Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $mathbb T$, we proved in a previous work}, see also the paper of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal $mathfrak m$ of $mathbb T$, appear to be free. In this work, we obtain the same result for $mathfrak m$ such that its associated galoisian $overline{mathbb F}_l$-representation $overline{rho_{mathfrak m}}$ is irreducible.
{"title":"Galois irreducibility implies cohomology freeness for KHT Shimura varieties","authors":"P. Boyer","doi":"10.5802/jep.216","DOIUrl":"https://doi.org/10.5802/jep.216","url":null,"abstract":"Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $mathbb T$, \u0000we proved in a previous work}, see also the paper of Caraiani-Scholze for other PEL Shimura \u0000varieties, that its localized cohomology groups at a generic maximal ideal $mathfrak m$ of \u0000$mathbb T$, appear to be free. \u0000In this work, we obtain the same result for $mathfrak m$ such that its associated \u0000galoisian $overline{mathbb F}_l$-representation $overline{rho_{mathfrak m}}$ is irreducible.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131489587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Homeomorphism groups of generalized Wa.zewski dendrites act on the infinite countable set of branch points of the dendrite and thus have a nice Polish topology. In this paper, we study them in the light of this Polish topology. The group of the universal Wa.zewski dendrite $D_infty$ is more characteristic than the others because it is the unique one with a dense conjugacy class. For this group $G_infty$, we show some of its topological properties like existence of a comeager conjugacy class, the Steinhaus property, automatic continuity and the small index subgroup property. Moreover, we identify the universal minimal flow of $G_infty$. This allows us to prove that point-stabilizers in $G_infty$ are amenable and to describe the universal Furstenberg boundary of $G_infty$.
{"title":"Topological properties of Ważewski dendrite groups","authors":"Bruno Duchesne","doi":"10.5802/jep.121","DOIUrl":"https://doi.org/10.5802/jep.121","url":null,"abstract":"Homeomorphism groups of generalized Wa.zewski dendrites act on the infinite countable set of branch points of the dendrite and thus have a nice Polish topology. In this paper, we study them in the light of this Polish topology. The group of the universal Wa.zewski dendrite $D_infty$ is more characteristic than the others because it is the unique one with a dense conjugacy class. For this group $G_infty$, we show some of its topological properties like existence of a comeager conjugacy class, the Steinhaus property, automatic continuity and the small index subgroup property. Moreover, we identify the universal minimal flow of $G_infty$. This allows us to prove that point-stabilizers in $G_infty$ are amenable and to describe the universal Furstenberg boundary of $G_infty$.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128649330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper introduces a new method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also known as Stasheff polytopes. We show that it satisfies an easy-to-state cellular formula. For the first time, we endow a family of realizations of the associahedra (the Loday realizations) with a topological and cellular operad structure; it is shown to be compatible with the diagonal maps.
{"title":"The diagonal of the associahedra","authors":"N. Masuda, H. Thomas, A. Tonks, B. Vallette","doi":"10.5802/jep.142","DOIUrl":"https://doi.org/10.5802/jep.142","url":null,"abstract":"This paper introduces a new method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also known as Stasheff polytopes. We show that it satisfies an easy-to-state cellular formula. For the first time, we endow a family of realizations of the associahedra (the Loday realizations) with a topological and cellular operad structure; it is shown to be compatible with the diagonal maps.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128823073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Orbital functions and heat kernels of Kleinian groups","authors":"Adrien Boulanger","doi":"10.5802/jep.200","DOIUrl":"https://doi.org/10.5802/jep.200","url":null,"abstract":"We study orbital functions associated to Kleinian groups through the heat kernel approach developed in cite{artmoiheatcounting1}.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123272283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Olivier Glorieux, Daniel Monclair, Nicolas Tholozan
We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in $mathbf{P}(mathbb{R}^{n}) times mathbf{P}({mathbb{R}^{n}}^*)$ is bounded between two critical exponents associated respectively to a highest weight and a simple root.
{"title":"Hausdorff dimension of limit sets for projective Anosov representations","authors":"Olivier Glorieux, Daniel Monclair, Nicolas Tholozan","doi":"10.5802/jep.241","DOIUrl":"https://doi.org/10.5802/jep.241","url":null,"abstract":"We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in $mathbf{P}(mathbb{R}^{n}) times mathbf{P}({mathbb{R}^{n}}^*)$ is bounded between two critical exponents associated respectively to a highest weight and a simple root.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"129 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132778414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We continue our study of the problem of mixing for a class of PDEs with very degenerate noise. As we established earlier, the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric holds under the hypothesis that the unperturbed equation has exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem. The proof uses a new idea that reduces the verification of a stability property to the investigation of a conditional random walk.
