Given cusp forms and of integral weight , the depth two holomorphic iterated Eichler–Shimura integral is defined by , where is the Eichler integral of and are formal variables. We provide an explicit vector‐valued modular form whose top components are given by . We show that this vector‐valued modular form gives rise to a scalar‐valued iterated Eichler integral of depth two, denoted by , that can be seen as a higher depth generalization of the scalar‐valued Eichler integral of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol–Popa. We show that can be expressed in terms of sums of products of components of vector‐valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form . This allows for effective computation of .
给定顶点形式和积分权值，定义深度二全纯迭代Eichler - shimura积分为，其中为和为形式变量的Eichler积分。我们提供了一个显式的向量值模形式，它的上分量由。我们证明了这个向量值模形式产生了深度2的标量值迭代Eichler积分，记为，它可以看作是深度1的标量值Eichler积分的更高深度推广。作为题外话，我们的论点提供了一个由周期多项式满足的正交关系的另一种解释，最初是由于pa ol - popa。我们证明了它可以用具有经典模形式的向量值爱森斯坦级数的分量的乘积的和来表示，这些分量是用判别模形式的一个合适的幂次相乘后表示的。这允许有效的计算。
{"title":"Scalar‐valued depth two Eichler–Shimura integrals of cusp forms","authors":"Tobias Magnusson, Martin Raum","doi":"10.1112/tlm3.12055","DOIUrl":"https://doi.org/10.1112/tlm3.12055","url":null,"abstract":"Given cusp forms and of integral weight , the depth two holomorphic iterated Eichler–Shimura integral is defined by , where is the Eichler integral of and are formal variables. We provide an explicit vector‐valued modular form whose top components are given by . We show that this vector‐valued modular form gives rise to a scalar‐valued iterated Eichler integral of depth two, denoted by , that can be seen as a higher depth generalization of the scalar‐valued Eichler integral of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol–Popa. We show that can be expressed in terms of sums of products of components of vector‐valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form . This allows for effective computation of .","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136341439","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}
Abstract A general method of producing correspondences and spectral categories out of symmetric ring objects in general categories is given. As an application, stable homotopy theory of spectra is recovered from modules over a commutative symmetric ring spectrum defined in terms of framed correspondences over an algebraically closed field. Another application recovers stable motivic homotopy theory from spectral modules over associated spectral categories.
{"title":"Correspondences and stable homotopy theory","authors":"Grigory Garkusha","doi":"10.1112/tlm3.12056","DOIUrl":"https://doi.org/10.1112/tlm3.12056","url":null,"abstract":"Abstract A general method of producing correspondences and spectral categories out of symmetric ring objects in general categories is given. As an application, stable homotopy theory of spectra is recovered from modules over a commutative symmetric ring spectrum defined in terms of framed correspondences over an algebraically closed field. Another application recovers stable motivic homotopy theory from spectral modules over associated spectral categories.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135425772","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}
Barbara Baumeister, Derek F. Holt, Georges Neaime, Sarah Rees
Abstract We provide a complete description of the presentations of the interval groups related to quasi‐Coxeter elements in finite Coxeter groups. In the simply laced cases, we show that each interval group is the quotient of the Artin group associated with the corresponding Carter diagram by the normal closure of a set of twisted cycle commutators, one for each 4‐cycle of the diagram. Our techniques also reprove an analogous result for the Artin groups of finite Coxeter groups, which are interval groups corresponding to Coxeter elements. We also analyse the situation in the non‐simply laced cases, where a new Garside structure is discovered. Furthermore, we obtain a complete classification of whether the interval group we consider is isomorphic or not to the related Artin group. Indeed, using methods of Tits, we prove that the interval groups of proper quasi‐Coxeter elements are not isomorphic to the Artin groups of the same type, in the case of when is even or in any of the exceptional cases. In Baumeister et al. (J. Algebra 629 (2023), 399–423), we show using different methods that this result holds for type for all .
