A transversal in an n×n$n times n$ latin square is a collection of n$n$ entries not repeating any row, column, or symbol. Kwan showed that almost every n×n$n times n$ latin square has (1+o(1))n/e2n$bigl ((1 + o(1)) n / e^2bigr )^n$ transversals as n→∞$n rightarrow infty$ . Using a loose variant of the circle method we sharpen this to (e−1/2+o(1))n!2/nn$(e^{-1/2} + o(1)) n!^2 / n^n$ . Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin squares with high probability as well as multiplication tables of quasirandom groups.
n×n $n times n$拉丁方格中的截线是n个$n$项的集合，不重复任何行、列或符号。Kwan证明了几乎每个n×n $n times n$拉丁方都有(1+o(1))n/e2n $bigl ((1 + o(1)) n / e^2bigr )^n$截线为n→∞$n rightarrow infty$。使用圆法的松散变体，我们将其锐化为(e - 1/2+o(1))n!2/nn $(e^{-1/2} + o(1)) n!^2 / n^n$。该方法适用于满足准随机条件的所有拉丁平方，既包括高概率随机拉丁平方，也包括准随机群的乘法表。
{"title":"Transversals in quasirandom latin squares","authors":"Sean Eberhard, Freddie Manners, Rudi Mrazovi'c","doi":"10.1112/plms.12538","DOIUrl":"https://doi.org/10.1112/plms.12538","url":null,"abstract":"A transversal in an n×n$n times n$ latin square is a collection of n$n$ entries not repeating any row, column, or symbol. Kwan showed that almost every n×n$n times n$ latin square has (1+o(1))n/e2n$bigl ((1 + o(1)) n / e^2bigr )^n$ transversals as n→∞$n rightarrow infty$ . Using a loose variant of the circle method we sharpen this to (e−1/2+o(1))n!2/nn$(e^{-1/2} + o(1)) n!^2 / n^n$ . Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin squares with high probability as well as multiplication tables of quasirandom groups.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48540155","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}
We give two applications of our prior work toward the Putman–Wieland conjecture. First, we deduce a strengthening of a result of Marković–Tošić on virtual mapping class group actions on the homology of covers. Second, let g⩾2$ggeqslant 2$ and let Σg′,n′→Σg,n$Sigma _{g^{prime },n^{prime }}rightarrow Sigma _{g, n}$ be a finite H$H$ ‐cover of topological surfaces. We show the virtual action of the mapping class group of Σg,n+1$Sigma _{g,n+1}$ on an H$H$ ‐isotypic component of H1(Σg′)$H^1(Sigma _{g^{prime }})$ has nonunitary image.
{"title":"Applications of the algebraic geometry of the Putman–Wieland conjecture","authors":"Aaron Landesman, Daniel Litt","doi":"10.1112/plms.12539","DOIUrl":"https://doi.org/10.1112/plms.12539","url":null,"abstract":"We give two applications of our prior work toward the Putman–Wieland conjecture. First, we deduce a strengthening of a result of Marković–Tošić on virtual mapping class group actions on the homology of covers. Second, let g⩾2$ggeqslant 2$ and let Σg′,n′→Σg,n$Sigma _{g^{prime },n^{prime }}rightarrow Sigma _{g, n}$ be a finite H$H$ ‐cover of topological surfaces. We show the virtual action of the mapping class group of Σg,n+1$Sigma _{g,n+1}$ on an H$H$ ‐isotypic component of H1(Σg′)$H^1(Sigma _{g^{prime }})$ has nonunitary image.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45239869","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}
{"title":"Issue Information","authors":"","doi":"10.1112/plms.12415","DOIUrl":"https://doi.org/10.1112/plms.12415","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43930625","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}
We prove that the localization C∼w$ widetilde{mathcal {C}}_w$ of the monoidal category Cw$ mathcal {C}_w$ is rigid, and the category Cw,v$ mathcal {C}_{w,v}$ admits a localization via a real commuting family of central objects. For a quiver Hecke algebra R$R$ and an element w$w$ in the Weyl group, the subcategory Cw$ mathcal {C}_w$ of the category R-gmod$Rtext{-}mathrm{gmod}$ of finite‐dimensional graded R$R$ ‐modules categorifies the quantum unipotent coordinate ring Aq(n(w))$A_q(mathfrak {n}(w))$ . In the previous paper, we constructed a monoidal category C∼w$ widetilde{mathcal {C}}_w$ such that it contains Cw$ mathcal {C}_w$ and the objects {M(wΛi,Λi)∣i∈I}$lbrace {{hspace*{0.6pt}mathsf {M}}(wLambda _i,Lambda _i)}mid {iin I}rbrace$ corresponding to the frozen variables are invertible. In this paper, we show that there