— To each finite-dimensional representation of a simple Lie algebra is associated a multiplicative graph in the sense of Kerov and Vershik defined from the decomposition of its tensor powers into irreducible components. It was shown in [11] and [12] that the conditioning of natural random Littelmann paths to stay in their corresponding Weyl chamber is controlled by central measures on this type of graphs. Using the K-theory of associated C∗-algebras, Handelman [8] established a homeomorphism between the set of central measures on these multiplicative graphs and the weight polytope of the underlying representation. In the present paper, we make explicit this homeomorphism independently of Handelman’s results by using Littelmann’s path model. As a by-product we also get an explicit parametrization of the weight polytope in terms of drifts of random Littelmann paths. This explicit parametrization yields a complete description of harmonic and c-harmonic functions for the Littelmann path model describing the iterated tensor product of an irreducible representation. Résumé. — Nous associons un graphe multiplicatif au sens de Vershik et Kerov à chaque représentation de dimension finie d’une algèbre de Lie simple en considérant la décomposition de ses produits tensoriels successifs en représentations irréductibles. Pour chacune de ces représentations de dimension finie, il a été montré en [11] et [12] que le conditionnement d’un chemin de Littelmann aléatoire à rester dans la chambre de Weyl est décrit par les mesures centrales sur le graphe multiplicatif associé. En utilisant la K-théorie des C∗-algèbres correspondantes, Handelman a établi un homéomorphisme entre l’ensemble des mesures centrales sur un de ces graphes multiplicatifs et le polytope des poids de la représentation sous-jacente. Dans cet article, nous rendons explicite l’homéomorphisme d’Handelman en utilisant les modèles de chemins de Littelmann. On obtient en conséquence une paramétrisation du polytope des poids en termes de dérives de chemins de Littelmann aléatoires. La paramétrisation explicite donne une description complète des fonctions harmoniques et c-harmoniques pour les modèles de chemins de Littelmann décrivant les itérations de produits tensoriels d’une représentation irréductible.
{"title":"Mesures centrales pour les graphes multiplicatifs, représentations d’algèbres de Lie et polytopes des poids","authors":"Cédric Lecouvey, Pierre Tarrago","doi":"10.5802/AIF.3350","DOIUrl":"https://doi.org/10.5802/AIF.3350","url":null,"abstract":"— To each finite-dimensional representation of a simple Lie algebra is associated a multiplicative graph in the sense of Kerov and Vershik defined from the decomposition of its tensor powers into irreducible components. It was shown in [11] and [12] that the conditioning of natural random Littelmann paths to stay in their corresponding Weyl chamber is controlled by central measures on this type of graphs. Using the K-theory of associated C∗-algebras, Handelman [8] established a homeomorphism between the set of central measures on these multiplicative graphs and the weight polytope of the underlying representation. In the present paper, we make explicit this homeomorphism independently of Handelman’s results by using Littelmann’s path model. As a by-product we also get an explicit parametrization of the weight polytope in terms of drifts of random Littelmann paths. This explicit parametrization yields a complete description of harmonic and c-harmonic functions for the Littelmann path model describing the iterated tensor product of an irreducible representation. Résumé. — Nous associons un graphe multiplicatif au sens de Vershik et Kerov à chaque représentation de dimension finie d’une algèbre de Lie simple en considérant la décomposition de ses produits tensoriels successifs en représentations irréductibles. Pour chacune de ces représentations de dimension finie, il a été montré en [11] et [12] que le conditionnement d’un chemin de Littelmann aléatoire à rester dans la chambre de Weyl est décrit par les mesures centrales sur le graphe multiplicatif associé. En utilisant la K-théorie des C∗-algèbres correspondantes, Handelman a établi un homéomorphisme entre l’ensemble des mesures centrales sur un de ces graphes multiplicatifs et le polytope des poids de la représentation sous-jacente. Dans cet article, nous rendons explicite l’homéomorphisme d’Handelman en utilisant les modèles de chemins de Littelmann. On obtient en conséquence une paramétrisation du polytope des poids en termes de dérives de chemins de Littelmann aléatoires. La paramétrisation explicite donne une description complète des fonctions harmoniques et c-harmoniques pour les modèles de chemins de Littelmann décrivant les itérations de produits tensoriels d’une représentation irréductible.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45165868","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}
{"title":"Corrigendum to “Differentiating the stochastic entropy in negatively curved spaces under conformal changes”","authors":"F. Ledrappier, Lin Shu","doi":"10.5802/AIF.3389","DOIUrl":"https://doi.org/10.5802/AIF.3389","url":null,"abstract":"","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47448192","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}
For the wave equation $partial_t^2-Delta+V$ on $mathbb{R}^d$ with compactly supported, real valued potential $V$, we establish a sharp relation between Sobolev regularity of $V$ and the existence of finite order expansions as $trightarrow 0$ for the relative trace of the wave group.
