Forum of Mathematics Sigma最新文献

{"title":"Polypositroids","authors":"Thomas Lam, Alexander Postnikov","doi":"10.1017/fms.2024.11","DOIUrl":"https://doi.org/10.1017/fms.2024.11","url":null,"abstract":"<p>We initiate the study of a class of polytopes, which we coin <span>polypositroids</span>, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian, polypositroids are “positive” polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We introduce a notion of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315142604314-0934:S2050509424000112:S2050509424000112_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(W,c)$</span></span></img></span></span><span>-polypositroid</span> for a finite Weyl group <span>W</span> and a choice of Coxeter element <span>c</span>. We connect the theory of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315142604314-0934:S2050509424000112:S2050509424000112_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$(W,c)$</span></span></img></span></span>-polypositroids to cluster algebras of finite type and to generalized associahedra. We discuss <span>membranes</span>, which are certain triangulated 2-dimensional surfaces inside polypositroids. Membranes extend the notion of plabic graphs from positroids to polypositroids.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140149752","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}

{"title":"Undecidability of polynomial inequalities in weighted graph homomorphism densities","authors":"Grigoriy Blekherman, Annie Raymond, Fan Wei","doi":"10.1017/fms.2024.19","DOIUrl":"https://doi.org/10.1017/fms.2024.19","url":null,"abstract":"<p>Many problems and conjectures in extremal combinatorics concern polynomial inequalities between homomorphism densities of graphs where we allow edges to have real weights. Using the theory of graph limits, we can equivalently evaluate polynomial expressions in homomorphism densities on <span>kernels W</span>, that is, symmetric, bounded and measurable functions <span>W</span> from <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$[0,1]^2 to mathbb {R}$</span></span></img></span></span>. In 2011, Hatami and Norin proved a fundamental result that it is undecidable to determine the validity of polynomial inequalities in homomorphism densities for graphons (i.e., the case where the range of <span>W</span> is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$[0,1]$</span></span></img></span></span>, which corresponds to unweighted graphs or, equivalently, to graphs with edge weights between <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$0$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$1$</span></span></img></span></span>). The corresponding problem for more general sets of kernels, for example, for all kernels or for kernels with range <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$[-1,1]$</span></span></img></span></span>, remains open. For any <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$a&gt; 0$</span></span></img></span></span>, we show undecidability of polynomial inequalities for any set of kernels which contains all kernels with range <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline7.png\"><span data-mathjax-type=\"texmath\"><span>${0,a}$</span></span></img></span></span>. This result also an","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140149592","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}

{"title":"On the Turán number of the hypercube","authors":"Oliver Janzer, Benny Sudakov","doi":"10.1017/fms.2024.27","DOIUrl":"https://doi.org/10.1017/fms.2024.27","url":null,"abstract":"In 1964, Erdős proposed the problem of estimating the Turán number of the <jats:italic>d</jats:italic>-dimensional hypercube <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline1.png\" /> <jats:tex-math> $Q_d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Since <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline2.png\" /> <jats:tex-math> $Q_d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a bipartite graph with maximum degree <jats:italic>d</jats:italic>, it follows from results of Füredi and Alon, Krivelevich, Sudakov that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline3.png\" /> <jats:tex-math> $mathrm {ex}(n,Q_d)=O_d(n^{2-1/d})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. A recent general result of Sudakov and Tomon implies the slightly stronger bound <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline4.png\" /> <jats:tex-math> $mathrm {ex}(n,Q_d)=o(n^{2-1/d})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We obtain the first power-improvement for this old problem by showing that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline5.png\" /> <jats:tex-math> $mathrm {ex}(n,Q_d)=O_dleft (n^{2-frac {1}{d-1}+frac {1}{(d-1)2^{d-1}}}right )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. This answers a question of Liu. Moreover, our techniques give a power improvement for a larger class of graphs than cubes. We use a similar method to prove that any <jats:italic>n</jats:italic>-vertex, properly edge-coloured graph without a rainbow cycle has at most <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline6.png\" /> <jats:tex-math> $O(n(log n)^2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> edges, improving the previous best bound of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline7.png\" /> <jats:tex-math> $n(log n)^{2+o(1)}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> by Tomon. Furthermore, we show that any properly edge-coloured <jats:italic>n</jats:italic>-vertex graph with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline8.png\" /> <jats:tex-math> $omega (nlog n)$ </jats:te","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140149744","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}

