。对于爱因斯坦真空方程的渐近平直解，其初始数据度量中有一个度数为 $-1$ 的同质项，即可能包括一个非各向同性的质量项，我们证明了辐射场和曲率分量在未来空无穷远处的新的详细行为。特别是，当曲率分量$rho =frac{1}{4}{R_{3434}}$ 乘以$r^3$时，沿着空超表面$C_u$的极限在$t to infty$上趋于函数$P(u, theta,phi)$ on $mathbb{R} times S^2$。当取极限 $u rightarrow + infty$（对应于空间无穷大处的极限）时，这个函数趋向于 $S^2$ 上的函数 $P^+(theta,phi)$。我们证明后一极限不具有任何 $l=1$ 模式。然而，它具有所有其他模式，即 $l = 0, l geq 2$。关键曲率分量的重要导数不在$u$中衰减，这是这些更一般的时空的一个特殊特征。我们证明，对于大数据，韦尔曲率分量在未来空无穷大处的剥离在$r^{-3}$阶停止，即$(r^{-4}|u|^{+1}$，而对于小数据，则在$r^{-frac{7}{2}}阶停止。尽管如此，我们还是证明，由于所涉及的 $l=1$ 模式行为良好，这些时空在未来空无穷大处的角动量定义良好。
{"title":"Radiation and Asymptotics for Spacetimes with Non-Isotropic Mass","authors":"Lydia Bieri","doi":"10.4310/pamq.2024.v20.n4.a4","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a4","url":null,"abstract":"We derive new results on radiation, angular momentum at future null infinity and peeling for a general class of spacetimes. For asymptotically-flat solutions of the Einstein vacuum equations with a term homogeneous of degree $-1$ in the initial data metric, that is it may include a non-isotropic mass term, we prove new detailed behavior of the radiation field and curvature components at future null infinity. In particular, the limit along the null hypersurface $C_u$ as $t to infty$ of the curvature component $rho =frac{1}{4}{R_{3434}}$ multiplied with $r^3$ tends to a function $P(u, theta, phi)$ on $mathbb{R} times S^2$. When taking the limit $u rightarrow + infty$ (which corresponds to the limit at spacelike infinity), this function tends to a function $P^+(theta, phi)$ on $S^2$. We prove that the latter limit does not have any $l=1$ modes. However, it has all the other modes, $l = 0, l geq 2$. Important derivatives of crucial curvature components do not decay in $u$, which is a special feature of these more general spacetimes We show that peeling of the Weyl curvature components at future null infinity stops at the order $r^{-3}$, that is $(r^{-4}|u|^{+1}$, for large data, and at order $r^{-frac{7}{2}}$ for small data. Despite this fact, we prove that angular momentum at future null infinity is well defined for these spacetimes, due to the good behavior of the $l=1$ modes involved.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/pamq.2024.v20.n4.a11
Xinliang An, Haoyang Chen, Silu Yin
In this paper, we prove that for the 2D elastic wave equations, a physical system with multiple wave-speeds, its Cauchy problem fails to be locally well-posed in $H frac{11}{4} (mathbb R^2)$. The ill-posedness here is driven by instantaneous shock formation. In 2D Smith-Tataru showed that the Cauchy problem for a single quasilinear wave equation is locally well-posed in $H^s$ with $s gt frac{11}{4}$. Hence our $H ^frac{11}{4}$ ill-posedness obtained here is a desired result. Our proof relies on combining a geometric method and an algebraic wave-decomposition approach, together with detailed analysis of the corresponding hyperbolic system.
