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Product structure and regularity theorem for totally nonnegative flag varieties 全非负旗变体的积结构和正则定理
IF 3.1 1区 数学
Inventiones mathematicae Pub Date : 2024-04-09 DOI: 10.1007/s00222-024-01256-2
Huanchen Bao, Xuhua He
{"title":"Product structure and regularity theorem for totally nonnegative flag varieties","authors":"Huanchen Bao, Xuhua He","doi":"10.1007/s00222-024-01256-2","DOIUrl":"https://doi.org/10.1007/s00222-024-01256-2","url":null,"abstract":"<p>The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) <span>(J)</span>-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.</p><p>We show that the <span>(J)</span>-totally nonnegative flag variety has a cellular decomposition into totally positive <span>(J)</span>-Richardson varieties. Moreover, each totally positive <span>(J)</span>-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive <span>(J)</span>-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the <span>(J)</span>-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of <span>(U^{-})</span> for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"145 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140572972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
$L^{2}$ -Cohomology of quasi-fibered boundary metrics 准纤维边界度量的 L^{2}$ -同调
IF 3.1 1区 数学
Inventiones mathematicae Pub Date : 2024-04-04 DOI: 10.1007/s00222-024-01253-5
Chris Kottke, Frédéric Rochon
{"title":"$L^{2}$ -Cohomology of quasi-fibered boundary metrics","authors":"Chris Kottke, Frédéric Rochon","doi":"10.1007/s00222-024-01253-5","DOIUrl":"https://doi.org/10.1007/s00222-024-01253-5","url":null,"abstract":"<p>We develop new techniques to compute the weighted <span>(L^{2})</span>-cohomology of quasi-fibered boundary metrics (QFB-metrics). Combined with the decay of <span>(L^{2})</span>-harmonic forms obtained in a companion paper, this allows us to compute the reduced <span>(L^{2})</span>-cohomology for various classes of QFB-metrics. Our results applies in particular to the Nakajima metric on the Hilbert scheme of <span>(n)</span> points on <span>(mathbb{C}^{2})</span>, for which we can show that the Vafa-Witten conjecture holds. Using the compactification of the monopole moduli space announced by Fritzsch, the first author and Singer, we can also give a proof of the Sen conjecture for the monopole moduli space of magnetic charge 3.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"35 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140572979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
$mathbf{C^{2}}$ -Lusin approximation of strongly convex functions $mathbf{C^{2}}$ -Lusin approximation of strongly convex functions（强凸函数的鲁辛近似
IF 3.1 1区 数学
Inventiones mathematicae Pub Date : 2024-04-03 DOI: 10.1007/s00222-024-01252-6
Daniel Azagra, Marjorie Drake, Piotr Hajłasz
{"title":"$mathbf{C^{2}}$ -Lusin approximation of strongly convex functions","authors":"Daniel Azagra, Marjorie Drake, Piotr Hajłasz","doi":"10.1007/s00222-024-01252-6","DOIUrl":"https://doi.org/10.1007/s00222-024-01252-6","url":null,"abstract":"<p>We prove that if <span>(u:mathbb{R}^{n}to mathbb{R})</span> is strongly convex, then for every <span>(varepsilon &gt;0)</span> there is a strongly convex function <span>(vin C^{2}(mathbb{R}^{n}))</span> such that <span>(|{uneq v}|&lt;varepsilon )</span> and <span>(Vert u-vVert _{infty}&lt;varepsilon )</span>.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"24 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140572975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Correction to “Anosov flows, growth rates on covers and group extensions of subshifts” 对 "阿诺索夫流、盖上增长率和子移的群扩展 "的更正
IF 3.1 1区 数学
Inventiones mathematicae Pub Date : 2024-03-28 DOI: 10.1007/s00222-024-01251-7
Rhiannon Dougall, Richard Sharp
{"title":"Correction to “Anosov flows, growth rates on covers and group extensions of subshifts”","authors":"Rhiannon Dougall, Richard Sharp","doi":"10.1007/s00222-024-01251-7","DOIUrl":"https://doi.org/10.1007/s00222-024-01251-7","url":null,"abstract":"<p>This note corrects an error in our paper <i>Anosov flows, growth rates on covers and group extensions of subshifts</i>, Invent. Math. 223:445–483, 2021. This leaves our main results, Theorem 1.1, Corollary 1.2, Theorem 1.3 and Theorem 5.1, unchanged. We also fill a gap in the arguments presented in Sect. 9; this requires a small modification to the results in this section.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"20 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140311603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sharp well-posedness for the Benjamin–Ono equation 本杰明-奥诺方程的夏普好拟性
IF 3.1 1区 数学
Inventiones mathematicae Pub Date : 2024-03-26 DOI: 10.1007/s00222-024-01250-8
Rowan Killip, Thierry Laurens, Monica Vişan
{"title":"Sharp well-posedness for the Benjamin–Ono equation","authors":"Rowan Killip, Thierry Laurens, Monica Vişan","doi":"10.1007/s00222-024-01250-8","DOIUrl":"https://doi.org/10.1007/s00222-024-01250-8","url":null,"abstract":"<p>The Benjamin–Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces <span>(H^{s})</span> for <span>(s&gt;-tfrac{1}{2})</span>. The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of Gérard’s explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"63 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140298399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Monodromy of the Casimir connection of a symmetrisable Kac–Moody algebra 可对称卡-莫迪代数的卡西米尔连接的单色性
IF 3.1 1区 数学
Inventiones mathematicae Pub Date : 2024-03-14 DOI: 10.1007/s00222-024-01242-8
Andrea Appel, Valerio Toledano Laredo