{"title":"Asymptotic expansions relating to the distribution of the length of longest increasing subsequences","authors":"Folkmar Bornemann","doi":"10.1017/fms.2024.13","DOIUrl":"https://doi.org/10.1017/fms.2024.13","url":null,"abstract":"We study the distribution of the length of longest increasing subsequences in random permutations of <jats:italic>n</jats:italic> integers as <jats:italic>n</jats:italic> grows large and establish an asymptotic expansion in powers of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000136_inline1.png\" /> <jats:tex-math> $n^{-1/3}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Whilst the limit law was already shown by Baik, Deift and Johansson to be the GUE Tracy–Widom distribution <jats:italic>F</jats:italic>, we find explicit analytic expressions of the first few finite-size correction terms as linear combinations of higher order derivatives of <jats:italic>F</jats:italic> with rational polynomial coefficients. Our proof replaces Johansson’s de-Poissonization, which is based on monotonicity as a Tauberian condition, by analytic de-Poissonization of Jacquet and Szpankowski, which is based on growth conditions in the complex plane; it is subject to a tameness hypothesis concerning complex zeros of the analytically continued Poissonized length distribution. In a preparatory step an expansion of the hard-to-soft edge transition law of LUE is studied, which is lifted to an expansion of the Poissonized length distribution for large intensities. Finally, expansions of Stirling-type approximations and of the expected value and variance of the length distribution are given.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140149591","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}

{"title":"Equivariant Hodge polynomials of heavy/light moduli spaces","authors":"Siddarth Kannan, Stefano Serpente, Claudia He Yun","doi":"10.1017/fms.2024.20","DOIUrl":"https://doi.org/10.1017/fms.2024.20","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$overline {mathcal {M}}_{g, m|n}$</span></span></img></span></span> denote Hassett’s moduli space of weighted pointed stable curves of genus <span>g</span> for the <span>heavy/light</span> weight data <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$begin{align*}left(1^{(m)}, 1/n^{(n)}right),end{align*}$$</span></span></img></span></p><p>and let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {M}_{g, m|n} subset overline {mathcal {M}}_{g, m|n}$</span></span></img></span></span> be the locus parameterizing smooth, not necessarily distinctly marked curves. We give a change-of-variables formula which computes the generating function for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$(S_mtimes S_n)$</span></span></img></span></span>-equivariant Hodge–Deligne polynomials of these spaces in terms of the generating functions for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$S_{n}$</span></span></img></span></span>-equivariant Hodge–Deligne polynomials of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$overline {mathcal {M}}_{g,n}$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {M}_{g,n}$</span></span></img></span></span>.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125511","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}

{"title":"Topology of moduli spaces of curves and anabelian geometry in positive characteristic","authors":"Zhi Hu, Yu Yang, Runhong Zong","doi":"10.1017/fms.2024.12","DOIUrl":"https://doi.org/10.1017/fms.2024.12","url":null,"abstract":"<p>In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topology of moduli spaces of curves can be understood from the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures concerning tame fundamental groups of curves over algebraically closed fields of characteristic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313103911284-0196:S2050509424000124:S2050509424000124_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$p&gt;0$</span></span></img></span></span> from the point of view of moduli spaces. The conjectures are generalized versions of the Weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields of characteristic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313103911284-0196:S2050509424000124:S2050509424000124_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$p&gt;0$</span></span></img></span></span> which was formulated by Tamagawa. Moreover, we prove that the conjectures hold for certain points lying in the moduli space of curves of genus <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313103911284-0196:S2050509424000124:S2050509424000124_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$0$</span></span></img></span></span>.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140129823","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}

