The Witt algebra ${mathfrak{W}}_{n}$ is the Lie algebra of all derivations of the $n$-variable polynomial ring $textbf{V}_{n}=textbf{C}[x_{1}, ldots , x_{n}]$ (or of algebraic vector fields on $textbf{A}^{n}$). A representation of ${mathfrak{W}}_{n}$ is polynomial if it arises as a subquotient of a sum of tensor powers of $textbf{V}_{n}$. Our main theorems assert that finitely generated polynomial representations of ${mathfrak{W}}_{n}$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $textbf{Fin}^{textrm{op}}$, where $textbf{Fin}$ is the category of finite sets. We also show that polynomial representations of ${mathfrak{W}}_{n}$ are equivalent to polynomial representations of the endomorphism monoid of $textbf{A}^{n}$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.
{"title":"Polynomial Representations of the Witt Lie Algebra","authors":"Steven V Sam, Andrew Snowden, Philip Tosteson","doi":"10.1093/imrn/rnae139","DOIUrl":"https://doi.org/10.1093/imrn/rnae139","url":null,"abstract":"The Witt algebra ${mathfrak{W}}_{n}$ is the Lie algebra of all derivations of the $n$-variable polynomial ring $textbf{V}_{n}=textbf{C}[x_{1}, ldots , x_{n}]$ (or of algebraic vector fields on $textbf{A}^{n}$). A representation of ${mathfrak{W}}_{n}$ is polynomial if it arises as a subquotient of a sum of tensor powers of $textbf{V}_{n}$. Our main theorems assert that finitely generated polynomial representations of ${mathfrak{W}}_{n}$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $textbf{Fin}^{textrm{op}}$, where $textbf{Fin}$ is the category of finite sets. We also show that polynomial representations of ${mathfrak{W}}_{n}$ are equivalent to polynomial representations of the endomorphism monoid of $textbf{A}^{n}$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"59 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let the circle group act on a compact oriented manifold $M$ with a non-empty discrete fixed point set. Then the dimension of $M$ is even. If $M$ has one fixed point, $M$ is the point. In any even dimension, such a manifold $M$ with two fixed points exists, a rotation of an even dimensional sphere. Suppose that $M$ has three fixed points. Then the dimension of $M$ is a multiple of 4. Under the assumption that each isotropy submanifold is orientable, we show that if $dim M=8$, then the weights at the fixed points agree with those of an action on the quaternionic projective space $mathbb{H}mathbb{P}^{2}$, and show that there is no such 12-dimensional manifold $M$.
{"title":"Circle Actions on Oriented Manifolds With 3 Fixed Points","authors":"Donghoon Jang","doi":"10.1093/imrn/rnae132","DOIUrl":"https://doi.org/10.1093/imrn/rnae132","url":null,"abstract":"Let the circle group act on a compact oriented manifold $M$ with a non-empty discrete fixed point set. Then the dimension of $M$ is even. If $M$ has one fixed point, $M$ is the point. In any even dimension, such a manifold $M$ with two fixed points exists, a rotation of an even dimensional sphere. Suppose that $M$ has three fixed points. Then the dimension of $M$ is a multiple of 4. Under the assumption that each isotropy submanifold is orientable, we show that if $dim M=8$, then the weights at the fixed points agree with those of an action on the quaternionic projective space $mathbb{H}mathbb{P}^{2}$, and show that there is no such 12-dimensional manifold $M$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"11 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $k$ be a field, let $H subset G$ be (possibly disconnected) reductive groups over $k$, and let $Gamma $ be a finitely generated group. Vinberg and Martin have shown that the induced morphism $underline{operatorname{Hom}}_{ktextrm{-gp}}(Gamma , H)//H to underline{operatorname{Hom}}_{ktextrm{-gp}}(Gamma , G)//G$ is finite. In this note, we generalize this result (with a significantly different proof) by replacing $k$ with an arbitrary locally Noetherian scheme, answering a question of Dat. Along the way, we use Bruhat–Tits theory to establish a few apparently new results about integral models of reductive groups over discrete valuation rings.