摘要给出了有限Coxeter群中与拟Coxeter元相关的区间群的完整描述。在简单加权的情况下，我们证明了每个区间群都是Artin群的商，通过一组扭环换向子的正规闭包与相应的卡特图相关联，图的每4个环对应一个扭环换向子。我们的方法也证明了有限Coxeter群的Artin群的一个类似结果，这些群是对应于Coxeter元的区间群。我们还分析了在非简单的情况下，一个新的Garside结构被发现的情况。进一步，我们得到了所考虑的区间群是否与相关的Artin群同构的完全分类。事实上，我们利用Tits的方法证明了在当为偶数或任何例外情况下，适当拟- Coxeter元的区间群与同类型的Artin群不同构。在Baumeister et al. (J. Algebra 629(2023)， 399-423)中，我们使用不同的方法表明该结果适用于所有类型。
{"title":"Interval groups related to finite Coxeter groups Part II","authors":"Barbara Baumeister, Derek F. Holt, Georges Neaime, Sarah Rees","doi":"10.1112/tlm3.12057","DOIUrl":"https://doi.org/10.1112/tlm3.12057","url":null,"abstract":"Abstract We provide a complete description of the presentations of the interval groups related to quasi‐Coxeter elements in finite Coxeter groups. In the simply laced cases, we show that each interval group is the quotient of the Artin group associated with the corresponding Carter diagram by the normal closure of a set of twisted cycle commutators, one for each 4‐cycle of the diagram. Our techniques also reprove an analogous result for the Artin groups of finite Coxeter groups, which are interval groups corresponding to Coxeter elements. We also analyse the situation in the non‐simply laced cases, where a new Garside structure is discovered. Furthermore, we obtain a complete classification of whether the interval group we consider is isomorphic or not to the related Artin group. Indeed, using methods of Tits, we prove that the interval groups of proper quasi‐Coxeter elements are not isomorphic to the Artin groups of the same type, in the case of when is even or in any of the exceptional cases. In Baumeister et al. (J. Algebra 629 (2023), 399–423), we show using different methods that this result holds for type for all .","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135538261","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":"The set of mildly regular boundary points has full caloric measure","authors":"N. Watson","doi":"10.1112/tlm3.12052","DOIUrl":"https://doi.org/10.1112/tlm3.12052","url":null,"abstract":"","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44145782","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":"Issue Information","authors":"","doi":"10.1112/tlm3.12033","DOIUrl":"https://doi.org/10.1112/tlm3.12033","url":null,"abstract":"","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43527572","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}
The recent paper The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups Gn,r$G_{n,r}$ of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterizes the automorphisms of the Higman–Thompson groups Gn,r$G_{n,r}$ . This characterization is as the specific subgroup of the rational group Rn,r$mathcal {R}_{n,r}$ of Grigorchuk, Nekrashevych and Suchanskiĭ consisting of elements which have the additional property of being bi‐synchronizing. This article extends the arguments of BCMNO to characterize the automorphism group of Tn,r$T_{n,r}$ as a subgroup of Aut(Gn,r)$mathop {mathrm{Aut}}({G_{n,r}})$ . We naturally also study the outer automorphism groups Out(Tn,r)$mathop {mathrm{Out}}({T_{n,r}})$ . We show that each group Out(Tn,r)$mathop {mathrm{Out}}({T_{n,r}})$ can be realized a subgroup of the group Out(Tn,n−1)$mathop {mathrm{Out}}({T_{n,n-1}})$ . Extending results of Brin and Guzman, we also show that the groups Out(Tn,r)$mathop {mathrm{Out}}({T_{n,r}})$ , for n>2$n,{>},2$ , are all infinite and contain an isomorphic copy of Thompson's group F$F$ . Our techniques for studying the groups Out(Tn,r)$mathop {mathrm{Out}}({T_{n,r}})$ work equally well for Out(Gn,r)$mathop {mathrm{Out}}({G_{n,r}})$ and we are able to prove some results for both families of groups. In particular, for X∈{T,G}$X in lbrace T,Grbrace$ , we show that the groups Out(Xn,r)$mathop {mathrm{Out}}({X_{n,r}})$ fit in a lattice structure where