{"title":"On the trace of the wave group and regularity of potentials","authors":"Hart F. Smith","doi":"10.5802/aif.3538","DOIUrl":"https://doi.org/10.5802/aif.3538","url":null,"abstract":"For the wave equation $partial_t^2-Delta+V$ on $mathbb{R}^d$ with compactly supported, real valued potential $V$, we establish a sharp relation between Sobolev regularity of $V$ and the existence of finite order expansions as $trightarrow 0$ for the relative trace of the wave group.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44092010","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}
In several space dimensions, scalar shock waves between two constant states u $pm$ are not necessarily planar. We describe them in detail. Then we prove their asymptotic stability, assuming that they are uniformly non-characteristic. Our result is conditional for a general flux, while unconditional for the multi-D Burgers equation.
{"title":"Asymptotic stability of scalar multi-D inviscid shock waves","authors":"D. Serre","doi":"10.5802/aif.3569","DOIUrl":"https://doi.org/10.5802/aif.3569","url":null,"abstract":"In several space dimensions, scalar shock waves between two constant states u $pm$ are not necessarily planar. We describe them in detail. Then we prove their asymptotic stability, assuming that they are uniformly non-characteristic. Our result is conditional for a general flux, while unconditional for the multi-D Burgers equation.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43625586","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}
We consider an action of the automorphism group $mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$ on the filtered vector space $A_d(n)$ of Jacobi diagrams of degree $d$ on $n$ oriented arcs. This action induces on the associated graded vector space of $A_d(n)$, which is identified with the space $B_d(n)$ of open Jacobi diagrams, an action of the general linear group $mathrm{GL}(n,Z)$ and an action of the graded Lie algebra of the IA-automorphism group of $F_n$ associated with its lower central series. We use these actions on $B_d(n)$ to study the $mathrm{Aut}(F_n)$-module structure of $A_d(n)$. In particular, we consider the case where $d=2$ in detail and give an indecomposable decomposition of $A_2(n)$. We also construct a polynomial functor $A_d$ of degree $2d$ from the opposite category of the category of finitely generated free groups to the category of filtered vector spaces, which includes the $mathrm{Aut}(F_n)$-module structure of $A_d(n)$ for all $ngeq 0$.
{"title":"Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I","authors":"Mai Katada","doi":"10.5802/aif.3544","DOIUrl":"https://doi.org/10.5802/aif.3544","url":null,"abstract":"We consider an action of the automorphism group $mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$ on the filtered vector space $A_d(n)$ of Jacobi diagrams of degree $d$ on $n$ oriented arcs. This action induces on the associated graded vector space of $A_d(n)$, which is identified with the space $B_d(n)$ of open Jacobi diagrams, an action of the general linear group $mathrm{GL}(n,Z)$ and an action of the graded Lie algebra of the IA-automorphism group of $F_n$ associated with its lower central series. We use these actions on $B_d(n)$ to study the $mathrm{Aut}(F_n)$-module structure of $A_d(n)$. In particular, we consider the case where $d=2$ in detail and give an indecomposable decomposition of $A_2(n)$. We also construct a polynomial functor $A_d$ of degree $2d$ from the opposite category of the category of finitely generated free groups to the category of filtered vector spaces, which includes the $mathrm{Aut}(F_n)$-module structure of $A_d(n)$ for all $ngeq 0$.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47506483","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}
F'elix Cabello S'anchez, Alberto Salguero-Alarc'on
The paper studies short exact sequences of Banach modules over the convolution algebra $L_1=L_1(G)$, where $G$ is a compact abelian group. The main tool is the notion of a nonlinear $L_1$-centralizer, which in combination with the Fourier transform, is used to produce sequences of $L_1$-modules $0rightarrow L_q rightarrow Z rightarrow L_p rightarrow 0$ that are nontrivial as long as the general theory allows it, namely for $pin (1,infty], qin[1,infty)$. Concrete examples are worked in detail for the circle group, with applications to the Hardy classes, and the Cantor group.