{"title":"Morphisms of Character Varieties","authors":"Sean Cotner","doi":"10.1093/imrn/rnae124","DOIUrl":"https://doi.org/10.1093/imrn/rnae124","url":null,"abstract":"Let $k$ be a field, let $H subset G$ be (possibly disconnected) reductive groups over $k$, and let $Gamma $ be a finitely generated group. Vinberg and Martin have shown that the induced morphism $underline{operatorname{Hom}}_{ktextrm{-gp}}(Gamma , H)//H to underline{operatorname{Hom}}_{ktextrm{-gp}}(Gamma , G)//G$ is finite. In this note, we generalize this result (with a significantly different proof) by replacing $k$ with an arbitrary locally Noetherian scheme, answering a question of Dat. Along the way, we use Bruhat–Tits theory to establish a few apparently new results about integral models of reductive groups over discrete valuation rings.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"22 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Without any continuity assumptions, a complete classification of $textrm{SL}(n)$ contravariant, matrix-valued valuations on convex polytopes is established. Furthermore, the constraint for matrix symmetry is removed. If $ngeq 4$, then such valuations are uniquely characterized by the generic Lutwak–Yang–Zhang matrix; in dimension three, a new function appears. The classification result in the 2-dimensional case is consistent with the established example of $textrm{SL}(2)$-equivariant matrix-valued valuation.
{"title":"SL n Contravariant Matrix-Valued Valuations on Polytopes","authors":"Chunna Zeng, Yuqi Zhou","doi":"10.1093/imrn/rnae122","DOIUrl":"https://doi.org/10.1093/imrn/rnae122","url":null,"abstract":"Without any continuity assumptions, a complete classification of $textrm{SL}(n)$ contravariant, matrix-valued valuations on convex polytopes is established. Furthermore, the constraint for matrix symmetry is removed. If $ngeq 4$, then such valuations are uniquely characterized by the generic Lutwak–Yang–Zhang matrix; in dimension three, a new function appears. The classification result in the 2-dimensional case is consistent with the established example of $textrm{SL}(2)$-equivariant matrix-valued valuation.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"38 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$th Gleason polynomial $G_{n}in{mathbb{Q}}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $operatorname{Per}_{n}(0)$ is the $1$-dimensional moduli space of quadratic rational maps on ${mathbb{P}}^{1}$ with an $n$-periodic critical point. We show that if $G_{n}$ is irreducible over ${mathbb{Q}}$, then $operatorname{Per}_{n}(0)$ is irreducible over ${mathbb{C}}$. To do this, we exhibit a ${mathbb{Q}}$-rational smooth point on a projective completion of $operatorname{Per}_{n}(0)$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $n$, $operatorname{Per}_{n}(0)$ itself has no ${mathbb{Q}}$-rational points.
{"title":"Moduli Spaces of Quadratic Maps: Arithmetic and Geometry","authors":"Rohini Ramadas","doi":"10.1093/imrn/rnae126","DOIUrl":"https://doi.org/10.1093/imrn/rnae126","url":null,"abstract":"We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$th Gleason polynomial $G_{n}in{mathbb{Q}}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $operatorname{Per}_{n}(0)$ is the $1$-dimensional moduli space of quadratic rational maps on ${mathbb{P}}^{1}$ with an $n$-periodic critical point. We show that if $G_{n}$ is irreducible over ${mathbb{Q}}$, then $operatorname{Per}_{n}(0)$ is irreducible over ${mathbb{C}}$. To do this, we exhibit a ${mathbb{Q}}$-rational smooth point on a projective completion of $operatorname{Per}_{n}(0)$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $n$, $operatorname{Per}_{n}(0)$ itself has no ${mathbb{Q}}$-rational points.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"50 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We determine the contact mapping class group of the standard contact structures on lens spaces. To prove the main result, we use the one-parametric convex surface theory to classify Legendrian and transverse rational unknots in any tight contact structure on lens spaces up to Legendrian and transverse isotopy.