Out(Xn,1)⊴Out(Xn,r)$mathop {mathrm{Out}}({X_{n,1}}) unlhd mathop {mathrm{Out}}({X_{n,r}})$ for all 1⩽r⩽n−1$1 leqslant r leqslant n-1$ and Out(Xn,r)⊴Out(Xn,n−1)$mathop {mathrm{Out}}({X_{n,r}}) unlhd mathop {mathrm{Out}}({X_{n,n-1}})$ . This gives a partial answer to a question in BCMNO concerning the normal subgroup structure of Out(Gn,n−1)$mathop {mathrm{Out}}({G_{n,n-1}})$ . Furthermore, we deduce that for 1⩽j,d⩽n−1$1leqslant j,d leqslant n-1$ such that d=gcd(j,n−1)$d = gcd (j, n-1)$ , Out(Xn,j)=Out(Xn,d)$mathop {mathrm{Out}}({X_{n,j}}) = mathop {mathrm{Out}}({X_{n,d}})$ extending a result of BCMNO for the groups Gn,r$G_{n,r}$ to the groups Tn,r$T_{n,r}$ . We give a negative answer to the question in BCMNO which asks whether Out(Gn,r)≅Out(Gn,s)$mathop {mathrm{Out}}({G_{n,r}}) cong mathop {mathrm{Out}}({G_{n,s}})$ if and only if gcd(n−1,r)=gcd(n−1,s)$gcd (n-1,r) = gcd (n-1,s)$ . Lastly, we show that the groups Tn,r$T_{n,r}$ have the R∞$R_{infty }$ property. This extends a result of Burillo, Matucci and Ventura and, independently, Gonçalves and Sankaran, for Thompson's group T$T$ .
{"title":"Automorphisms of the generalized Thompson's group Tn,r$T_{n,r}$","authors":"F. Olukoya","doi":"10.1112/tlm3.12044","DOIUrl":"https://doi.org/10.1112/tlm3.12044","url":null,"abstract":"The recent paper The further chameleon groups of Richard Thompson and Graham Higman: automorphisms via dynamics for the Higman groups Gn,r$G_{n,r}$ of Bleak, Cameron, Maissel, Navas and Olukoya (BCMNO) characterizes the automorphisms of the Higman–Thompson groups Gn,r$G_{n,r}$ . This characterization is as the specific subgroup of the rational group Rn,r$mathcal {R}_{n,r}$ of Grigorchuk, Nekrashevych and Suchanskiĭ consisting of elements which have the additional property of being bi‐synchronizing. This article extends the arguments of BCMNO to characterize the automorphism group of Tn,r$T_{n,r}$ as a subgroup of Aut(Gn,r)$mathop {mathrm{Aut}}({G_{n,r}})$ . We naturally also study the outer automorphism groups Out(Tn,r)$mathop {mathrm{Out}}({T_{n,r}})$ . We show that each group Out(Tn,r)$mathop {mathrm{Out}}({T_{n,r}})$ can be realized a subgroup of the group Out(Tn,n−1)$mathop {mathrm{Out}}({T_{n,n-1}})$ . Extending results of Brin and Guzman, we also show that the groups Out(Tn,r)$mathop {mathrm{Out}}({T_{n,r}})$ , for n>2$n,{>},2$ , are all infinite and contain an isomorphic copy of Thompson's group F$F$ . Our techniques for studying the groups Out(Tn,r)$mathop {mathrm{Out}}({T_{n,r}})$ work equally well for Out(Gn,r)$mathop {mathrm{Out}}({G_{n,r}})$ and we are able to prove some results for both families of groups. In particular, for X∈{T,G}$X in lbrace T,Grbrace$ , we show that the groups Out(Xn,r)$mathop {mathrm{Out}}({X_{n,r}})$ fit in a lattice structure where Out(Xn,1)⊴Out(Xn,r)$mathop {mathrm{Out}}({X_{n,1}}) unlhd mathop {mathrm{Out}}({X_{n,r}})$ for all 1⩽r⩽n−1$1 leqslant r leqslant n-1$ and Out(Xn,r)⊴Out(Xn,n−1)$mathop {mathrm{Out}}({X_{n,r}}) unlhd mathop {mathrm{Out}}({X_{n,n-1}})$ . This gives a partial answer to a question in BCMNO concerning the normal subgroup structure of Out(Gn,n−1)$mathop {mathrm{Out}}({G_{n,n-1}})$ . Furthermore, we deduce that for 1⩽j,d⩽n−1$1leqslant j,d leqslant n-1$ such that d=gcd(j,n−1)$d = gcd (j, n-1)$ , Out(Xn,j)=Out(Xn,d)$mathop {mathrm{Out}}({X_{n,j}}) = mathop {mathrm{Out}}({X_{n,d}})$ extending a result of BCMNO for the groups Gn,r$G_{n,r}$ to the groups Tn,r$T_{n,r}$ . We give a negative answer to the question in BCMNO which asks whether Out(Gn,r)≅Out(Gn,s)$mathop {mathrm{Out}}({G_{n,r}}) cong mathop {mathrm{Out}}({G_{n,s}})$ if and only if gcd(n−1,r)=gcd(n−1,s)$gcd (n-1,r) = gcd (n-1,s)$ . Lastly, we show that the groups Tn,r$T_{n,r}$ have the R∞$R_{infty }$ property. This extends a result of Burillo, Matucci and Ventura and, independently, Gonçalves and Sankaran, for Thompson's group T$T$ .","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45491590","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}
The transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the $p$-local structures of finite groups. Other than the transporter categories and localities of finite groups, important examples include centric, quasicentric, and subcentric linking systems for saturated fusion systems. These examples are however not defined in general on the full collection of subgroups of the Sylow group. We study here punctured groups, a short name for transporter systems or localities on the collection of nonidentity subgroups of a finite $p$-group. As an application of the existence of a punctured group, we show that the subgroup homology decomposition on the centric collection is sharp for the fusion system. We also prove a Signalizer Functor Theorem for punctured groups and use it to show that the smallest Benson-Solomon exotic fusion system at the prime $2$ has a punctured group, while the others do not. As for exotic fusion systems at odd primes $p$, we survey several classes and find that in almost all cases, either the subcentric linking system is a punctured group for the system, or the system has no punctured group because the normalizer of some subgroup of order $p$ is exotic. Finally, we classify punctured groups restricting to the centric linking system for certain fusion systems on extraspecial $p$-groups of order $p^3$.
{"title":"Punctured groups for exotic fusion systems","authors":"Ellen Henke, Assaf Libman, J. Lynd","doi":"10.1112/tlm3.12054","DOIUrl":"https://doi.org/10.1112/tlm3.12054","url":null,"abstract":"The transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the $p$-local structures of finite groups. Other than the transporter categories and localities of finite groups, important examples include centric, quasicentric, and subcentric linking systems for saturated fusion systems. These examples are however not defined in general on the full collection of subgroups of the Sylow group. We study here punctured groups, a short name for transporter systems or localities on the collection of nonidentity subgroups of a finite $p$-group. As an application of the existence of a punctured group, we show that the subgroup homology decomposition on the centric collection is sharp for the fusion system. We also prove a Signalizer Functor Theorem for punctured groups and use it to show that the smallest Benson-Solomon exotic fusion system at the prime $2$ has a punctured group, while the others do not. As for exotic fusion systems at odd primes $p$, we survey several classes and find that in almost all cases, either the subcentric linking system is a punctured group for the system, or the system has no punctured group because the normalizer of some subgroup of order $p$ is exotic. Finally, we classify punctured groups restricting to the centric linking system for certain fusion systems on extraspecial $p$-groups of order $p^3$.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43134354","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}
For a 3‐manifold M$M$ and an acyclic SL(2,C)$mathit {SL}(2,mathbb {C})$ ‐representation ρ$rho$ of its fundamental group, the SL(2,C)$mathit {SL}(2,mathbb {C})$ ‐Reidemeister torsion τρ(M)∈C×$tau _rho (M) in mathbb {C}^times$ is defined. If there are only finitely many conjugacy classes of irreducible representations, then the Reidemeister torsions are known to be algebraic numbers. Furthermore, we prove that the Reidemeister torsions are not only algebraic numbers but also algebraic integers for most Seifert fibered spaces and infinitely many hyperbolic 3‐manifolds. Also, for a knot exterior E(K)$E(K)$ , we discuss the behavior of τρ(E(K))$tau _rho (E(K))$ when the restriction of ρ$rho$ to the boundary torus is fixed.