{"title":"Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions","authors":"Ruiyuan Chen","doi":"10.1017/fms.2024.25","DOIUrl":"https://doi.org/10.1017/fms.2024.25","url":null,"abstract":"<p>We extend the Becker–Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker–Kechris theorems, as well as Sami’s and Hjorth’s sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms and the equivalence of ‘potentially open’ versus ‘orbitwise open’ Borel sets. We also characterize ‘potentially open’ <span>n</span>-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions and prove a result subsuming Lupini’s Becker–Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures.</p><p>Our proof method is new even in the classical case of Polish groups and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids and more generally in the point-free context, for open localic groupoids.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125528","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}

{"title":"Hyperfiniteness of boundary actions of acylindrically hyperbolic groups","authors":"Koichi Oyakawa","doi":"10.1017/fms.2024.24","DOIUrl":"https://doi.org/10.1017/fms.2024.24","url":null,"abstract":"<p>We prove that, for any countable acylindrically hyperbolic group <span>G</span>, there exists a generating set <span>S</span> of <span>G</span> such that the corresponding Cayley graph <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240309072942358-0895:S2050509424000240:S2050509424000240_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$Gamma (G,S)$</span></span></img></span></span> is hyperbolic, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240309072942358-0895:S2050509424000240:S2050509424000240_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$|partial Gamma (G,S)|&gt;2$</span></span></img></span></span>, the natural action of <span>G</span> on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240309072942358-0895:S2050509424000240:S2050509424000240_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$Gamma (G,S)$</span></span></img></span></span> is acylindrical and the natural action of <span>G</span> on the Gromov boundary <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240309072942358-0895:S2050509424000240:S2050509424000240_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$partial Gamma (G,S)$</span></span></img></span></span> is hyperfinite. This result broadens the class of groups that admit a non-elementary acylindrical action on a hyperbolic space with a hyperfinite boundary action.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140099772","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}

{"title":"Components of moduli stacks of two-dimensional Galois representations","authors":"Ana Caraiani, Matthew Emerton, Toby Gee, David Savitt","doi":"10.1017/fms.2024.4","DOIUrl":"https://doi.org/10.1017/fms.2024.4","url":null,"abstract":"<p>In the article [CEGS20b], we introduced various moduli stacks of two-dimensional tamely potentially Barsotti–Tate representations of the absolute Galois group of a <span>p</span>-adic local field, as well as related moduli stacks of Breuil–Kisin modules with descent data. We study the irreducible components of these stacks, establishing, in particular, that the components of the former are naturally indexed by certain Serre weights.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140099612","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}

{"title":"Delta and Theta Operator Expansions","authors":"Alessandro Iraci, Marino Romero","doi":"10.1017/fms.2024.14","DOIUrl":"https://doi.org/10.1017/fms.2024.14","url":null,"abstract":"<p>We give an elementary symmetric function expansion for the expressions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306134500664-0603:S2050509424000148:S2050509424000148_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$MDelta _{m_gamma e_1}Pi e_lambda ^{ast }$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306134500664-0603:S2050509424000148:S2050509424000148_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$MDelta _{m_gamma e_1}Pi s_lambda ^{ast }$</span></span></img></span></span> when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306134500664-0603:S2050509424000148:S2050509424000148_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$t=1$</span></span></img></span></span> in terms of what we call <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306134500664-0603:S2050509424000148:S2050509424000148_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$gamma $</span></span></img></span></span>-parking functions and lattice <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306134500664-0603:S2050509424000148:S2050509424000148_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$gamma $</span></span></img></span></span>-parking functions. Here, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306134500664-0603:S2050509424000148:S2050509424000148_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$Delta _F$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306134500664-0603:S2050509424000148:S2050509424000148_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$Pi $</span></span></img></span></span> are certain eigenoperators of the modified Macdonald basis and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306134500664-0603:S2050509424000148:S2050509424000148_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$M=(1-q)(1-t)$</span></span></img></span></span>. Our main results, in turn, give an elementary basis expansion at <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240306134500664-0603:S2050509424000148:S2050509424000148_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$t=1$</span></span></img></span></span> for symmetric functions of the form <span><sp","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140053940","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}