Pub Date : 2023-09-15DOI: 10.1007/s00029-023-00877-4
Guglielmo Nocera, Marco Volpe
Abstract The notion of conically smooth structure on a stratified space was introduced by Ayala, Francis and Tanaka. This is a very well behaved analogue of a differential structure in the context of stratified topological spaces, satisfying good properties such as the existence of resolutions of singularities and handlebody decompositions. In this paper we prove Ayala, Francis and Tanaka’s conjecture that any Whitney stratified space admits a canonical conically smooth structure. We thus establish a connection between the theory of conically smooth spaces and the classical examples of stratified spaces from differential topology.
{"title":"Whitney stratifications are conically smooth","authors":"Guglielmo Nocera, Marco Volpe","doi":"10.1007/s00029-023-00877-4","DOIUrl":"https://doi.org/10.1007/s00029-023-00877-4","url":null,"abstract":"Abstract The notion of conically smooth structure on a stratified space was introduced by Ayala, Francis and Tanaka. This is a very well behaved analogue of a differential structure in the context of stratified topological spaces, satisfying good properties such as the existence of resolutions of singularities and handlebody decompositions. In this paper we prove Ayala, Francis and Tanaka’s conjecture that any Whitney stratified space admits a canonical conically smooth structure. We thus establish a connection between the theory of conically smooth spaces and the classical examples of stratified spaces from differential topology.","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135353865","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}
Pub Date : 2023-07-31DOI: 10.1007/s00029-023-00863-w
O. Pochinka, Valeriya I. Shmukler, E. Talanova
{"title":"Bifurcation of a disappearance of a non-compact heteroclinic curve","authors":"O. Pochinka, Valeriya I. Shmukler, E. Talanova","doi":"10.1007/s00029-023-00863-w","DOIUrl":"https://doi.org/10.1007/s00029-023-00863-w","url":null,"abstract":"","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":"32 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80001791","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}
Pub Date : 2023-04-20DOI: 10.1007/s00029-023-00842-1
T. Heidersdorf, R. Weissauer
{"title":"On classical tensor categories attached to the irreducible representations of the general linear supergroups GL(n|n)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setle","authors":"T. Heidersdorf, R. Weissauer","doi":"10.1007/s00029-023-00842-1","DOIUrl":"https://doi.org/10.1007/s00029-023-00842-1","url":null,"abstract":"","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":"34 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79767518","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}
Pub Date : 2023-01-16DOI: 10.1007/s00029-022-00824-9
W. Kucharz, Krzysztof Kurdyka
{"title":"Analytic functions and Nash functions along curves","authors":"W. Kucharz, Krzysztof Kurdyka","doi":"10.1007/s00029-022-00824-9","DOIUrl":"https://doi.org/10.1007/s00029-022-00824-9","url":null,"abstract":"","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":"16 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88206653","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}
Pub Date : 2023-01-01Epub Date: 2022-11-17DOI: 10.1007/s00029-022-00805-y
Elie Casbi, Jian-Rong Li
<p><p>For any simply-laced type simple Lie algebra <math><mi>g</mi></math> and any height function <math><mi>ξ</mi></math> adapted to an orientation <i>Q</i> of the Dynkin diagram of <math><mi>g</mi></math> , Hernandez-Leclerc introduced a certain category <math> <msup><mrow><mi>C</mi></mrow> <mrow><mo>≤</mo> <mi>ξ</mi></mrow> </msup> </math> of representations of the quantum affine algebra <math> <mrow><msub><mi>U</mi> <mi>q</mi></msub> <mrow><mo>(</mo> <mover><mi>g</mi> <mo>^</mo></mover> <mo>)</mo></mrow> </mrow> </math> , as well as a subcategory <math><msub><mi>C</mi> <mi>Q</mi></msub> </math> of <math> <msup><mrow><mi>C</mi></mrow> <mrow><mo>≤</mo> <mi>ξ</mi></mrow> </msup> </math> whose complexified Grothendieck ring is isomorphic to the coordinate ring <math><mrow><mi>C</mi> <mo>[</mo> <mi>N</mi> <mo>]</mo></mrow> </math> of a maximal unipotent subgroup. In this paper, we define an algebraic morphism <math> <msub><mover><mi>D</mi> <mo>~</mo></mover> <mi>ξ</mi></msub> </math> on a torus <math> <msup><mrow><mi>Y</mi></mrow> <mrow><mo>≤</mo> <mi>ξ</mi></mrow> </msup> </math> containing the image of <math> <mrow><msub><mi>K</mi> <mn>0</mn></msub> <mrow><mo>(</mo> <msup><mrow><mi>C</mi></mrow> <mrow><mo>≤</mo> <mi>ξ</mi></mrow> </msup> <mo>)</mo></mrow> </mrow> </math> under the truncated <i>q</i>-character morphism. We prove that the restriction of <math> <msub><mover><mi>D</mi> <mo>~</mo></mover> <mi>ξ</mi></msub> </math> to <math> <mrow><msub><mi>K</mi> <mn>0</mn></msub> <mrow><mo>(</mo> <msub><mi>C</mi> <mi>Q</mi></msub> <mo>)</mo></mrow> </mrow> </math> coincides