is a monoidal equivalence between the category C∼w$ widetilde{mathcal {C}}_w$ and (C∼w−1)rev$(widetilde{mathcal {C}}_{w^{-1}})^{hspace*{0.6pt}mathrm{rev}}$ . Together with the already known left‐rigidity of C∼w$ widetilde{mathcal {C}}_w$ , it follows that the monoidal category C∼w$ widetilde{mathcal {C}}_w$ is rigid. If v≼w$vpreccurlyeq w$ in the Bruhat order, there is a subcategory Cw,v$ mathcal {C}_{w,v}$ of Cw$ mathcal {C}_w$ that categorifies the doubly‐invariant algebra N′(w)C[N]N(v)$^{N^{prime }(w)} {mathbb {C}}[N]^{N(v)}$ . We prove that the family M(wΛi,vΛi)i∈I$bigl ({hspace*{0.6pt}mathsf {M}}(wLambda _i,vLambda _i)bigr )_{iin I}$ of simple R$R$ ‐module forms a real commuting family of graded central objects in the category Cw,v$ mathcal {C}_{w,v}$ so that there is a localization C∼w,v$ widetilde{mathcal {C}}_{w,v}$ of Cw,v$ mathcal {C}_{w,v}$ in which M(wΛi,vΛi)${hspace*{0.6pt}mathsf {M}}(wLambda _i,vLambda _i)$ are invertible. Since the localization of the algebra N′(w)C[N]N(v)$^{N^{prime }(w)} {mathbb {C}}[N]^{N(v)}$ by the family of the isomorphism classes of M(wΛi,vΛi)${hspace*{0.6pt}mathsf {M}}(wLambda _i,vLambda _i)$ is isomorphic to the coordinate ring C[Rw,v]${mathbb {C}}[R_{w,v}]$ of the open Richardson variety associated with w$w$ and v$v$ , the localization C∼w,v$ widetilde{mathcal {C}}_{w,v}$ categorifies the coordinate ring C[Rw,v]${mathbb {C}}[R_{w,v}]$ .
{"title":"Localizations for quiver Hecke algebras II","authors":"M. Kashiwara, Myungho Kim, Se-jin Oh, E. Park","doi":"10.1112/plms.12558","DOIUrl":"https://doi.org/10.1112/plms.12558","url":null,"abstract":"We prove that the localization C∼w$ widetilde{mathcal {C}}_w$ of the monoidal category Cw$ mathcal {C}_w$ is rigid, and the category Cw,v$ mathcal {C}_{w,v}$ admits a localization via a real commuting family of central objects. For a quiver Hecke algebra R$R$ and an element w$w$ in the Weyl group, the subcategory Cw$ mathcal {C}_w$ of the category R-gmod$Rtext{-}mathrm{gmod}$ of finite‐dimensional graded R$R$ ‐modules categorifies the quantum unipotent coordinate ring Aq(n(w))$A_q(mathfrak {n}(w))$ . In the previous paper, we constructed a monoidal category C∼w$ widetilde{mathcal {C}}_w$ such that it contains Cw$ mathcal {C}_w$ and the objects {M(wΛi,Λi)∣i∈I}$lbrace {{hspace*{0.6pt}mathsf {M}}(wLambda _i,Lambda _i)}mid {iin I}rbrace$ corresponding to the frozen variables are invertible. In this paper, we show that there is a monoidal equivalence between the category C∼w$ widetilde{mathcal {C}}_w$ and (C∼w−1)rev$(widetilde{mathcal {C}}_{w^{-1}})^{hspace*{0.6pt}mathrm{rev}}$ . Together with the already known left‐rigidity of C∼w$ widetilde{mathcal {C}}_w$ , it follows that the monoidal category C∼w$ widetilde{mathcal {C}}_w$ is rigid. If v≼w$vpreccurlyeq w$ in the Bruhat order, there is a subcategory Cw,v$ mathcal {C}_{w,v}$ of Cw$ mathcal {C}_w$ that categorifies the doubly‐invariant algebra N′(w)C[N]N(v)$^{N^{prime }(w)} {mathbb {C}}[N]^{N(v)}$ . We prove that the family M(wΛi,vΛi)i∈I$bigl ({hspace*{0.6pt}mathsf {M}}(wLambda _i,vLambda _i)bigr )_{iin I}$ of simple R$R$ ‐module forms a real commuting family of graded central objects in the category Cw,v$ mathcal {C}_{w,v}$ so that there is a localization C∼w,v$ widetilde{mathcal {C}}_{w,v}$ of Cw,v$ mathcal {C}_{w,v}$ in which M(wΛi,vΛi)${hspace*{0.6pt}mathsf {M}}(wLambda _i,vLambda _i)$ are invertible. Since the localization of the algebra N′(w)C[N]N(v)$^{N^{prime }(w)} {mathbb {C}}[N]^{N(v)}$ by the family of the isomorphism classes of M(wΛi,vΛi)${hspace*{0.6pt}mathsf {M}}(wLambda _i,vLambda _i)$ is isomorphic to the coordinate ring C[Rw,v]${mathbb {C}}[R_{w,v}]$ of the open Richardson variety associated with w$w$ and v$v$ , the localization C∼w,v$ widetilde{mathcal {C}}_{w,v}$ categorifies the coordinate ring C[Rw,v]${mathbb {C}}[R_{w,v}]$ .","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46969228","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}
Pub Date : 2022-08-01Epub Date: 2022-05-12DOI: 10.1112/plms.12448
Jernej Činč, Piotr Oprocha
