In cite{Lusztig}, Lusztig gives an explicit formula for the bijection between the set of bipartitions and the set $mathcal{N}$ of unipotent classes in a spin group which carry irreducible local systems equivariant for the spin group but not equivariant for the special orthogonal group. The set $mathcal{N}$ has a natural partial order and therefore induces a partial order on bipartitions. We use the explicit formula given in cite{Lusztig} to prove that this partial order on bipartitions is the same as the dominance order appeared in Dipper-James-Murphy's work.
在 cite｛Lusztig｝中，Lusztigg给出了自旋群中单能类的二分集和集$mathcal｛N｝$之间的双射的显式，该双射对自旋群具有不可约局部系统等变，但对特殊正交群不具有等变。集合$mathcal｛N｝$具有自然偏序，因此在二分上引发偏序。我们使用在｛Lusztig｝中给出的显式公式来证明这个关于二分的偏序与Dipper James Murphy的工作中出现的支配序是相同的。
{"title":"A partial order on bipartitions from the generalized Springer correspondence","authors":"Jianqiao Xia","doi":"10.4171/JCA/2-3-4","DOIUrl":"https://doi.org/10.4171/JCA/2-3-4","url":null,"abstract":"In cite{Lusztig}, Lusztig gives an explicit formula for the bijection between the set of bipartitions and the set $mathcal{N}$ of unipotent classes in a spin group which carry irreducible local systems equivariant for the spin group but not equivariant for the special orthogonal group. The set $mathcal{N}$ has a natural partial order and therefore induces a partial order on bipartitions. We use the explicit formula given in cite{Lusztig} to prove that this partial order on bipartitions is the same as the dominance order appeared in Dipper-James-Murphy's work.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/2-3-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43747294","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}
A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalgamation of four copies of the Baumslag--Solitar group, is a candidate. Here we contribute to the discussion of the problem of its soficity in two ways. We construct variations on Higman's group replacing the Baumslag--Solitar group by other groups $G$. We give an elementary condition on $G$, enjoyed for example by $mathbb{Z} wr mathbb{Z}$ and the integral Heisenberg group, under which the resulting group is sofic. We then use soficity to deduce that there exist permutations of $mathbb{Z} / nmathbb{Z}$ that are seemingly pathological in that they have order dividing four and yet locally they behave like exponential functions over most of their domains. Our approach is based on that of Helfgott and Juschenko, who recently showed the soficity of Higman's group would imply some the existence of some similarly pathological functions. Our results call into question their suggestion that this might be a step towards proving the existence of a non-sofic group.
{"title":"Soficity and variations on Higman’s group","authors":"M. Kassabov, Vivian Kuperberg, T. Riley","doi":"10.4171/JCA/26","DOIUrl":"https://doi.org/10.4171/JCA/26","url":null,"abstract":"A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalgamation of four copies of the Baumslag--Solitar group, is a candidate. Here we contribute to the discussion of the problem of its soficity in two ways. \u0000We construct variations on Higman's group replacing the Baumslag--Solitar group by other groups $G$. We give an elementary condition on $G$, enjoyed for example by $mathbb{Z} wr mathbb{Z}$ and the integral Heisenberg group, under which the resulting group is sofic. \u0000We then use soficity to deduce that there exist permutations of $mathbb{Z} / nmathbb{Z}$ that are seemingly pathological in that they have order dividing four and yet locally they behave like exponential functions over most of their domains. Our approach is based on that of Helfgott and Juschenko, who recently showed the soficity of Higman's group would imply some the existence of some similarly pathological functions. Our results call into question their suggestion that this might be a step towards proving the existence of a non-sofic group.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/26","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45544477","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}
We prove that a large class of Gromov hyperbolic groups $G$, including all torsion-free hyperbolic groups, embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanskiu{i}. The proof involves assigning a system of binary addresses to points in the Gromov boundary of $G$, and proving that elements of $G$ act on these addresses by transducers. These addresses derive from a certain self-similar tree of subsets of $G$, whose boundary is naturally homeomorphic to the horofunction boundary of $G$.