with the morphism <math><mover><mi>D</mi> <mo>¯</mo></mover> </math> recently introduced by Baumann-Kamnitzer-Knutson in their study of equivariant multiplicities of Mirković-Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov-Reshetikhin modules in <math><msub><mi>C</mi> <mi>Q</mi></msub> </math> , as well as certain results by Brundan-Kleshchev-McNamara on the representation theory of quiver Hecke algebras. This alternative description of <math><mover><mi>D</mi> <mo>¯</mo></mover> </math> allows us to prove a conjecture by the first author on the distinguished values of <math><mover><mi>D</mi> <mo>¯</mo></mover> </math> on the flag minors of <math><mrow><mi>C</mi> <mo>[</mo> <mi>N</mi> <mo>]</mo></mrow> </math> . We also provide applications of our results from the perspective of Kang-Kashiwara-Kim-Oh's generalized Schur-Weyl duality. Finally, we use Kashiwara-Kim-Oh-Park's recent constructions to define a cluster algebra <math> <msub><mover><mi>A</mi> <mo>¯</mo></mover> <mi>Q</mi></msub> </math> as a subquotient of <math> <mrow><msub><mi>K</mi> <mn>0</mn></msub> <mrow><mo>(</mo> <msup><mrow><mi>C</mi></mrow> <mrow><mo>≤</mo> <mi>ξ</mi></mrow> </msup> <mo>)</mo></mrow> </mrow> </math> naturally containing <math><mrow><mi>C</mi> <mo>[</mo> <mi>N</mi> <mo>]</mo></mrow> </math> , and suggest the existence of an analogue of the Mirković-Vilo
{"title":"Equivariant multiplicities via representations of quantum affine algebras.","authors":"Elie Casbi, Jian-Rong Li","doi":"10.1007/s00029-022-00805-y","DOIUrl":"https://doi.org/10.1007/s00029-022-00805-y","url":null,"abstract":"<p><p>For any simply-laced type simple Lie algebra <math><mi>g</mi></math> and any height function <math><mi>ξ</mi></math> adapted to an orientation <i>Q</i> of the Dynkin diagram of <math><mi>g</mi></math> , Hernandez-Leclerc introduced a certain category <math> <msup><mrow><mi>C</mi></mrow> <mrow><mo>≤</mo> <mi>ξ</mi></mrow> </msup> </math> of representations of the quantum affine algebra <math> <mrow><msub><mi>U</mi> <mi>q</mi></msub> <mrow><mo>(</mo> <mover><mi>g</mi> <mo>^</mo></mover> <mo>)</mo></mrow> </mrow> </math> , as well as a subcategory <math><msub><mi>C</mi> <mi>Q</mi></msub> </math> of <math> <msup><mrow><mi>C</mi></mrow> <mrow><mo>≤</mo> <mi>ξ</mi></mrow> </msup> </math> whose complexified Grothendieck ring is isomorphic to the coordinate ring <math><mrow><mi>C</mi> <mo>[</mo> <mi>N</mi> <mo>]</mo></mrow> </math> of a maximal unipotent subgroup. In this paper, we define an algebraic morphism <math> <msub><mover><mi>D</mi> <mo>~</mo></mover> <mi>ξ</mi></msub> </math> on a torus <math> <msup><mrow><mi>Y</mi></mrow> <mrow><mo>≤</mo> <mi>ξ</mi></mrow> </msup> </math> containing the image of <math> <mrow><msub><mi>K</mi> <mn>0</mn></msub> <mrow><mo>(</mo> <msup><mrow><mi>C</mi></mrow> <mrow><mo>≤</mo> <mi>ξ</mi></mrow> </msup> <mo>)</mo></mrow> </mrow> </math> under the truncated <i>q</i>-character morphism. We prove that the restriction of <math> <msub><mover><mi>D</mi> <mo>~</mo></mover> <mi>ξ</mi></msub> </math> to <math> <mrow><msub><mi>K</mi> <mn>0</mn></msub> <mrow><mo>(</mo> <msub><mi>C</mi> <mi>Q</mi></msub> <mo>)</mo></mrow> </mrow> </math> coincides with the morphism <math><mover><mi>D</mi> <mo>¯</mo></mover> </math> recently introduced by Baumann-Kamnitzer-Knutson in their study of equivariant multiplicities of Mirković-Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov-Reshetikhin modules in <math><msub><mi>C</mi> <mi>Q</mi></msub> </math> , as well as certain results by Brundan-Kleshchev-McNamara on the representation theory of quiver Hecke algebras. This alternative description of <math><mover><mi>D</mi> <mo>¯</mo></mover> </math> allows us to prove a conjecture by the first author on the distinguished values of <math><mover><mi>D</mi> <mo>¯</mo></mover> </math> on the flag minors of <math><mrow><mi>C</mi> <mo>[</mo> <mi>N</mi> <mo>]</mo></mrow> </math> . We also provide applications of our results from the perspective of Kang-Kashiwara-Kim-Oh's generalized Schur-Weyl duality. Finally, we use Kashiwara-Kim-Oh-Park's recent constructions to define a cluster algebra <math> <msub><mover><mi>A</mi> <mo>¯</mo></mover> <mi>Q</mi></msub> </math> as a subquotient of <math> <mrow><msub><mi>K</mi> <mn>0</mn></msub> <mrow><mo>(</mo> <msup><mrow><mi>C</mi></mrow> <mrow><mo>≤</mo> <mi>ξ</mi></mrow> </msup> <mo>)</mo></mrow> </mrow> </math> naturally containing <math><mrow><mi>C</mi> <mo>[</mo> <mi>N</mi> <mo>]</mo></mrow> </math> , and suggest the existence of an analogue of the Mirković-Vilo","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":"29 1","pages":"9"},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9707734/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"35207479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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