{"title":"When Kalton and Peck met Fourier","authors":"F'elix Cabello S'anchez, Alberto Salguero-Alarc'on","doi":"10.5802/aif.3562","DOIUrl":"https://doi.org/10.5802/aif.3562","url":null,"abstract":"The paper studies short exact sequences of Banach modules over the convolution algebra $L_1=L_1(G)$, where $G$ is a compact abelian group. The main tool is the notion of a nonlinear $L_1$-centralizer, which in combination with the Fourier transform, is used to produce sequences of $L_1$-modules $0rightarrow L_q rightarrow Z rightarrow L_p rightarrow 0$ that are nontrivial as long as the general theory allows it, namely for $pin (1,infty], qin[1,infty)$. Concrete examples are worked in detail for the circle group, with applications to the Hardy classes, and the Cantor group.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42120332","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}
— Based on our previous work we develop a stationary scattering theory for the Schrödinger operator on a manifold possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends. Scattering by obstacles, possibly non-smooth and/or unbounded in a certain manner, is included in the theory. We develop the theory largely along the classical lines of Jäger, Saitō and Constantin, and derive in particular WKB-asymptotics of minimal generalized eigenfunctions. As an application we prove a conjecture of Hempel, Post and Weder on cross-ends transmissions in its natural and strong form within the framework of our theory. Résumé. — Sur la base de nos travaux antérieurs, nous développons une théorie stationnaire de la diffusion pour l’opérateur de Schrödinger sur une variété possédant une fonction d’échappement. Une classe particulière d’exemples sont les variétés à extrémités euclidiennes et/ou hyperboliques. La diffusion par des obstacles, éventuellement non lisses et/ou non bornés d’une certaine manière, est incluse dans la théorie. Nous développons la théorie en grande partie selon les idées classiques de Jäger, Saitō et Constantin, et dérivons en particulier les asymptotiques WKB des fonctions propres généralisées minimales. Comme application, nous prouvons une conjecture de Hempel, Post et Weder sur les transmissions transversales sous sa forme naturelle et forte dans le cadre de notre théorie.