{"title":"Rational embeddings of hyperbolic groups","authors":"James M. Belk, C. Bleak, Francesco Matucci","doi":"10.4171/JCA/52","DOIUrl":"https://doi.org/10.4171/JCA/52","url":null,"abstract":"We prove that a large class of Gromov hyperbolic groups $G$, including all torsion-free hyperbolic groups, embed into the asynchronous rational group defined by Grigorchuk, Nekrashevych and Sushchanskiu{i}. The proof involves assigning a system of binary addresses to points in the Gromov boundary of $G$, and proving that elements of $G$ act on these addresses by transducers. These addresses derive from a certain self-similar tree of subsets of $G$, whose boundary is naturally homeomorphic to the horofunction boundary of $G$.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47375448","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}
We study (tame) frieze patterns over subsets of the complex numbers, with particular emphasis on the corresponding quiddity cycles. We provide new general transformations for quiddity cycles of frieze patterns. As one application, we present a combinatorial model for obtaining the quiddity cycles of all tame frieze patterns over the integers (with zero entries allowed), generalising the classic Conway-Coxeter theory. This model is thus also a model for the set of specializations of cluster algebras of Dynkin type $A$ in which all cluster variables are integers. Moreover, we address the question of whether for a given height there are only finitely many non-zero frieze patterns over a given subset $R$ of the complex numbers. Under certain conditions on $R$, we show upper bounds for the absolute values of entries in the quiddity cycles. As a consequence, we obtain that if $R$ is a discrete subset of the complex numbers then for every height there are only finitely many non-zero frieze patterns over $R$. Using this, we disprove a conjecture of Fontaine, by showing that for a complex $d$-th root of unity $zeta_d$ there are only finitely many non-zero frieze patterns for a given height over $R=mathbb{Z}[zeta_d]$ if and only if $din {1,2,3,4,6}$.
{"title":"Frieze patterns over integers and other subsets of the complex numbers","authors":"M. Cuntz, T. Holm","doi":"10.4171/JCA/29","DOIUrl":"https://doi.org/10.4171/JCA/29","url":null,"abstract":"We study (tame) frieze patterns over subsets of the complex numbers, with particular emphasis on the corresponding quiddity cycles. We provide new general transformations for quiddity cycles of frieze patterns. As one application, we present a combinatorial model for obtaining the quiddity cycles of all tame frieze patterns over the integers (with zero entries allowed), generalising the classic Conway-Coxeter theory. This model is thus also a model for the set of specializations of cluster algebras of Dynkin type $A$ in which all cluster variables are integers. \u0000Moreover, we address the question of whether for a given height there are only finitely many non-zero frieze patterns over a given subset $R$ of the complex numbers. Under certain conditions on $R$, we show upper bounds for the absolute values of entries in the quiddity cycles. As a consequence, we obtain that if $R$ is a discrete subset of the complex numbers then for every height there are only finitely many non-zero frieze patterns over $R$. Using this, we disprove a conjecture of Fontaine, by showing that for a complex $d$-th root of unity $zeta_d$ there are only finitely many non-zero frieze patterns for a given height over $R=mathbb{Z}[zeta_d]$ if and only if $din {1,2,3,4,6}$.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/29","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45312908","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}
On the one hand, it is well known that the only subquadratic Dehn function of finitely presented groups is the linear one. On the other hand there is a huge class of Dehn functions $d(n)$ with growth at least $n^4$ (essentially all possible such Dehn functions) constructed in cite{SBR} and based on the time functions of Turing machines and S-machines. The class of Dehn functions $n^{alpha}$ with $alphain (2; 4)$ remained more mysterious even though it has attracted quite a bit of attention (see, for example, cite{BB}). We fill the gap obtaining Dehn functions of the form $n^{alpha}$ (and much more) for all real $alphage 2$ computable in reasonable time, for example, $alpha=pi$ or $alpha= e$, or $alpha$ is any algebraic number. As in cite{SBR}, we use S-machines but new tools and new way of proof are needed for the best possible lower bound $d(n)ge n^2$.