{"title":"The Contact Mapping Class Group and Rational Unknots in Lens Spaces","authors":"Hyunki Min","doi":"10.1093/imrn/rnae121","DOIUrl":"https://doi.org/10.1093/imrn/rnae121","url":null,"abstract":"We determine the contact mapping class group of the standard contact structures on lens spaces. To prove the main result, we use the one-parametric convex surface theory to classify Legendrian and transverse rational unknots in any tight contact structure on lens spaces up to Legendrian and transverse isotopy.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"38 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Braverman, Finkelberg, and Nakajima define Kac-Moody affine Grassmannian slices as Coulomb branches of $3d$ ${mathcal{N}}=4$ quiver gauge theories and prove that their Coulomb branch construction agrees with the usual loop group definition in finite ADE types. The Coulomb branch construction has good algebraic properties, but its geometry is hard to understand in general. In finite types, an essential geometric feature is that slices embed into one another. We show that these embeddings are compatible with the fundamental monopole operators (FMOs), remarkable regular functions arising from the Coulomb branch construction. Beyond finite type these embeddings were not known, and our second result is to construct them for all symmetric Kac-Moody types. We show that these embeddings respect Poisson structures under a mild “goodness” hypothesis. These results give an affirmative answer to a question posed by Finkelberg in his 2018 ICM address and demonstrate the utility of FMOs in studying the geometry of Kac-Moody affine Grassmannian slices, even in finite types.
{"title":"Fundamental Monopole Operators and Embeddings of Kac-Moody Affine Grassmannian Slices","authors":"Dinakar Muthiah, Alex Weekes","doi":"10.1093/imrn/rnae115","DOIUrl":"https://doi.org/10.1093/imrn/rnae115","url":null,"abstract":"Braverman, Finkelberg, and Nakajima define Kac-Moody affine Grassmannian slices as Coulomb branches of $3d$ ${mathcal{N}}=4$ quiver gauge theories and prove that their Coulomb branch construction agrees with the usual loop group definition in finite ADE types. The Coulomb branch construction has good algebraic properties, but its geometry is hard to understand in general. In finite types, an essential geometric feature is that slices embed into one another. We show that these embeddings are compatible with the fundamental monopole operators (FMOs), remarkable regular functions arising from the Coulomb branch construction. Beyond finite type these embeddings were not known, and our second result is to construct them for all symmetric Kac-Moody types. We show that these embeddings respect Poisson structures under a mild “goodness” hypothesis. These results give an affirmative answer to a question posed by Finkelberg in his 2018 ICM address and demonstrate the utility of FMOs in studying the geometry of Kac-Moody affine Grassmannian slices, even in finite types.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"91 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a Cohen-Dimca-Orlik-type theorem for rank one ${mathbb{Z}}$-local systems on complex hyperplane arrangement complements. This settles a recent conjecture of S. Sugawara.
我们证明了复超平面排列补集上秩一 ${mathbb{Z}}$ 局域系统的科恩-迪姆卡-奥利克型定理。这解决了 S. Sugawara 最近的一个猜想。
{"title":"Cohomology of ℤ-Local Systems on Complex Hyperplane Arrangement Complements","authors":"Yongqiang Liu, Laurenţiu Maxim, Botong Wang","doi":"10.1093/imrn/rnae111","DOIUrl":"https://doi.org/10.1093/imrn/rnae111","url":null,"abstract":"We prove a Cohen-Dimca-Orlik-type theorem for rank one ${mathbb{Z}}$-local systems on complex hyperplane arrangement complements. This settles a recent conjecture of S. Sugawara.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"32 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a family of multi-dimensional Askey–Wilson signed measures. We offer an explicit description of the stationary measure of the open asymmetric simple exclusion process (ASEP) in the full phase diagram, in terms of integrations with respect to these Askey–Wilson signed measures. Using our description, we provide a rigorous derivation of the density profile and limit fluctuations of open ASEP in the entire shock region, including the high and low density phases as well as the coexistence line. This in particular confirms the existing physics postulations of the density profile.