{"title":"Monodromy of the Casimir connection of a symmetrisable Kac–Moody algebra","authors":"Andrea Appel, Valerio Toledano Laredo","doi":"10.1007/s00222-024-01242-8","DOIUrl":"https://doi.org/10.1007/s00222-024-01242-8","url":null,"abstract":"<p>Let <span>(mathfrak {g})</span> be a symmetrisable Kac–Moody algebra and <span>(V)</span> an integrable <span>(mathfrak {g})</span>–module in category <span>(mathcal {O})</span>. We show that the monodromy of the (normally ordered) rational Casimir connection on <span>(V)</span> can be made equivariant with respect to the Weyl group <span>(W)</span> of <span>(mathfrak {g})</span>, and therefore defines an action of the braid group <span>(mathcal {B}_{W})</span> on <span>(V)</span>. We then prove that this action is canonically equivalent to the quantum Weyl group action of <span>(mathcal {B}_{W})</span> on a quantum deformation of <span>(V)</span>, that is an integrable, category <span>(mathcal {O})</span> module <span>(mathcal {V})</span> over the quantum group <span>(U_{hbar }mathfrak {g})</span> such that <span>(mathcal {V}/hbar mathcal {V})</span> is isomorphic to <span>(V)</span>. This extends a result of the second author which is valid for <span>(mathfrak {g})</span> semisimple.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"9 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140126591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Monoidal categorification and quantum affine algebras II 单义分类和量子仿射代数 II
IF 3.1 1区 数学
Inventiones mathematicae Pub Date : 2024-03-12 DOI: 10.1007/s00222-024-01249-1
Masaki Kashiwara, Myungho Kim, Se-jin Oh, Euiyong Park
{"title":"Monoidal categorification and quantum affine algebras II","authors":"Masaki Kashiwara, Myungho Kim, Se-jin Oh, Euiyong Park","doi":"10.1007/s00222-024-01249-1","DOIUrl":"https://doi.org/10.1007/s00222-024-01249-1","url":null,"abstract":"<p>We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of <span>(i)</span>-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories <span>(mathscr{C}_{{mathfrak{g}}}^{0})</span> and <span>(mathscr{C}_{{mathfrak{g}}}^{-})</span> provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"40 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140126047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Demazure crystals and the Schur positivity of Catalan functions 德马祖尔晶体和加泰罗尼亚函数的舒尔正性
IF 3.1 1区 数学
Inventiones mathematicae Pub Date : 2024-03-08 DOI: 10.1007/s00222-024-01237-5
Jonah Blasiak, Jennifer Morse, Anna Pun
{"title":"Demazure crystals and the Schur positivity of Catalan functions","authors":"Jonah Blasiak, Jennifer Morse, Anna Pun","doi":"10.1007/s00222-024-01237-5","DOIUrl":"https://doi.org/10.1007/s00222-024-01237-5","url":null,"abstract":"<p>Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include <span>(k)</span>-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of <span>(U_{q}(widehat{mathfrak{sl}}_{ell }))</span>-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"29 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stability of the Faber-Krahn inequality for the short-time Fourier transform 短时傅立叶变换的法布尔-克拉恩不等式的稳定性
IF 3.1 1区 数学
Inventiones mathematicae Pub Date : 2024-03-01 DOI: 10.1007/s00222-024-01248-2
Jaime Gómez, André Guerra, João P. G. Ramos, Paolo Tilli
{"title":"Stability of the Faber-Krahn inequality for the short-time Fourier transform","authors":"Jaime Gómez, André Guerra, João P. G. Ramos, Paolo Tilli","doi":"10.1007/s00222-024-01248-2","DOIUrl":"https://doi.org/10.1007/s00222-024-01248-2","url":null,"abstract":"<p>We prove a sharp quantitative version of the Faber–Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit <span>(delta (f;Omega ))</span> which measures by how much the STFT of a function <span>(fin L^{2}(mathbb{R}))</span> fails to be optimally concentrated on an arbitrary set <span>(Omega subset mathbb{R}^{2})</span> of positive, finite measure. We then show that an optimal power of the deficit <span>(delta (f;Omega ))</span> controls both the <span>(L^{2})</span>-distance of <span>(f)</span> to an appropriate class of Gaussians and the distance of <span>(Omega )</span> to a ball, through the Fraenkel asymmetry of <span>(Omega )</span>. Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"103 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140008704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Uniform negative immersions and the coherence of one-relator groups 均匀负沉浸和单链组的一致性
IF 3.1 1区 数学
Inventiones mathematicae Pub Date : 2024-02-29 DOI: 10.1007/s00222-024-01246-4
Larsen Louder, Henry Wilton
{"title":"Uniform negative immersions and the coherence of one-relator groups","authors":"Larsen Louder, Henry Wilton","doi":"10.1007/s00222-024-01246-4","DOIUrl":"https://doi.org/10.1007/s00222-024-01246-4","url":null,"abstract":"<p>Previously, the authors proved that the presentation complex of a one-relator group <span>(G)</span> satisfies a geometric condition called <i>negative immersions</i> if every two-generator, one-relator subgroup of <span>(G)</span> is free. Here, we prove that one-relator groups with negative immersions are coherent, answering a question of Baumslag in this case. Other strong constraints on the finitely generated subgroups also follow such as, for example, the co-Hopf property. The main new theorem strengthens negative immersions to <i>uniform negative immersions</i>, using a rationality theorem proved with linear-programming techniques.</p>","PeriodicalId":14429,"journal":{"name":"Inventiones mathematicae","volume":"8 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140008770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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