The main goal of this paper is to study topological and measure-theoretic properties of an intriguing family of strange planar attractors. Building toward these results, we first show that any generic Lebesgue measure-preserving map generates the pseudo-arc as inverse limit with as a single bonding map. These maps can be realized as attractors of disc homeomorphisms in such a way that the attractors vary continuously (in Hausdorff distance on the disc) with the change of bonding map as a parameter. Furthermore, for generic Lebesgue measure-preserving maps the background Oxtoby-Ulam measures induced by Lebesgue measure for on the interval are physical on the disc and in addition there is a dense set of maps defining a unique physical measure. Moreover, the family of physical measures on the attractors varies continuously in the weak* topology; that is, the parametrized family is statistically stable. We also find an arc in the generic Lebesgue measure-preserving set of maps and construct a family of disk homeomorphisms parametrized by this arc which induces a continuously varying family of pseudo-arc attractors with prime ends rotation numbers varying continuously in . It follows that there are uncountably many dynamically non-equivalent embeddings of the pseudo-arc in this family of attractors.
{"title":"Parametrized family of pseudo-arc attractors: Physical measures and prime end rotations.","authors":"Jernej Činč, Piotr Oprocha","doi":"10.1112/plms.12448","DOIUrl":"https://doi.org/10.1112/plms.12448","url":null,"abstract":"<p><p>The main goal of this paper is to study topological and measure-theoretic properties of an intriguing family of strange planar attractors. Building toward these results, we first show that any generic Lebesgue measure-preserving map <math><mrow><mi>f</mi></mrow> </math> generates the pseudo-arc as inverse limit with <math><mrow><mi>f</mi></mrow> </math> as a single bonding map. These maps can be realized as attractors of disc homeomorphisms in such a way that the attractors vary continuously (in Hausdorff distance on the disc) with the change of bonding map as a parameter. Furthermore, for generic Lebesgue measure-preserving maps <math><mrow><mi>f</mi></mrow> </math> the background Oxtoby-Ulam measures induced by Lebesgue measure for <math><mrow><mi>f</mi></mrow> </math> on the interval are physical on the disc and in addition there is a dense set of maps <math><mrow><mi>f</mi></mrow> </math> defining a unique physical measure. Moreover, the family of physical measures on the attractors varies continuously in the weak* topology; that is, the parametrized family is statistically stable. We also find an arc in the generic Lebesgue measure-preserving set of maps and construct a family of disk homeomorphisms parametrized by this arc which induces a continuously varying family of pseudo-arc attractors with prime ends rotation numbers varying continuously in <math> <mrow><mrow><mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>]</mo></mrow> </mrow> </math> . It follows that there are uncountably many dynamically non-equivalent embeddings of the pseudo-arc in this family of attractors.</p>","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"125 2","pages":"318-357"},"PeriodicalIF":1.8,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9544952/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"33515433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Issue Information","authors":"","doi":"10.1112/plms.12414","DOIUrl":"https://doi.org/10.1112/plms.12414","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49592719","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}
Given integers d$d$ and m$m$ , satisfying 1⩽m⩽d/2$1leqslant mleqslant d/2$ , and an arbitrary base field, let Xm$X_m$ be the m$m$ th Grassmannian of a generic d$d$ ‐dimensional quadratic form of trivial discriminant and Clifford invariant. The index of Xm$X_m$ , defined as the g.c.d. of degrees of its closed points, is a 2‐power 2i(m)$2^{mathrm{i}(m)}$ . We find a strong lower bound on the exponent i(m)$mathrm{i}(m)$ which is its exact value for most d,m$d,m$ and which is always within 1 from the exact value.
