In [14], Jacquet–Piatetskii-Shapiro–Shalika defined a family of compact open subgroups of p-adic general linear groups indexed by nonnegative integers and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of nonnegative integers. For the proof, we introduce the Rankin–Selberg integrals for Speh representations.
{"title":"Local newforms for the general linear groups over a non-archimedean local field","authors":"Hiraku Atobe, S. Kondo, S. Yasuda","doi":"10.1017/fmp.2022.17","DOIUrl":"https://doi.org/10.1017/fmp.2022.17","url":null,"abstract":"\u0000 In [14], Jacquet–Piatetskii-Shapiro–Shalika defined a family of compact open subgroups of p-adic general linear groups indexed by nonnegative integers and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of nonnegative integers. For the proof, we introduce the Rankin–Selberg integrals for Speh representations.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47538220","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}
Abstract We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing and reflexive differentials associated to Z.
{"title":"Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension","authors":"M. Mustaţă, M. Popa","doi":"10.1017/fmp.2022.15","DOIUrl":"https://doi.org/10.1017/fmp.2022.15","url":null,"abstract":"Abstract We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing and reflexive differentials associated to Z.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41639009","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}
Abstract We prove two theorems about the Malcev Lie algebra associated to the Torelli group of a surface of genus g: Stably, it is Koszul and the kernel of the Johnson homomorphism consists only of trivial $mathrm {Sp}_{2g}(mathbb {Z})$ -representations lying in the centre.
{"title":"On the Torelli Lie algebra","authors":"A. Kupers, O. Randal-Williams","doi":"10.1017/fmp.2023.10","DOIUrl":"https://doi.org/10.1017/fmp.2023.10","url":null,"abstract":"Abstract We prove two theorems about the Malcev Lie algebra associated to the Torelli group of a surface of genus g: Stably, it is Koszul and the kernel of the Johnson homomorphism consists only of trivial \u0000$mathrm {Sp}_{2g}(mathbb {Z})$\u0000 -representations lying in the centre.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41982208","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}
Abstract A finite set of integers A tiles the integers by translations if $mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have $Aoplus B=mathbb {Z}_M$ for some $Min mathbb {N}$ and $Bsubset mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with A and B. In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set A containing certain configurations can tile a cyclic group $mathbb {Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period $(pqr)^2$, where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].
{"title":"Combinatorial and harmonic-analytic methods for integer tilings","authors":"I. Łaba, Itay Londner","doi":"10.1017/fmp.2022.3","DOIUrl":"https://doi.org/10.1017/fmp.2022.3","url":null,"abstract":"Abstract A finite set of integers A tiles the integers by translations if $mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have $Aoplus B=mathbb {Z}_M$ for some $Min mathbb {N}$ and $Bsubset mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with A and B. In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set A containing certain configurations can tile a cyclic group $mathbb {Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period $(pqr)^2$, where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43909505","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}
Daniel Cristofaro-Gardiner, Vincent Humilière, C. Y. Mak, Sobhan Seyfaddini, I. Smith
Abstract We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: We show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of hameomorphisms constructed by Oh and Müller, and we construct an infinite-dimensional family of quasi-morphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants, via orbifold Floer homology, for links composed of parallel circles in the two-sphere. A particular feature of our work is that it avoids the orbifold setting and relies instead on ‘classical’ Floer homology. This not only substantially simplifies the technical background but seems essential for some aspects (such as the application to constructing quasi-morphisms).
{"title":"Quantitative Heegaard Floer cohomology and the Calabi invariant","authors":"Daniel Cristofaro-Gardiner, Vincent Humilière, C. Y. Mak, Sobhan Seyfaddini, I. Smith","doi":"10.1017/fmp.2022.18","DOIUrl":"https://doi.org/10.1017/fmp.2022.18","url":null,"abstract":"Abstract We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: We show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of hameomorphisms constructed by Oh and Müller, and we construct an infinite-dimensional family of quasi-morphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants, via orbifold Floer homology, for links composed of parallel circles in the two-sphere. A particular feature of our work is that it avoids the orbifold setting and relies instead on ‘classical’ Floer homology. This not only substantially simplifies the technical background but seems essential for some aspects (such as the application to constructing quasi-morphisms).","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44250367","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}
Abstract We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $gamma $ in $mathbb R^d$ , $dge 3$ . Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal $L^p$ Sobolev regularity estimates, which settle the conjecture raised by Beltran–Guo–Hickman–Seeger [1]. Besides, we show the sharp local smoothing estimates on a range of p for every $dge 3$ . As a result, we establish, for the first time, nontrivial $L^p$ boundedness of the maximal average over dilations of $gamma $ for $dge 4$ .