—前估算our work we开发了文娱scattering theory for the Schr em五个接线生据流形an escape的功能。A具有class of实例are with Euclidean and / or流形hyperbolic end。= =地理= =根据美国人口普查，这个县的面积为，其中土地面积为。(We the theory largely不得不“马勒”lines of Jōger,知宗(Constantin,最小的谈话WKB-asymptotics上拖累》generalized eigenfunctions。prove we As an应用a Post and Weder猜想of Hempel、变速箱cross-ends in its natural and strong form within the framework of our theory)。摘要。-在我们之前工作的基础上，我们发展了schrodinger算子在具有排气函数的流形上的稳定扩散理论。一类特殊的例子是具有欧几里得和/或双曲末端的变种。通过障碍物的扩散，可能是非平滑的和/或在某种程度上不受限制的，包括在理论中。我们一般是按照常规思维理论[Jōger,知和Constantin,尤其是稳当的发生普遍wbk自身功能的最低。作为一个应用，我们在我们的理论框架内证明了亨佩尔、波斯特和维德关于横向传输的自然而有力的猜想。
{"title":"Stationary scattering theory on manifolds","authors":"K. Ito, E. Skibsted","doi":"10.5802/aif.3417","DOIUrl":"https://doi.org/10.5802/aif.3417","url":null,"abstract":"— Based on our previous work we develop a stationary scattering theory for the Schrödinger operator on a manifold possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends. Scattering by obstacles, possibly non-smooth and/or unbounded in a certain manner, is included in the theory. We develop the theory largely along the classical lines of Jäger, Saitō and Constantin, and derive in particular WKB-asymptotics of minimal generalized eigenfunctions. As an application we prove a conjecture of Hempel, Post and Weder on cross-ends transmissions in its natural and strong form within the framework of our theory. Résumé. — Sur la base de nos travaux antérieurs, nous développons une théorie stationnaire de la diffusion pour l’opérateur de Schrödinger sur une variété possédant une fonction d’échappement. Une classe particulière d’exemples sont les variétés à extrémités euclidiennes et/ou hyperboliques. La diffusion par des obstacles, éventuellement non lisses et/ou non bornés d’une certaine manière, est incluse dans la théorie. Nous développons la théorie en grande partie selon les idées classiques de Jäger, Saitō et Constantin, et dérivons en particulier les asymptotiques WKB des fonctions propres généralisées minimales. Comme application, nous prouvons une conjecture de Hempel, Post et Weder sur les transmissions transversales sous sa forme naturelle et forte dans le cadre de notre théorie.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71209712","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}
We show that the real Cremona group of the plane is a nontrivial amalgam of two groups amalgamated along their intersection and give an alternative proof of its abelianisation.
{"title":"The real plane Cremona group is an amalgamated product","authors":"Susanna Zimmermann","doi":"10.5802/aif.3415","DOIUrl":"https://doi.org/10.5802/aif.3415","url":null,"abstract":"We show that the real Cremona group of the plane is a nontrivial amalgam of two groups amalgamated along their intersection and give an alternative proof of its abelianisation.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71209832","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}
We analyze the Moyal star product in deformation quantization from the resurgence theory perspective. By putting algebraic conditions on Borel transforms, one can define the space of ``algebro-resurgent series'' (a subspace of $1$-Gevrey formal series in $ihbar/2$ with coefficients in $C{q,p}$), which we show is stable under Moyal star product.
{"title":"On the Moyal Star Product of Resurgent Series","authors":"Yong Li, D. Sauzin, Shanzhong Sun","doi":"10.5802/aif.3565","DOIUrl":"https://doi.org/10.5802/aif.3565","url":null,"abstract":"We analyze the Moyal star product in deformation quantization from the resurgence theory perspective. By putting algebraic conditions on Borel transforms, one can define the space of ``algebro-resurgent series'' (a subspace of $1$-Gevrey formal series in $ihbar/2$ with coefficients in $C{q,p}$), which we show is stable under Moyal star product.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43157496","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}
Cylinders in projective varieties play an important role in connection with unipotent group actions on certain affine algebraic varieties. The previous work due to Dubouloz and Kishimoto deals with the condition for a del Pezzo fibration to contain a vertical cylinder. In the present work, as a generalization in the sense of singularities, we shall determine the condition under which a del Pezzo fibration with canonical singularities admits a vertical cylinder by means of degree and type of singularities found on the corresponding generic fiber.
{"title":"Cylinders in canonical del Pezzo fibrations","authors":"Masatomo Sawahara","doi":"10.5802/aif.3573","DOIUrl":"https://doi.org/10.5802/aif.3573","url":null,"abstract":"Cylinders in projective varieties play an important role in connection with unipotent group actions on certain affine algebraic varieties. The previous work due to Dubouloz and Kishimoto deals with the condition for a del Pezzo fibration to contain a vertical cylinder. In the present work, as a generalization in the sense of singularities, we shall determine the condition under which a del Pezzo fibration with canonical singularities admits a vertical cylinder by means of degree and type of singularities found on the corresponding generic fiber.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46096552","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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