{"title":"Polynomially-bounded Dehn functions of groups","authors":"A. Olshanskii","doi":"10.4171/JCA/2-4-1","DOIUrl":"https://doi.org/10.4171/JCA/2-4-1","url":null,"abstract":"On the one hand, it is well known that the only subquadratic Dehn function of finitely presented groups is the linear one. On the other hand there is a huge class of Dehn functions $d(n)$ with growth at least $n^4$ (essentially all possible such Dehn functions) constructed in cite{SBR} and based on the time functions of Turing machines and S-machines. The class of Dehn functions $n^{alpha}$ with $alphain (2; 4)$ remained more mysterious even though it has attracted quite a bit of attention (see, for example, cite{BB}). We fill the gap obtaining Dehn functions of the form $n^{alpha}$ (and much more) for all real $alphage 2$ computable in reasonable time, for example, $alpha=pi$ or $alpha= e$, or $alpha$ is any algebraic number. As in cite{SBR}, we use S-machines but new tools and new way of proof are needed for the best possible lower bound $d(n)ge n^2$.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/2-4-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47898430","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}
We carry out a detailed investigation of congruence half-factorial Krull monoids with finite cyclic class group and related problems. Specifically, we determine precisely all relatively large values that can occur as a minimal distance of a Krull monoid with finite cyclic class group, as well as the exact distribution of prime divisors over the ideal classes in these cases. Our results apply to various classical objects, including maximal orders and certain semi-groups of modules. In addition, we present applications to quantitative problems in factorization theory. More specifically, we determine exponents in the asymptotic formulas for the number of algebraic integers whose sets of lengths have a large difference.
{"title":"On congruence half-factorial Krull monoids with cyclic class group","authors":"A. Plagne, W. Schmid","doi":"10.4171/jca/34","DOIUrl":"https://doi.org/10.4171/jca/34","url":null,"abstract":"We carry out a detailed investigation of congruence half-factorial Krull monoids with finite cyclic class group and related problems. Specifically, we determine precisely all relatively large values that can occur as a minimal distance of a Krull monoid with finite cyclic class group, as well as the exact distribution of prime divisors over the ideal classes in these cases. Our results apply to various classical objects, including maximal orders and certain semi-groups of modules. In addition, we present applications to quantitative problems in factorization theory. More specifically, we determine exponents in the asymptotic formulas for the number of algebraic integers whose sets of lengths have a large difference.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jca/34","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47458040","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}
We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define the almost positive Schur roots $Phi_c$ and a compatibility degree, given by a formula that is new even in finite type. The clusters define a complete fan $operatorname{Fan}_c(Phi)$. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of $operatorname{Fan}_c(Phi)$ induced by real roots to the ${mathbf g}$-vector fan of the associated cluster algebra. We show that $Phi_c$ is the set of denominator vectors of the associated acyclic cluster algebra and conjecture that the compatibility degree also describes denominator vectors for non-acyclic initial seeds. We extend results on exchangeability of roots to the affine case.
{"title":"An affine almost positive roots model","authors":"Nathan Reading, Salvatore Stella","doi":"10.4171/jca/37","DOIUrl":"https://doi.org/10.4171/jca/37","url":null,"abstract":"We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define the almost positive Schur roots $Phi_c$ and a compatibility degree, given by a formula that is new even in finite type. The clusters define a complete fan $operatorname{Fan}_c(Phi)$. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of $operatorname{Fan}_c(Phi)$ induced by real roots to the ${mathbf g}$-vector fan of the associated cluster algebra. We show that $Phi_c$ is the set of denominator vectors of the associated acyclic cluster algebra and conjecture that the compatibility degree also describes denominator vectors for non-acyclic initial seeds. We extend results on exchangeability of roots to the affine case.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jca/37","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44753749","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}
Inspired by work of Cartwright and Sturmfels, in a previous paper we introduced two classes of multigraded ideals named after them. These ideals are defined in terms of properties of their multigraded generic initial ideals. The goal of this paper is showing that three families of ideals that have recently attracted the attention of researchers are Cartwright-Sturmfels ideals. More specifically, we prove that binomial edge ideals, multigraded homogenizations of linear spaces, and multiview ideals are Cartwright-Sturmfels ideals, hence recovering and extending recent results of Herzog-Hibi-Hreinsdottir-Kahle-Rauh, Ohtani, Ardila-Boocher, Aholt-Sturmfels-Thomas, and Binglin Li. We also propose a conjecture on the rigidity of local cohomology modules of Cartwright-Sturmfels ideals, that was inspired by a theorem of Brion. We provide some evidence for the conjecture by proving it in the monomial case.