{"title":"Askey–Wilson Signed Measures and Open ASEP in the Shock Region","authors":"Yizao Wang, Jacek Wesołowski, Zongrui Yang","doi":"10.1093/imrn/rnae116","DOIUrl":"https://doi.org/10.1093/imrn/rnae116","url":null,"abstract":"We introduce a family of multi-dimensional Askey–Wilson signed measures. We offer an explicit description of the stationary measure of the open asymmetric simple exclusion process (ASEP) in the full phase diagram, in terms of integrations with respect to these Askey–Wilson signed measures. Using our description, we provide a rigorous derivation of the density profile and limit fluctuations of open ASEP in the entire shock region, including the high and low density phases as well as the coexistence line. This in particular confirms the existing physics postulations of the density profile.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"41 2 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A (non-commutative) Ulam quasimorphism is a map $q$ from a group $Gamma $ to a topological group $G$ such that $q(xy)q(y)^{-1}q(x)^{-1}$ belongs to a fixed compact subset of $G$. Generalizing the construction of Barge and Ghys, we build a family of quasimorphisms on a fundamental group of a closed manifold $M$ of negative sectional curvature, taking values in an arbitrary Lie group. This construction, which generalizes the Barge-Ghys quasimorphisms, associates a quasimorphism to any principal $G$-bundle with connection on $M$. Kapovich and Fujiwara have shown that all quasimorphisms taking values in a discrete group can be constructed from group homomorphisms and quasimorphisms taking values in a commutative group. We construct Barge-Ghys type quasimorphisms taking prescribed values on a given subset in $Gamma $, producing counterexamples to the Kapovich and Fujiwara theorem for quasimorphisms taking values in a Lie group. Our construction also generalizes a result proven by D. Kazhdan in his paper “On $varepsilon $-representations”. Kazhdan has proved that for any $varepsilon>0$, there exists an $varepsilon $-representation of the fundamental group of a Riemann surface of genus 2 which cannot be $1/10$-approximated by a representation. We generalize his result by constructing an $varepsilon $-representation of the fundamental group of a closed manifold of negative sectional curvature taking values in an arbitrary Lie group.
{"title":"Non-commutative Barge-Ghys Quasimorphisms","authors":"Michael Brandenbursky, Misha Verbitsky","doi":"10.1093/imrn/rnae119","DOIUrl":"https://doi.org/10.1093/imrn/rnae119","url":null,"abstract":"A (non-commutative) Ulam quasimorphism is a map $q$ from a group $Gamma $ to a topological group $G$ such that $q(xy)q(y)^{-1}q(x)^{-1}$ belongs to a fixed compact subset of $G$. Generalizing the construction of Barge and Ghys, we build a family of quasimorphisms on a fundamental group of a closed manifold $M$ of negative sectional curvature, taking values in an arbitrary Lie group. This construction, which generalizes the Barge-Ghys quasimorphisms, associates a quasimorphism to any principal $G$-bundle with connection on $M$. Kapovich and Fujiwara have shown that all quasimorphisms taking values in a discrete group can be constructed from group homomorphisms and quasimorphisms taking values in a commutative group. We construct Barge-Ghys type quasimorphisms taking prescribed values on a given subset in $Gamma $, producing counterexamples to the Kapovich and Fujiwara theorem for quasimorphisms taking values in a Lie group. Our construction also generalizes a result proven by D. Kazhdan in his paper “On $varepsilon $-representations”. Kazhdan has proved that for any $varepsilon>0$, there exists an $varepsilon $-representation of the fundamental group of a Riemann surface of genus 2 which cannot be $1/10$-approximated by a representation. We generalize his result by constructing an $varepsilon $-representation of the fundamental group of a closed manifold of negative sectional curvature taking values in an arbitrary Lie group.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"41 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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