{"title":"An algebraic property of Reidemeister torsion","authors":"Teruaki Kitano, Yuta Nozaki","doi":"10.1112/tlm3.12049","DOIUrl":"https://doi.org/10.1112/tlm3.12049","url":null,"abstract":"For a 3‐manifold M$M$ and an acyclic SL(2,C)$mathit {SL}(2,mathbb {C})$ ‐representation ρ$rho$ of its fundamental group, the SL(2,C)$mathit {SL}(2,mathbb {C})$ ‐Reidemeister torsion τρ(M)∈C×$tau _rho (M) in mathbb {C}^times$ is defined. If there are only finitely many conjugacy classes of irreducible representations, then the Reidemeister torsions are known to be algebraic numbers. Furthermore, we prove that the Reidemeister torsions are not only algebraic numbers but also algebraic integers for most Seifert fibered spaces and infinitely many hyperbolic 3‐manifolds. Also, for a knot exterior E(K)$E(K)$ , we discuss the behavior of τρ(E(K))$tau _rho (E(K))$ when the restriction of ρ$rho$ to the boundary torus is fixed.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46302947","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":"Issue Information","authors":"","doi":"10.1112/tlm3.12021","DOIUrl":"https://doi.org/10.1112/tlm3.12021","url":null,"abstract":"","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48071636","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}
P. Caines, F. Mohammadi, E. Sáenz-de-Cabezón, H. Wynn
Lattice conditional independence models [Andersson and Perlman, Lattice models for conditional independence in a multivariate normal distribution, Ann. Statist. 21 (1993), 1318–1358] are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be described via a transitive directed acyclic graph (TDAG) in which, as is normal for causal models, the conditional independence is in terms of conditioning on ancestors in the graph. We demonstrate that a parallel stream of research in algebra, the theory of Hibi ideals, not only maps directly to the lattice conditional independence models but gives a vehicle to generalise the theory from the linear Gaussian case. Given a distributive lattice (i) each conditional independence statement is associated with a Hibi relation defined on the lattice, (ii) the directed graph is given by chains in the lattice which correspond to chains of conditional independence, (iii) the elimination ideal of product terms in the chains gives the Hibi ideal and (iv) the TDAG can be recovered from a special bipartite graph constructed via the Alexander dual of the Hibi ideal. It is briefly demonstrated that there are natural applications to statistical log‐linear models, time series and Shannon information flow.
{"title":"Lattice conditional independence models and Hibi ideals","authors":"P. Caines, F. Mohammadi, E. Sáenz-de-Cabezón, H. Wynn","doi":"10.1112/tlm3.12041","DOIUrl":"https://doi.org/10.1112/tlm3.12041","url":null,"abstract":"Lattice conditional independence models [Andersson and Perlman, Lattice models for conditional independence in a multivariate normal distribution, Ann. Statist. 21 (1993), 1318–1358] are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be described via a transitive directed acyclic graph (TDAG) in which, as is normal for causal models, the conditional independence is in terms of conditioning on ancestors in the graph. We demonstrate that a parallel stream of research in algebra, the theory of Hibi ideals, not only maps directly to the lattice conditional independence models but gives a vehicle to generalise the theory from the linear Gaussian case. Given a distributive lattice (i) each conditional independence statement is associated with a Hibi relation defined on the lattice, (ii) the directed graph is given by chains in the lattice which correspond to chains of conditional independence, (iii) the elimination ideal of product terms in the chains gives the Hibi ideal and (iv) the TDAG can be recovered from a special bipartite graph constructed via the Alexander dual of the Hibi ideal. It is briefly demonstrated that there are natural applications to statistical log‐linear models, time series and Shannon information flow.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47051545","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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