{"title":"Mixing via controllability for randomly forced nonlinear dissipative PDEs","authors":"S. Kuksin, V. Nersesyan, A. Shirikyan","doi":"10.5802/jep.130","DOIUrl":"https://doi.org/10.5802/jep.130","url":null,"abstract":"We continue our study of the problem of mixing for a class of PDEs with very degenerate noise. As we established earlier, the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric holds under the hypothesis that the unperturbed equation has exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem. The proof uses a new idea that reduces the verification of a stability property to the investigation of a conditional random walk.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114414277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we give an alternative proof of Schreiber's theorem which says that an infinite discrete approximate subgroup in $mathbb{R}^d$ is relatively dense around a subspace. We also deduce from Schreiber's theorem two new results. The first one says that any infinite discrete approximate subgroup in $mathbb{R}^d$ is a restriction of a Meyer set to a thickening of a linear subspace in $mathbb{R}^d$, and the second one provides an extension of Schreiber's theorem to the case of the Heisenberg group.
{"title":"Extensions of Schreiber’s theorem on discrete approximate subgroups in $protect mathbb{R}^d$","authors":"A. Fish","doi":"10.5802/JEP.90","DOIUrl":"https://doi.org/10.5802/JEP.90","url":null,"abstract":"In this paper we give an alternative proof of Schreiber's theorem which says that an infinite discrete approximate subgroup in $mathbb{R}^d$ is relatively dense around a subspace. We also deduce from Schreiber's theorem two new results. The first one says that any infinite discrete approximate subgroup in $mathbb{R}^d$ is a restriction of a Meyer set to a thickening of a linear subspace in $mathbb{R}^d$, and the second one provides an extension of Schreiber's theorem to the case of the Heisenberg group.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125521224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite--dimensional representations of a quantum affine algebra of type $A$. We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type $A$ with coefficients.
{"title":"Tensor products and $q$-characters of HL-modules and monoidal categorifications","authors":"Matheus Brito, Vyjayanthi Chari","doi":"10.5802/jep.101","DOIUrl":"https://doi.org/10.5802/jep.101","url":null,"abstract":"We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite--dimensional representations of a quantum affine algebra of type $A$. We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type $A$ with coefficients.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"105 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133267970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we construct a bar-cobar adjunction and a Koszul duality theory for protoperads, which are an operadic type notion encoding faithfully some categories of bialgebras with diagonal symmetries, like double Lie algebras (DLie). We give a criterion to show that a binary quadratic protoperad is Koszul and we apply it successfully to the protoperad DLie. As a corollary, we deduce that the properad DPois which encodes double Poisson algebras is Koszul. This allows us to describe the homotopy properties of double Poisson algebras which play a key role in non commutative geometry.
{"title":"Protoperads II: Koszul duality","authors":"J. Leray","doi":"10.5802/jep.131","DOIUrl":"https://doi.org/10.5802/jep.131","url":null,"abstract":"In this paper, we construct a bar-cobar adjunction and a Koszul duality theory for protoperads, which are an operadic type notion encoding faithfully some categories of bialgebras with diagonal symmetries, like double Lie algebras (DLie). We give a criterion to show that a binary quadratic protoperad is Koszul and we apply it successfully to the protoperad DLie. As a corollary, we deduce that the properad DPois which encodes double Poisson algebras is Koszul. This allows us to describe the homotopy properties of double Poisson algebras which play a key role in non commutative geometry.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124825361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Debarre-Voisin hyperk"ahler fourfolds are built from alternating $3$-forms on a $10$-dimensional complex vector space, which we call trivectors. They are analogous to the Beauville-Donagi fourfolds associated with cubic fourfolds. In this article, we study several trivectors whose associated Debarre-Voisin variety is degenerate, in the sense that it is either reducible or has excessive dimension. We show that the Debarre-Voisin varieties specialize, along general $1$-parameter degenerations to these trivectors, to varieties isomorphic or birationally isomorphic to the Hilbert square of a K3 surface.
{"title":"Hilbert squares of K3 surfaces and Debarre-Voisin varieties","authors":"O. Debarre, Fr'ed'eric Han, K. O’Grady, C. Voisin","doi":"10.5802/jep.125","DOIUrl":"https://doi.org/10.5802/jep.125","url":null,"abstract":"The Debarre-Voisin hyperk\"ahler fourfolds are built from alternating $3$-forms on a $10$-dimensional complex vector space, which we call trivectors. They are analogous to the Beauville-Donagi fourfolds associated with cubic fourfolds. In this article, we study several trivectors whose associated Debarre-Voisin variety is degenerate, in the sense that it is either reducible or has excessive dimension. We show that the Debarre-Voisin varieties specialize, along general $1$-parameter degenerations to these trivectors, to varieties isomorphic or birationally isomorphic to the Hilbert square of a K3 surface.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124493462","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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