Journal of Combinatorial Algebra最新文献_第5页

Presentations of categories of modules using the Cautis–Kamnitzer–Morrison principle 使用Cautis–Kamnitzer–Morrison原理演示模块类别

IF 0.9 2区数学

Journal of Combinatorial Algebra Pub Date : 2018-03-08 DOI: 10.4171/JCA/27

Giulian Wiggins

{"title":"Presentations of categories of modules using the Cautis–Kamnitzer–Morrison principle","authors":"Giulian Wiggins","doi":"10.4171/JCA/27","DOIUrl":"https://doi.org/10.4171/JCA/27","url":null,"abstract":"We use duality theorems to obtain presentations of some categories of modules. To derive these presentations we generalize a result of Cautis-Kamnitzer-Morrison [arXiv:1210.6437v4]: \u0000Let $mathfrak{g}$ be a reductive Lie algebra, and $A$ an algebra, both over $mathbb{C}$. Consider a $(mathfrak{g} , A)$-bimodule $P$ in which \u0000(a) $P$ has a multiplicity free decomposition into irreducible $(mathfrak{g} , A)$-bimodules. \u0000(b) $P$ is \"saturated\" i.e. for any irreducible $mathfrak{g}$-module $V$, if every weight of $V$ is a weight of $P$, then $V$ is a submodule of $P$. \u0000We show that statements (a) and (b) are necessary and sufficient conditions for the existence of an isomorphism of categories between the full subcategory of $mathcal{R}ep A$ whose objects are $mathfrak{g}$-weight spaces of $P$, and a quotient of the category version of Lusztig's idempotented form, $dot{{mathcal{U}}} mathfrak{g}$, formed by setting to zero all morphisms factoring through a collection of objects in $dot{{mathcal{U}}} mathfrak{g}$ depending on $P$. This is essentially a categorical version of the identification of generalized Schur algebras with quotients of Lusztig's idempotented forms given by Doty in [arXiv:math/0305208]. \u0000Applied to Schur-Weyl Duality we obtain a diagrammatic presentation of the full subcategory of $mathcal{R}ep S_d$ whose objects are direct sums of permutation modules, as well as an explicit description of the $otimes$-product of morphisms between permutation modules. Applied to Brauer-Schur-Weyl Duality we obtain diagrammatic presentations of subcategories of $mathcal{R}ep mathcal{B}_{d}^{(- 2n)}$ and $mathcal{R}ep mathcal{B}_{r,s}^{(n)}$ whose Karoubi completion is the whole of $mathcal{R}ep mathcal{B}_{d}^{(- 2n)}$ and $mathcal{R}ep mathcal{B}_{r,s}^{(n)}$ respectively.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2018-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JCA/27","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46229215","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}

引用次数: 0

A partial order on bipartitions from the generalized Springer correspondence 广义施普林格对应的二分上的偏序

IF 0.9 2区数学

Journal of Combinatorial Algebra Pub Date : 2018-01-29 DOI: 10.4171/JCA/2-3-4

Jianqiao Xia

引用次数: 1

Soficity and variations on Higman’s group 希格曼群的稳定性和变异性

IF 0.9 2区数学

Journal of Combinatorial Algebra Pub Date : 2017-12-19 DOI: 10.4171/JCA/26

M. Kassabov, Vivian Kuperberg, T. Riley

引用次数: 2

Rational embeddings of hyperbolic groups 双曲群的有理嵌入

IF 0.9 2区数学

Journal of Combinatorial Algebra Pub Date : 2017-11-22 DOI: 10.4171/JCA/52

James M. Belk, C. Bleak, Francesco Matucci

引用次数: 9

Frieze patterns over integers and other subsets of the complex numbers 整数和复数的其他子集上的Frieze模式

IF 0.9 2区数学

Journal of Combinatorial Algebra Pub Date : 2017-11-10 DOI: 10.4171/JCA/29

M. Cuntz, T. Holm

{"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":null,"pages":null},"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}

引用次数: 23

Polynomially-bounded Dehn functions of groups 群的多项式有界Dehn函数

IF 0.9 2区数学

Journal of Combinatorial Algebra Pub Date : 2017-10-02 DOI: 10.4171/JCA/2-4-1

A. Olshanskii

引用次数: 7

On congruence half-factorial Krull monoids with cyclic class group 关于具有循环子群的同余半阶乘Krull半群

IF 0.9 2区数学

Journal of Combinatorial Algebra Pub Date : 2017-09-04 DOI: 10.4171/jca/34

A. Plagne, W. Schmid

引用次数: 13

An affine almost positive roots model 仿射几乎正根模型