{"title":"Cartwright–Sturmfels ideals associated to graphs and linear spaces","authors":"Aldo Conca, E. D. Negri, Elisa Gorla","doi":"10.4171/JCA/2-3-2","DOIUrl":"https://doi.org/10.4171/JCA/2-3-2","url":null,"abstract":"Inspired by work of Cartwright and Sturmfels, in a previous paper we introduced two classes of multigraded ideals named after them. These ideals are defined in terms of properties of their multigraded generic initial ideals. The goal of this paper is showing that three families of ideals that have recently attracted the attention of researchers are Cartwright-Sturmfels ideals. More specifically, we prove that binomial edge ideals, multigraded homogenizations of linear spaces, and multiview ideals are Cartwright-Sturmfels ideals, hence recovering and extending recent results of Herzog-Hibi-Hreinsdottir-Kahle-Rauh, Ohtani, Ardila-Boocher, Aholt-Sturmfels-Thomas, and Binglin Li. We also propose a conjecture on the rigidity of local cohomology modules of Cartwright-Sturmfels ideals, that was inspired by a theorem of Brion. We provide some evidence for the conjecture by proving it in the monomial case.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/2-3-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42592903","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}
C. Bleak, Matthew G. Brin, M. Kassabov, J. Moore, Matthew C. B. Zaremsky
We adapt the Ping-Pong Lemma, which historically was used to study free products of groups, to the setting of the homeomorphism group of the unit interval. As a consequence, we isolate a large class of generating sets for subgroups of $mathrm{Homeo}_+(I)$ for which certain finite dynamical data can be used to determine the marked isomorphism type of the groups which they generate. As a corollary, we will obtain a criteria for embedding subgroups of $mathrm{Homeo}_+(I)$ into Richard Thompson's group $F$. In particular, every member of our class of generating sets generates a group which embeds into $F$ and in particular is not a free product. An analogous abstract theory is also developed for groups of permutations of an infinite set.
{"title":"Groups of fast homeomorphisms of the interval and the ping-pong argument","authors":"C. Bleak, Matthew G. Brin, M. Kassabov, J. Moore, Matthew C. B. Zaremsky","doi":"10.4171/JCA/25","DOIUrl":"https://doi.org/10.4171/JCA/25","url":null,"abstract":"We adapt the Ping-Pong Lemma, which historically was used to study free products of groups, to the setting of the homeomorphism group of the unit interval. As a consequence, we isolate a large class of generating sets for subgroups of $mathrm{Homeo}_+(I)$ for which certain finite dynamical data can be used to determine the marked isomorphism type of the groups which they generate. As a corollary, we will obtain a criteria for embedding subgroups of $mathrm{Homeo}_+(I)$ into Richard Thompson's group $F$. In particular, every member of our class of generating sets generates a group which embeds into $F$ and in particular is not a free product. An analogous abstract theory is also developed for groups of permutations of an infinite set.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/25","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43535797","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}
Berenstein and Kirillov have studied the action of Bender-Knuth moves on semistandard tableaux. Losev has studied a cactus group action in Kazhdan-Lusztig theory; in type $A$ this action can also be identified in the work of Henriques and Kamnitzer. We establish the relationship between the two actions. We show that the Berenstein-Kirillov group is a quotient of the cactus group. We use this to derive previously unknown relations in the Berenstein-Kirillov group. We also determine precise implications between subsets of relations in the two groups, which yields a presentation for cactus groups in terms of Bender-Knuth generators.
{"title":"The Berenstein–Kirillov group and cactus groups","authors":"Michael Chmutov, Max Glick, P. Pylyavskyy","doi":"10.4171/jca/36","DOIUrl":"https://doi.org/10.4171/jca/36","url":null,"abstract":"Berenstein and Kirillov have studied the action of Bender-Knuth moves on semistandard tableaux. Losev has studied a cactus group action in Kazhdan-Lusztig theory; in type $A$ this action can also be identified in the work of Henriques and Kamnitzer. We establish the relationship between the two actions. We show that the Berenstein-Kirillov group is a quotient of the cactus group. We use this to derive previously unknown relations in the Berenstein-Kirillov group. We also determine precise implications between subsets of relations in the two groups, which yields a presentation for cactus groups in terms of Bender-Knuth generators.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/jca/36","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70871072","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}
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