{"title":"$H^{frac{11}{4}}(mathbb{R}^2)$ Ill-Posedness for 2D Elastic Wave System","authors":"Xinliang An, Haoyang Chen, Silu Yin","doi":"10.4310/pamq.2024.v20.n4.a11","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a11","url":null,"abstract":"In this paper, we prove that for the 2D elastic wave equations, a physical system with multiple wave-speeds, its Cauchy problem fails to be locally well-posed in $H frac{11}{4} (mathbb R^2)$. The ill-posedness here is driven by instantaneous shock formation. In 2D Smith-Tataru showed that the Cauchy problem for a single quasilinear wave equation is locally well-posed in $H^s$ with $s gt frac{11}{4}$. Hence our $H ^frac{11}{4}$ ill-posedness obtained here is a desired result. Our proof relies on combining a geometric method and an algebraic wave-decomposition approach, together with detailed analysis of the corresponding hyperbolic system.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/pamq.2024.v20.n4.a2
Hans Ringström
In a recent article, we propose a general geometric notion of initial data on big bang singularities. This notion is of interest in its own right. However, it also serves the purpose of giving a unified perspective on many of the results in the literature. In the present article, we give a partial justification of this statement by rephrasing the results concerning Bianchi class A orthogonal stiff fluid solutions and solutions in the $mathbb{T}^3$-Gowdy symmetric vacuum setting in terms of our general geometric notion of initial data on the big bang singularity.
在最近的一篇文章中，我们提出了大爆炸奇点初始数据的一般几何概念。这个概念本身就很有趣。不过，它也可以为许多文献中的结果提供一个统一的视角。在本文中，我们根据我们关于大爆炸奇点初始数据的一般几何概念，重新表述了关于边奇类 A 正交刚性流体解和 $mathbb{T}^3$-Gowdy 对称真空环境中的解的结果，从而给出了这一表述的部分理由。
{"title":"Initial data on big bang singularities in symmetric settings","authors":"Hans Ringström","doi":"10.4310/pamq.2024.v20.n4.a2","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a2","url":null,"abstract":"In a recent article, we propose a general geometric notion of initial data on big bang singularities. This notion is of interest in its own right. However, it also serves the purpose of giving a unified perspective on many of the results in the literature. In the present article, we give a partial justification of this statement by rephrasing the results concerning Bianchi class A orthogonal stiff fluid solutions and solutions in the $mathbb{T}^3$-Gowdy symmetric vacuum setting in terms of our general geometric notion of initial data on the big bang singularity.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.4310/pamq.2024.v20.n4.a8
Sergiu Klainerman, Jérémie Szeftel
This a brief introduction to the sequence of works $href{https://doi.org/10.48550/arXiv.2104.11857}{[65]}$, $href{https://doi.org/10.48550/arXiv.2205.14808}{[41]}$, $href{https://doi.org/10.1007/s40818-022-00131-8}{[63]}$, $href{https://doi.org/10.1007/s40818-022-00132-7}{[64]}$, and $href{https://doi.org/10.1007/s40818-023-00152-x}{[85]}$ which establish the nonlinear stability of Kerr black holes with small angular momentum. We are delighted to dedicate this article to Demetrios Christodoulou for whom we both have great admiration. The first author would also like to thank Demetrios for the magic moments of friendship, discussions and collaboration he enjoyed together with him.
{"title":"Brief introduction to the nonlinear stability of Kerr","authors":"Sergiu Klainerman, Jérémie Szeftel","doi":"10.4310/pamq.2024.v20.n4.a8","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n4.a8","url":null,"abstract":"This a brief introduction to the sequence of works $href{https://doi.org/10.48550/arXiv.2104.11857}{[65]}$, $href{https://doi.org/10.48550/arXiv.2205.14808}{[41]}$, $href{https://doi.org/10.1007/s40818-022-00131-8}{[63]}$, $href{https://doi.org/10.1007/s40818-022-00132-7}{[64]}$, and $href{https://doi.org/10.1007/s40818-023-00152-x}{[85]}$ which establish the nonlinear stability of Kerr black holes with small angular momentum. We are delighted to dedicate this article to Demetrios Christodoulou for whom we both have great admiration. The first author would also like to thank Demetrios for the magic moments of friendship, discussions and collaboration he enjoyed together with him.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-15DOI: 10.4310/pamq.2024.v20.n3.a1
Anders Björner, Mark Goresky, Robert MacPherson
We discuss ways in which tools from topology can be used to derive lower bounds for the circuit complexity of Boolean functions.
