We prove the convergence of (solid) ellipsoids to a Gaussian space in Gromov's concentration/weak topology as the dimension diverges to infinity. This gives the first discovered example of an irreducible nontrivial convergent sequence in the concentration topology, where ‘irreducible nontrivial’ roughly means to be not constructed from Lévy families nor box convergent sequences.
{"title":"High-dimensional ellipsoids converge to Gaussian spaces","authors":"Daisuke Kazukawa, Takashi Shioya","doi":"10.2969/jmsj/86648664","DOIUrl":"https://doi.org/10.2969/jmsj/86648664","url":null,"abstract":"We prove the convergence of (solid) ellipsoids to a Gaussian space in Gromov's concentration/weak topology as the dimension diverges to infinity. This gives the first discovered example of an irreducible nontrivial convergent sequence in the concentration topology, where ‘irreducible nontrivial’ roughly means to be not constructed from Lévy families nor box convergent sequences.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135944107","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 this paper, we study the parabolic BGG categories for graded Lie superalgebras of Cartan type over the field of complex numbers. The gradation of such a Lie superalgebra $mathfrak{g}$ naturally arises, with the zero component $mathfrak{g}_{0}$ being a reductive Lie algebra. We first show that there are only two proper parabolic subalgebras containing Levi subalgebra $mathfrak{g}_{0}$: the “maximal one” $mathsf{P}_{max}$ and the “minimal one” $mathsf{P}_{min}$. Furthermore, the parabolic BGG category arising from $mathsf{P}_{max}$ essentially turns out to be a subcategory of the one arising from $mathsf{P}_{min}$. Such a priority of $mathsf{P}_{min}$ in the sense of representation theory reduces the question to the study of the “minimal parabolic” BGG category $mathcal{O}^{min}$ associated with $mathsf{P}_{min}$. We prove the existence of projective covers of simple objects in these categories, which enables us to establish a satisfactory block theory. Most notably, our main results are as follows. (1) We classify and obtain a precise description of the blocks of $mathcal{O}^{min}$. (2) We investigate indecomposable tilting and indecomposable projective modules in $mathcal{O}^{min}$, and compute their character formulas.
{"title":"Parabolic BGG categories and their block decomposition for Lie superalgebras of Cartan type","authors":"Fei-Fei DUAN, Bin SHU, Yu-Feng YAO","doi":"10.2969/jmsj/90439043","DOIUrl":"https://doi.org/10.2969/jmsj/90439043","url":null,"abstract":"In this paper, we study the parabolic BGG categories for graded Lie superalgebras of Cartan type over the field of complex numbers. The gradation of such a Lie superalgebra $mathfrak{g}$ naturally arises, with the zero component $mathfrak{g}_{0}$ being a reductive Lie algebra. We first show that there are only two proper parabolic subalgebras containing Levi subalgebra $mathfrak{g}_{0}$: the “maximal one” $mathsf{P}_{max}$ and the “minimal one” $mathsf{P}_{min}$. Furthermore, the parabolic BGG category arising from $mathsf{P}_{max}$ essentially turns out to be a subcategory of the one arising from $mathsf{P}_{min}$. Such a priority of $mathsf{P}_{min}$ in the sense of representation theory reduces the question to the study of the “minimal parabolic” BGG category $mathcal{O}^{min}$ associated with $mathsf{P}_{min}$. We prove the existence of projective covers of simple objects in these categories, which enables us to establish a satisfactory block theory. Most notably, our main results are as follows. (1) We classify and obtain a precise description of the blocks of $mathcal{O}^{min}$. (2) We investigate indecomposable tilting and indecomposable projective modules in $mathcal{O}^{min}$, and compute their character formulas.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136210218","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 a continuum percolation built over stationary ergodic point processes. Assuming that the occupied region has a unique unbounded cluster and the cluster satisfies volume regularity and isoperimetric condition, we prove a quenched invariance principle for reflecting diffusions on the cluster.
{"title":"Quenched invariance principle for a reflecting diffusion in a continuum percolation cluster","authors":"Yutaka TAKEUCHI","doi":"10.2969/jmsj/89198919","DOIUrl":"https://doi.org/10.2969/jmsj/89198919","url":null,"abstract":"We consider a continuum percolation built over stationary ergodic point processes. Assuming that the occupied region has a unique unbounded cluster and the cluster satisfies volume regularity and isoperimetric condition, we prove a quenched invariance principle for reflecting diffusions on the cluster.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695911","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}
An element of a Weyl group of classical type is skew if it is the left factor in a reduced factorization of a Grassmannian element. The skew Grothendieck polynomials are those which are indexed by skew elements of the Weyl group. We define set-valued tableaux which are fillings of the associated skew Young diagrams and use them to prove tableau formulas for the skew double Grothendieck polynomials in all four classical Lie types. We deduce tableau formulas for the Grassmannian Grothendieck polynomials and the $K$-theoretic analogues of the (double mixed) skew Stanley functions in the respective Lie types.