{"title":"Indexes of generic Grassmannians for spin groups","authors":"N. Karpenko, A. Merkurjev","doi":"10.1112/plms.12471","DOIUrl":"https://doi.org/10.1112/plms.12471","url":null,"abstract":"Given integers d$d$ and m$m$ , satisfying 1⩽m⩽d/2$1leqslant mleqslant d/2$ , and an arbitrary base field, let Xm$X_m$ be the m$m$ th Grassmannian of a generic d$d$ ‐dimensional quadratic form of trivial discriminant and Clifford invariant. The index of Xm$X_m$ , defined as the g.c.d. of degrees of its closed points, is a 2‐power 2i(m)$2^{mathrm{i}(m)}$ . We find a strong lower bound on the exponent i(m)$mathrm{i}(m)$ which is its exact value for most d,m$d,m$ and which is always within 1 from the exact value.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46757097","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}
{"title":"Issue Information","authors":"","doi":"10.1111/phpr.12793","DOIUrl":"https://doi.org/10.1111/phpr.12793","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"125 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43239062","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}
We construct finite‐dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite‐dimensional pointed Hopf algebra over a nonabelian group with nonsimple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch–Schneider. Our starting point is the classification of finite‐dimensional Nichols algebras over nonabelian groups by Heckenberger–Vendramin, which consist of low‐rank exceptions and large‐rank families. We prove that the large‐rank families are cocycle twists of Nichols algebras constructed by the second author as foldings of Nichols algebras of Cartan type over abelian groups by outer automorphisms. This enables us to give uniform Lie‐theoretic descriptions of the large‐rank families, prove generation in degree 1, and construct liftings. We also show that every lifting is a cocycle deformation of the corresponding coradically graded Hopf algebra using an explicit presentation by generators and relations of the Nichols algebra. On the level of tensor categories, we construct families of graded extensions of the representation category of a quantum group by a group of diagram automorphism.
{"title":"Pointed Hopf algebras over nonabelian groups with nonsimple standard braidings","authors":"I. Angiono, S. Lentner, Guillermo Sanmarco","doi":"10.1112/plms.12559","DOIUrl":"https://doi.org/10.1112/plms.12559","url":null,"abstract":"We construct finite‐dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite‐dimensional pointed Hopf algebra over a nonabelian group with nonsimple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch–Schneider. Our starting point is the classification of finite‐dimensional Nichols algebras over nonabelian groups by Heckenberger–Vendramin, which consist of low‐rank exceptions and large‐rank families. We prove that the large‐rank families are cocycle twists of Nichols algebras constructed by the second author as foldings of Nichols algebras of Cartan type over abelian groups by outer automorphisms. This enables us to give uniform Lie‐theoretic descriptions of the large‐rank families, prove generation in degree 1, and construct liftings. We also show that every lifting is a cocycle deformation of the corresponding coradically graded Hopf algebra using an explicit presentation by generators and relations of the Nichols algebra. On the level of tensor categories, we construct families of graded extensions of the representation category of a quantum group by a group of diagram automorphism.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"127 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42186089","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}
{"title":"Issue Information","authors":"","doi":"10.1112/plms.12412","DOIUrl":"https://doi.org/10.1112/plms.12412","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43187667","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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