我们将讨论如何利用拓扑学工具来推导布尔函数的电路复杂度下限。
{"title":"Topological aspects of Boolean functions","authors":"Anders Björner, Mark Goresky, Robert MacPherson","doi":"10.4310/pamq.2024.v20.n3.a1","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a1","url":null,"abstract":"We discuss ways in which tools from topology can be used to derive lower bounds for the circuit complexity of Boolean functions.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-15DOI: 10.4310/pamq.2024.v20.n3.a3
Rocco Chirivì, Xin Fang, Peter Littelmann
The existence of a Seshadri stratification on an embedded projective variety provides a flat degeneration of the variety to a union of projective toric varieties, called a semi-toric variety. Such a stratification is said to be normal when each irreducible component of the semi-toric variety is a normal toric variety. In this case, we show that a Gröbner basis of the defining ideal of the semi-toric variety can be lifted to define the embedded projective variety. Applications to Koszul and Gorenstein properties are discussed. Relations between LS‑algebras and certain Seshadri stratifications are studied.
内嵌射影变种上存在的塞沙德里分层提供了变种的平面退化，使其成为射影环变种的结合，称为半oric变种。当半oric 变的每个不可还原分量都是一个正常的 toric 变时，这种分层就被称为正常分层。在这种情况下，我们证明了半oric 变的定义理想的格罗伯纳基可以被提升以定义嵌入的射影变。我们还讨论了科斯祖尔和戈伦斯坦性质的应用。此外，我们还研究了 LS 后代数与某些 Seshadri 分层之间的关系。
{"title":"On normal Seshadri stratifications","authors":"Rocco Chirivì, Xin Fang, Peter Littelmann","doi":"10.4310/pamq.2024.v20.n3.a3","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a3","url":null,"abstract":"The existence of a Seshadri stratification on an embedded projective variety provides a flat degeneration of the variety to a union of projective toric varieties, called a semi-toric variety. Such a stratification is said to be normal when each irreducible component of the semi-toric variety is a normal toric variety. In this case, we show that a Gröbner basis of the defining ideal of the semi-toric variety can be lifted to define the embedded projective variety. Applications to Koszul and Gorenstein properties are discussed. Relations between LS‑algebras and certain Seshadri stratifications are studied.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-15DOI: 10.4310/pamq.2024.v20.n3.a9
Gus Lehrer, Yang Zhang
Let $W$ be a finite Coxeter group. We give an algebraic presentation of what we refer to as “the non-crossing algebra”, which is associated to the hyperplane complement of $W$ and to the cohomology of its Milnor fibre. This is used to produce simpler and more general chain (and cochain) complexes which compute the integral homology and cohomology groups of the Milnor fibre $F$ of $W$. In the process we define a new, larger algebra $tilde{A}$, which seems to be “dual” to the Fomin–Kirillov algebra, and in low ranks is linearly isomorphic to it. There is also a mysterious connection between $tilde{A}$ and the Orlik–Solomon algebra, in analogy with the fact that the Fomin–Kirillov algebra contains the coinvariant algebra of $W$. This analysis is applied to compute the multiplicities ${langle rho, H^k (F, mathbb{C}) rangle}_W$ and ${langle rho, H^k (M, mathbb{C}) rangle}_W$, where $M$ and $F$ are respectively the hyperplane complement and Milnor fibre associated to $W$ and $rho$ is a representation of $W$.