{"title":"Sharp smoothing properties of averages over curves","authors":"Hyerim Ko, Sanghyuk Lee, Sewook Oh","doi":"10.1017/fmp.2023.2","DOIUrl":"https://doi.org/10.1017/fmp.2023.2","url":null,"abstract":"Abstract We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve \u0000$gamma $\u0000 in \u0000$mathbb R^d$\u0000 , \u0000$dge 3$\u0000 . Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal \u0000$L^p$\u0000 Sobolev regularity estimates, which settle the conjecture raised by Beltran–Guo–Hickman–Seeger [1]. Besides, we show the sharp local smoothing estimates on a range of p for every \u0000$dge 3$\u0000 . As a result, we establish, for the first time, nontrivial \u0000$L^p$\u0000 boundedness of the maximal average over dilations of \u0000$gamma $\u0000 for \u0000$dge 4$\u0000 .","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42934109","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}
Harold Blum, Yuchen Liu, Chenyang Xu, Ziquan Zhuang
Abstract We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field .
{"title":"The existence of the Kähler–Ricci soliton degeneration","authors":"Harold Blum, Yuchen Liu, Chenyang Xu, Ziquan Zhuang","doi":"10.1017/fmp.2023.5","DOIUrl":"https://doi.org/10.1017/fmp.2023.5","url":null,"abstract":"Abstract We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field .","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49564253","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}
J. Blasiak, M. Haiman, J. Morse, Anna Y. Pun, G. Seelinger
Abstract We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $Delta _{h_l}Delta ' _{e_k} e_{n}$ , where $Delta ' _{e_k}$ and $Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $operatorname {mathrm {GL}}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.
{"title":"A Proof of the Extended Delta Conjecture","authors":"J. Blasiak, M. Haiman, J. Morse, Anna Y. Pun, G. Seelinger","doi":"10.1017/fmp.2023.3","DOIUrl":"https://doi.org/10.1017/fmp.2023.3","url":null,"abstract":"Abstract We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for \u0000$Delta _{h_l}Delta ' _{e_k} e_{n}$\u0000 , where \u0000$Delta ' _{e_k}$\u0000 and \u0000$Delta _{h_l}$\u0000 are Macdonald eigenoperators and \u0000$e_n$\u0000 is an elementary symmetric function. We actually prove a stronger identity of infinite series of \u0000$operatorname {mathrm {GL}}_m$\u0000 characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48116026","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}
J. Blasiak, M. Haiman, J. Morse, Anna Y. Pun, G. Seelinger
Abstract We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $operatorname {mathrm {GL}}_{l}$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials.
{"title":"A Shuffle Theorem for Paths Under Any Line","authors":"J. Blasiak, M. Haiman, J. Morse, Anna Y. Pun, G. Seelinger","doi":"10.1017/fmp.2023.4","DOIUrl":"https://doi.org/10.1017/fmp.2023.4","url":null,"abstract":"Abstract We generalize the shuffle theorem and its \u0000$(km,kn)$\u0000 version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the \u0000$(km,kn)$\u0000 Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of \u0000$operatorname {mathrm {GL}}_{l}$\u0000 characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44931716","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}
Abstract We study the fields of values of the irreducible characters of a finite group of degree not divisible by a prime p. In the case where $p=2$, we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups.
{"title":"The fields of values of characters of degree not divisible by p","authors":"G. Navarro, P. Tiep","doi":"10.1017/fmp.2021.1","DOIUrl":"https://doi.org/10.1017/fmp.2021.1","url":null,"abstract":"Abstract We study the fields of values of the irreducible characters of a finite group of degree not divisible by a prime p. In the case where $p=2$, we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2021.1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46503978","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}
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