{"title":"A rim-hook rule for quiver flag varieties","authors":"Wei Gu, Elana Kalashnikov","doi":"10.1007/s00029-024-00936-4","DOIUrl":"https://doi.org/10.1007/s00029-024-00936-4","url":null,"abstract":"<p>The Abelian/non-Abelian correspondence for cohomology (Martin in Symplectic quotients by a nonabelian group and by its maximal torus. arXiv:math/0001002 [math.SG], 2000; Ellingsrud–Strømme in On the chow ring of a geometric quotient, 1989) gives a novel description of the cohomology ring of the Grassmannian. We show that the natural generalization of this result to small quantum cohomology applies to Fano quiver flag varieties. Quiver flag varieties are generalisations of type A flag varieties. As a corollary, we see that the Gu–Sharpe mirror to a Fano quiver flag variety computes its quantum cohomology. The second focus of the paper is on applying this description to computations inside the classical and quantum cohomology rings. The rim-hook rule for quantum cohomology of the Grassmannian allows one to reduce quantum calculations to classical calculations in the cohomology of the Grassmannian. We use the Abelian/non-Abelian correspondence to prove a rim-hook removal rule for the cohomology and quantum cohomology (in the Fano case) of quiver flag varieties. This result is new even in the flag case. This gives an effective way of computing products in the (quantum) cohomology ring, reducing computations to that in the cohomology ring of the Grassmannian.</p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Building data for stacky covers","authors":"Eric Ahlqvist","doi":"10.1007/s00029-024-00939-1","DOIUrl":"https://doi.org/10.1007/s00029-024-00939-1","url":null,"abstract":"<p>We define <i>stacky building data</i> for <i>stacky covers</i> in the spirit of Pardini and give an equivalence of (2,1)-categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne–Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme <i>S</i> is defined over a field, we give a criterion for when a <i>birational</i> building datum comes from a tamely ramified cover for a finite abelian group scheme, generalizing a result of Biswas–Borne.\u0000</p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Prismatization","authors":"Vladimir Drinfeld","doi":"10.1007/s00029-024-00937-3","DOIUrl":"https://doi.org/10.1007/s00029-024-00937-3","url":null,"abstract":"<p>The eventual goal is to construct three related “prismatization” functors from the category of <i>p</i>-adic formal schemes to that of formal stacks. This should provide a good category of coefficients for prismatic cohomology in the spirit of <i>F</i>-gauges. In this article we define and study the three versions of the prismatization of <span>({{,mathrm{{Spf}},}}{mathbb {Z}}_p)</span>.</p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Relative poset polytopes and semitoric degenerations","authors":"Evgeny Feigin, Igor Makhlin","doi":"10.1007/s00029-024-00935-5","DOIUrl":"https://doi.org/10.1007/s00029-024-00935-5","url":null,"abstract":"<p>The two best studied toric degenerations of the flag variety are those given by the Gelfand–Tsetlin and FFLV polytopes. Each of them degenerates further into a particular monomial variety which raises the problem of describing the degenerations intermediate between the toric and the monomial ones. Using a theorem of Zhu one may show that every such degeneration is semitoric with irreducible components given by a regular subdivision of the corresponding polytope. This leads one to study the parts that appear in such subdivisions as well as the associated toric varieties. It turns out that these parts lie in a certain new family of poset polytopes which we term <i>relative poset polytopes</i>: each is given by a poset and a weakening of its order relation. In this paper we give an in depth study of (both common and marked) relative poset polytopes and their toric varieties in the generality of an arbitrary poset. We then apply these results to degenerations of flag varieties. We also show that our family of polytopes generalizes the family studied in a series of papers by Fang, Fourier, Litza and Pegel while sharing their key combinatorial properties such as pairwise Ehrhart-equivalence and Minkowski-additivity.</p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140839350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Oriented matroids from triangulations of products of simplices","authors":"Marcel Celaya, Georg Loho, Chi Ho Yuen","doi":"10.1007/s00029-024-00938-2","DOIUrl":"https://doi.org/10.1007/s00029-024-00938-2","url":null,"abstract":"<p>We introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex, which might be of independent interest. In particular, we generalize this using the language of matroids over hyperfields, which gives a new approach to construct matroids over hyperfields. A recurring theme in our work is that various tropical constructions can be extended beyond tropicalization with new formulations and proof methods.</p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"129 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140809397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Weil zeta functions of group representations over finite fields","authors":"Ged Corob Cook, Steffen Kionke, Matteo Vannacci","doi":"10.1007/s00029-024-00934-6","DOIUrl":"https://doi.org/10.1007/s00029-024-00934-6","url":null,"abstract":"<p>In this article we define and study a zeta function <span>(zeta _G)</span>—similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group <i>G</i>. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value <span>(zeta _G(k)^{-1})</span> at a sufficiently large integer <i>k</i> coincides with the probability that <i>k</i> random elements generate the completed group ring of <i>G</i>. The explicit formulas obtained so far suggest that <span>(zeta _G)</span> is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence <i>a</i>(<i>G</i>) of <span>(zeta _G)</span>. We calculate the Weil abscissae for free abelian, free abelian pro-<i>p</i>, free pro-<i>p</i>, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro-<span>({mathfrak {C}})</span> groups, where <span>({mathfrak {C}})</span> is a class of finite groups with prescribed composition factors. We prove that every real number <span>(a ge 1)</span> is the Weil abscissa <i>a</i>(<i>G</i>) of some profinite group <i>G</i>. In addition, we show that the Euler factors of <span>(zeta _G)</span> are rational functions in <span>(p^{-s})</span> if <i>G</i> is virtually abelian. For finite groups <i>G</i> we calculate <span>(zeta _G)</span> using the rational representation theory of <i>G</i>.</p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"102 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140617485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cut-and-join operators in cohomological field theory and topological recursion","authors":"Alexander Alexandrov","doi":"10.1007/s00029-024-00933-7","DOIUrl":"https://doi.org/10.1007/s00029-024-00933-7","url":null,"abstract":"<p>We construct a cubic cut-and-join operator description for the partition function of the Chekhov–Eynard–Orantin topological recursion for a local spectral curve with simple ramification points. In particular, this class contains partition functions of all semi-simple cohomological field theories. The cut-and-join description leads to an algebraic version of topological recursion. For the same partition functions we also derive N families of the Virasoro constraints and prove that these constraints, supplemented by a deformed dimension constraint, imply the cut-and-join description.</p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"96 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140597973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lie groupoids and logarithmic connections","authors":"Francis Bischoff","doi":"10.1007/s00029-024-00929-3","DOIUrl":"https://doi.org/10.1007/s00029-024-00929-3","url":null,"abstract":"<p>Using tools from the theory of Lie groupoids, we study the category of logarithmic flat connections on principal <i>G</i>-bundles, where <i>G</i> is a complex reductive structure group. Flat connections on the affine line with a logarithmic singularity at the origin are equivalent to representations of a groupoid associated to the exponentiated action of <span>(mathbb {C})</span>. We show that such representations admit a canonical Jordan–Chevalley decomposition and may be linearized by converting the <span>({mathbb {C}})</span>-action to a <span>({mathbb {C}}^{*})</span>-action. We then apply these results to give a functorial classification. Flat connections on a complex manifold with logarithmic singularities along a hypersurface are equivalent to representations of a twisted fundamental groupoid. Using a Morita equivalence, whose construction is inspired by Deligne’s notion of paths with tangential basepoints, we prove a van Kampen type theorem for this groupoid. This allows us to show that the category of representations of the twisted fundamental groupoid can be localized to the normal bundle of the hypersurface. As a result, we obtain a functorial Riemann–Hilbert correspondence for logarithmic connections in terms of generalized monodromy data.</p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140597975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the physical rigidity of Frenkel-Gross connection","authors":"Lingfei Yi","doi":"10.1007/s00029-024-00931-9","DOIUrl":"https://doi.org/10.1007/s00029-024-00931-9","url":null,"abstract":"<p>We show that the Frenkel-Gross connection on <span>({mathbb {G}}_m)</span> is physically rigid as <span>(check{G})</span>-connection, thus confirming the de Rham version of a conjecture of Heinloth-Ngô-Yun. The proof is based on the construction of the Hecke eigensheaf of a <span>(check{G})</span>-connection with only generic oper structure, using the localization of Weyl modules.</p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"97 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140602030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"From hypertoric geometry to bordered Floer homology via the $$m=1$$ amplituhedron","authors":"","doi":"10.1007/s00029-024-00932-8","DOIUrl":"https://doi.org/10.1007/s00029-024-00932-8","url":null,"abstract":"<h3>Abstract</h3> <p>We relate the Fukaya category of the symmetric power of a genus zero surface to deformed category <span> <span>(mathcal {O})</span> </span> of a cyclic hypertoric variety by establishing an isomorphism between algebras defined by Ozsváth–Szabó in Heegaard–Floer theory and Braden–Licata–Proudfoot–Webster in hypertoric geometry. The proof extends work of Karp–Williams on sign variation and the combinatorics of the <span> <span>(m=1)</span> </span> amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of <span> <span>(mathfrak {gl}(1|1))</span> </span>, and generalize our isomorphism to give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement. </p>","PeriodicalId":501600,"journal":{"name":"Selecta Mathematica","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140598024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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