{"title":"Tableau formulas for skew Grothendieck polynomials","authors":"Harry TAMVAKIS","doi":"10.2969/jmsj/89928992","DOIUrl":"https://doi.org/10.2969/jmsj/89928992","url":null,"abstract":"An element of a Weyl group of classical type is skew if it is the left factor in a reduced factorization of a Grassmannian element. The skew Grothendieck polynomials are those which are indexed by skew elements of the Weyl group. We define set-valued tableaux which are fillings of the associated skew Young diagrams and use them to prove tableau formulas for the skew double Grothendieck polynomials in all four classical Lie types. We deduce tableau formulas for the Grassmannian Grothendieck polynomials and the $K$-theoretic analogues of the (double mixed) skew Stanley functions in the respective Lie types.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135536861","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 this paper, we establish a refined transversality theorem on linear perturbations from a new perspective of Hausdorff measures. Furthermore, we give its applications not only to singularity theory but also to multiobjective optimization.
{"title":"A refined transversality theorem on linear perturbations and its applications","authors":"Shunsuke ICHIKI","doi":"10.2969/jmsj/90559055","DOIUrl":"https://doi.org/10.2969/jmsj/90559055","url":null,"abstract":"In this paper, we establish a refined transversality theorem on linear perturbations from a new perspective of Hausdorff measures. Furthermore, we give its applications not only to singularity theory but also to multiobjective optimization.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135436906","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":"Generalized Bott–Cattaneo–Rossi invariants in terms of Alexander polynomials","authors":"David Leturcq","doi":"10.2969/jmsj/88168816","DOIUrl":"https://doi.org/10.2969/jmsj/88168816","url":null,"abstract":"","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44669527","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}
T. Shibuya, Tatsuya Tsukamoto, Y. Uchida, T. Ishikawa
{"title":"Characterizations of pretzel knots which are simple-ribbon","authors":"T. Shibuya, Tatsuya Tsukamoto, Y. Uchida, T. Ishikawa","doi":"10.2969/jmsj/89438943","DOIUrl":"https://doi.org/10.2969/jmsj/89438943","url":null,"abstract":"","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45063042","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}
The region select game, introduced by Ayaka Shimizu, Akio Kawauchi and Kengo Kishimoto, is a game that is played on knot diagrams whose crossings are endowed with two colors. The game is based on the region crossing change moves that induce an unknotting operation on knot diagrams. We generalize the region select game to be played on a knot diagram endowed with $k$-colors at its vertices for $2 leq k leq infty$.
清水绫香、川内昭夫和岸本健吾推出的区域选择游戏，是一种以结图为基础的游戏，结图的交叉点被赋予了两种颜色。该游戏是基于区域交叉变化的移动，诱导结图上的解结操作。我们将区域选择游戏推广到一个结点图上，在$2 leq k leq infty$的顶点上赋予$k$ -颜色。
{"title":"$k$-color region select game","authors":"Ahmet BATAL, Neslihan GÜGÜMCÜ","doi":"10.2969/jmsj/89408940","DOIUrl":"https://doi.org/10.2969/jmsj/89408940","url":null,"abstract":"The region select game, introduced by Ayaka Shimizu, Akio Kawauchi and Kengo Kishimoto, is a game that is played on knot diagrams whose crossings are endowed with two colors. The game is based on the region crossing change moves that induce an unknotting operation on knot diagrams. We generalize the region select game to be played on a knot diagram endowed with $k$-colors at its vertices for $2 leq k leq infty$.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136380014","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":"On the Fourier coefficients of Siegel Eisenstein series of degree 3 of an odd prime level with the quadratic character","authors":"Keiichi Gunji","doi":"10.2969/jmsj/88988898","DOIUrl":"https://doi.org/10.2969/jmsj/88988898","url":null,"abstract":"","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41761840","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":"A topological product decomposition of Busemann space","authors":"Tomohiro Fukaya","doi":"10.2969/jmsj/89738973","DOIUrl":"https://doi.org/10.2969/jmsj/89738973","url":null,"abstract":"","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42672394","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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