{"title":"Milnor fibre homology complexes","authors":"Gus Lehrer, Yang Zhang","doi":"10.4310/pamq.2024.v20.n3.a9","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a9","url":null,"abstract":"Let $W$ be a finite Coxeter group. We give an algebraic presentation of what we refer to as “the non-crossing algebra”, which is associated to the hyperplane complement of $W$ and to the cohomology of its Milnor fibre. This is used to produce simpler and more general chain (and cochain) complexes which compute the integral homology and cohomology groups of the Milnor fibre $F$ of $W$. In the process we define a new, larger algebra $tilde{A}$, which seems to be “dual” to the Fomin–Kirillov algebra, and in low ranks is linearly isomorphic to it. There is also a mysterious connection between $tilde{A}$ and the Orlik–Solomon algebra, in analogy with the fact that the Fomin–Kirillov algebra contains the coinvariant algebra of $W$. This analysis is applied to compute the multiplicities ${langle rho, H^k (F, mathbb{C}) rangle}_W$ and ${langle rho, H^k (M, mathbb{C}) rangle}_W$, where $M$ and $F$ are respectively the hyperplane complement and Milnor fibre associated to $W$ and $rho$ is a representation of $W$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-15DOI: 10.4310/pamq.2024.v20.n3.a8
E. Haus, B. Langella, A. Maspero, M. Procesi
Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schrödinger equation (NLS) on the two dimensional torus $mathbb{T}^2 := (mathbb{R}/2 pi mathbb{Z})^2$, we consider a quasi-periodically forced NLS equation on $mathbb{T}^2$ arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.
受二维环$mathbb{T}^2 := (mathbb{R}/2 pi mathbb{Z})^2$上的非线性三次薛定谔方程（NLS）的 KAM 环的长期稳定性与不稳定性问题的启发，我们考虑了在 KAM 环上由 NLS 的线性化引起的 $mathbb{T}^2$ 上的准周期强迫 NLS 方程。我们证明了还原性结果以及原点的长期稳定性。主要的新颖之处在于获得了频率的精确渐近展开，这使得我们可以在任意阶施加梅尔尼科夫条件。
{"title":"Reducibility and nonlinear stability for a quasi-periodically forced NLS","authors":"E. Haus, B. Langella, A. Maspero, M. Procesi","doi":"10.4310/pamq.2024.v20.n3.a8","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a8","url":null,"abstract":"Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schrödinger equation (NLS) on the two dimensional torus $mathbb{T}^2 := (mathbb{R}/2 pi mathbb{Z})^2$, we consider a quasi-periodically forced NLS equation on $mathbb{T}^2$ arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-15DOI: 10.4310/pamq.2024.v20.n3.a2
Michel Brion
Every action of a finite group scheme $G$ on a variety admits a projective equivariant model, but not necessarily a normal one. As a remedy, we introduce and explore the notion of $G$-normalization. In particular, every curve equipped with a $G$-action has a unique projective $G$-normal model, characterized by the invertibility of ideal sheaves of all orbits. Also, $G$-normal curves occur naturally in some questions on surfaces in positive characteristics.
{"title":"Actions of finite group schemes on curves","authors":"Michel Brion","doi":"10.4310/pamq.2024.v20.n3.a2","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a2","url":null,"abstract":"Every action of a finite group scheme $G$ on a variety admits a projective equivariant model, but not necessarily a normal one. As a remedy, we introduce and explore the notion of $G$-normalization. In particular, every curve equipped with a $G$-action has a unique projective $G$-normal model, characterized by the invertibility of ideal sheaves of all orbits. Also, $G$-normal curves occur naturally in some questions on surfaces in positive characteristics.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-15DOI: 10.4310/pamq.2024.v20.n3.a10
Valery Lunts, Špela Špenko, Michel Van Den Bergh
We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $mathbb{R}^n$, combinatorially equivalent to the permutahedron, and the set of m-parking functions of length n. This bijection restricts to a bijection between the regular $S_n$-orbits and $(m, n)$-Dyck paths, the number of which is given by the Fuss–Catalan number $A_n (m, 1)$. Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes.
{"title":"Catalan numbers and noncommutative Hilbert schemes","authors":"Valery Lunts, Špela Špenko, Michel Van Den Bergh","doi":"10.4310/pamq.2024.v20.n3.a10","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a10","url":null,"abstract":"We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $mathbb{R}^n$, combinatorially equivalent to the permutahedron, and the set of m-parking functions of length n. This bijection restricts to a bijection between the regular $S_n$-orbits and $(m, n)$-Dyck paths, the number of which is given by the Fuss–Catalan number $A_n (m, 1)$. Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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