Pub Date : 2023-11-14DOI: 10.1007/s00029-023-00890-7
Tom Ducat
Abstract We study log Calabi–Yau pairs of the form $$(mathbb {P}^3,Delta )$$ (P3,Δ) , where $$Delta $$ Δ is a quartic surface, and classify all such pairs of coregularity less than or equal to one, up to volume preserving equivalence. In particular, if $$(mathbb {P}^3,Delta )$$ (P3,Δ) is a maximal log Calabi–Yau pair then we show that it has a toric model.
{"title":"Quartic surfaces up to volume preserving equivalence","authors":"Tom Ducat","doi":"10.1007/s00029-023-00890-7","DOIUrl":"https://doi.org/10.1007/s00029-023-00890-7","url":null,"abstract":"Abstract We study log Calabi–Yau pairs of the form $$(mathbb {P}^3,Delta )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $$Delta $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>Δ</mml:mi> </mml:math> is a quartic surface, and classify all such pairs of coregularity less than or equal to one, up to volume preserving equivalence. In particular, if $$(mathbb {P}^3,Delta )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a maximal log Calabi–Yau pair then we show that it has a toric model.","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134901344","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 : 2023-11-13DOI: 10.1007/s00029-023-00888-1
Steven N. Karp
{"title":"Wronskians, total positivity, and real Schubert calculus","authors":"Steven N. Karp","doi":"10.1007/s00029-023-00888-1","DOIUrl":"https://doi.org/10.1007/s00029-023-00888-1","url":null,"abstract":"","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134993022","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 : 2023-11-01DOI: 10.1007/s00029-023-00889-0
Johannes Schleischitz
{"title":"Exact uniform approximation and Dirichlet spectrum in dimension at least two","authors":"Johannes Schleischitz","doi":"10.1007/s00029-023-00889-0","DOIUrl":"https://doi.org/10.1007/s00029-023-00889-0","url":null,"abstract":"","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135371434","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 : 2023-10-26DOI: 10.1007/s00029-023-00884-5
Zhenkun Li, Yi Xie, Boyu Zhang
{"title":"Instanton homology and knot detection on thickened surfaces","authors":"Zhenkun Li, Yi Xie, Boyu Zhang","doi":"10.1007/s00029-023-00884-5","DOIUrl":"https://doi.org/10.1007/s00029-023-00884-5","url":null,"abstract":"","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136376344","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 : 2023-10-25DOI: 10.1007/s00029-023-00883-6
van Diejen, Jan Felipe, Görbe, Tamás
The fusion ring for $widehat{mathfrak{su}}(n)_m$ Wess-Zumino-Witten conformal field theories is known to be isomorphic to a factor ring of the ring of symmetric polynomials presented by Schur polynomials. We introduce a deformation of this factor ring associated with eigenpolynomials for the elliptic Ruijsenaars difference operators. The corresponding Littlewood-Richardson coefficients are governed by a Pieri rule stemming from the eigenvalue equation. The orthogonality of the eigenbasis gives rise to an analog of the Verlinde formula. In the trigonometric limit, our construction recovers the refined $widehat{mathfrak{su}}(n)_m$ Wess-Zumino-Witten fusion ring associated with the Macdonald polynomials.
{"title":"Elliptic Ruijsenaars difference operators, symmetric polynomials, and Wess–Zumino–Witten fusion rings","authors":"van Diejen, Jan Felipe, Görbe, Tamás","doi":"10.1007/s00029-023-00883-6","DOIUrl":"https://doi.org/10.1007/s00029-023-00883-6","url":null,"abstract":"The fusion ring for $widehat{mathfrak{su}}(n)_m$ Wess-Zumino-Witten conformal field theories is known to be isomorphic to a factor ring of the ring of symmetric polynomials presented by Schur polynomials. We introduce a deformation of this factor ring associated with eigenpolynomials for the elliptic Ruijsenaars difference operators. The corresponding Littlewood-Richardson coefficients are governed by a Pieri rule stemming from the eigenvalue equation. The orthogonality of the eigenbasis gives rise to an analog of the Verlinde formula. In the trigonometric limit, our construction recovers the refined $widehat{mathfrak{su}}(n)_m$ Wess-Zumino-Witten fusion ring associated with the Macdonald polynomials.","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134972236","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 : 2023-10-24DOI: 10.1007/s00029-023-00878-3
Cédric Bonnafé, Ulrich Thiel
Abstract We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero–Moser spaces and rational Cherednik algebras associated with complex reflection groups. In particular, we are concerned with Calogero–Moser families (which correspond to the $$mathbb {C}^times $$ C× -fixed points of the Calogero–Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig’s constructible characters based on a Galois covering of the Calogero–Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a $$mathbb {Q}$$ Q -factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity. Making possible these computations was also a source of inspiration for the first author to propose several conjectures about the geometry of Calogero–Moser spaces (cohomology, fixed points, symplectic leaves), often in relation with the representation theory of finite reductive groups.
摘要给出了一系列计算Calogero-Moser空间和与复反射群相关的有理Cherednik代数的几何不变量和表示论不变量的算法。特别地，我们关注Calogero-Moser族(对应于Calogero-Moser空间的$$mathbb {C}^times $$ C x不动点)和元胞特征(由Rouquier和Lusztig基于Calogero-Moser空间的伽罗瓦覆盖的可构造特征的第一作者提出的推广)。为了计算前者，我们设计了一种算法来确定理性Cherednik代数中心的生成器(该算法有几个进一步的应用)，为了计算后者，我们开发了一种通过Gaudin算子构建细胞特征的算法方法。我们已经在第二作者的Cherednik代数岩浆包中实现了我们所有的算法，并使用它来确认几个新情况下的开放猜想。作为双几何中的一个有趣的应用，我们能够确定许多特殊的复杂反射群的$$mathbb {Q}$$ Q -factorial终端的可动锥的腔室分解(从而确定相关辛奇点的非同构相对最小模型的数量)。使这些计算成为可能也是第一作者提出关于Calogero-Moser空间几何(上同调，不动点，辛叶)的几个猜想的灵感来源，这些猜想通常与有限约化群的表示理论有关。
{"title":"Computational aspects of Calogero–Moser spaces","authors":"Cédric Bonnafé, Ulrich Thiel","doi":"10.1007/s00029-023-00878-3","DOIUrl":"https://doi.org/10.1007/s00029-023-00878-3","url":null,"abstract":"Abstract We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero–Moser spaces and rational Cherednik algebras associated with complex reflection groups. In particular, we are concerned with Calogero–Moser families (which correspond to the $$mathbb {C}^times $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mo>×</mml:mo> </mml:msup> </mml:math> -fixed points of the Calogero–Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig’s constructible characters based on a Galois covering of the Calogero–Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a $$mathbb {Q}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>Q</mml:mi> </mml:math> -factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity. Making possible these computations was also a source of inspiration for the first author to propose several conjectures about the geometry of Calogero–Moser spaces (cohomology, fixed points, symplectic leaves), often in relation with the representation theory of finite reductive groups.","PeriodicalId":49551,"journal":{"name":"Selecta Mathematica-New Series","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135220364","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}
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