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Algebras and Representation Theory最新文献

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Reflection Representations of Coxeter Groups and Homology of Coxeter Graphs Coxeter 群的反射表示和 Coxeter 图的同源性
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-12-07 DOI: 10.1007/s10468-023-10242-w
Hongsheng Hu
{"title":"Reflection Representations of Coxeter Groups and Homology of Coxeter Graphs","authors":"Hongsheng Hu","doi":"10.1007/s10468-023-10242-w","DOIUrl":"10.1007/s10468-023-10242-w","url":null,"abstract":"<div><p>We study and classify a class of representations (called generalized geometric representations) of a Coxeter group of finite rank. These representations can be viewed as a natural generalization of the geometric representation. The classification is achieved by using characters of the integral homology group of certain graphs closely related to the Coxeter graph. On this basis, we also provide an explicit description of those representations on which the defining generators of the Coxeter group act by reflections.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"961 - 994"},"PeriodicalIF":0.5,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138545711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Frobenius Kernels of Algebraic Supergroups and Steinberg’s Tensor Product Theorem 代数超群的Frobenius核与Steinberg张量积定理
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-12-04 DOI: 10.1007/s10468-023-10240-y
Taiki Shibata
{"title":"Frobenius Kernels of Algebraic Supergroups and Steinberg’s Tensor Product Theorem","authors":"Taiki Shibata","doi":"10.1007/s10468-023-10240-y","DOIUrl":"10.1007/s10468-023-10240-y","url":null,"abstract":"<div><p>For a split quasireductive supergroup <span>(mathbbm {G})</span> defined over a field, we study structure and representation of Frobenius kernels <span>(mathbbm {G}_r)</span> of <span>(mathbbm {G})</span> and we give a necessary and sufficient condition for <span>(mathbbm {G}_r)</span> to be unimodular in terms of the root system of <span>(mathbbm {G})</span>. We also establish Steinberg’s tensor product theorem for <span>(mathbbm {G})</span> under some natural assumptions.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"927 - 959"},"PeriodicalIF":0.5,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-023-10240-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Middle Terms of AR-sequences of Graded Kronecker Modules 分级Kronecker模ar序列的中项
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-11-29 DOI: 10.1007/s10468-023-10241-x
Jie Liu
{"title":"Middle Terms of AR-sequences of Graded Kronecker Modules","authors":"Jie Liu","doi":"10.1007/s10468-023-10241-x","DOIUrl":"10.1007/s10468-023-10241-x","url":null,"abstract":"<div><p>Let <span>((T(n),Omega ))</span> be the covering of the generalized Kronecker quiver <i>K</i>(<i>n</i>), where <span>(Omega )</span> is a bipartite orientation. Then there exists a reflection functor <span>(sigma )</span> on the category <span>({{,textrm{mod},}}(T(n),Omega ))</span>. Suppose that <span>(0rightarrow Xrightarrow Yrightarrow Zrightarrow 0)</span> is an AR-sequence in the regular component <span>(mathcal {D})</span> of <span>({{,textrm{mod},}}(T(n),Omega ))</span>, and <i>b</i>(<i>Z</i>) is the number of flow modules in the <span>(sigma )</span>-orbit of <i>Z</i>. Then the middle term <i>Y</i> is a sink (source or flow) module if and only if <span>(sigma Z)</span> is a sink (source or flow) module. Moreover, their radii and centers satisfy <span>(r(Y)=r(sigma Z)+1)</span> and <span>(C(Y)=C(sigma Z))</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"911 - 926"},"PeriodicalIF":0.5,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-023-10241-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Braid Group Action on Exceptional Sequences for Weighted Projective Lines 加权投影线例外序列上的辫群作用
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-11-23 DOI: 10.1007/s10468-023-10243-9
Edson Ribeiro Alvares, Eduardo Nascimento Marcos, Hagen Meltzer
{"title":"On the Braid Group Action on Exceptional Sequences for Weighted Projective Lines","authors":"Edson Ribeiro Alvares,&nbsp;Eduardo Nascimento Marcos,&nbsp;Hagen Meltzer","doi":"10.1007/s10468-023-10243-9","DOIUrl":"10.1007/s10468-023-10243-9","url":null,"abstract":"<div><p>We give a new and intrinsic proof of the transitivity of the braid group action on the set of full exceptional sequences of coherent sheaves on a weighted projective line. We do not use the corresponding result of Crawley-Boevey for modules over hereditary algebras. As an application we prove that the strongest global dimension of the category of coherent sheaves on a weighted projective line <span>(mathbb {X})</span> does not depend on the parameters of <span>(mathbb {X})</span>. Finally we prove that the determinant of the matrix obtained by taking the values of <i>n</i> <span>(mathbb {Z})</span>-linear functions defined on the Grothendieck group <span>(textrm{K}_0(mathbb {X}) simeq mathbb {Z}^n )</span> of the elements of a full exceptional sequence is an invariant, up to sign.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"897 - 909"},"PeriodicalIF":0.5,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
High Order Free Hyperplane Arrangements in 3-Dimensional Vector Spaces 三维向量空间中的高阶自由超平面排列
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-11-04 DOI: 10.1007/s10468-023-10237-7
Norihiro Nakashima
{"title":"High Order Free Hyperplane Arrangements in 3-Dimensional Vector Spaces","authors":"Norihiro Nakashima","doi":"10.1007/s10468-023-10237-7","DOIUrl":"10.1007/s10468-023-10237-7","url":null,"abstract":"<div><p>Holm introduced <i>m</i>-free <span>(ell )</span>-arrangements which is a generalization of free arrangements, while he asked whether all <span>(ell )</span>-arrangements are <i>m</i>-free for <i>m</i> large enough. Recently Abe and the author gave a negative answer to this question when <span>(ell ge 4)</span>. In this paper we verify that 3-arrangements <span>(mathscr {A})</span> are <i>m</i>-free and compute the <i>m</i>-exponents for all <span>(mge |mathscr {A}|+2)</span>, where <span>(|mathscr {A}|)</span> is the cardinality of <span>(mathscr {A})</span>. Hence Holm’s question has a positive answer when <span>(ell =3)</span>. Finally we prove that 3-dimensional Weyl arrangements of types A and B are <i>m</i>-free for all <span>(mge 0)</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"877 - 896"},"PeriodicalIF":0.5,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135726755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Silting Reduction in Exact Categories 精确分类的淤积减少量
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-10-28 DOI: 10.1007/s10468-023-10238-6
Yu Liu, Panyue Zhou, Yu Zhou, Bin Zhu
{"title":"Silting Reduction in Exact Categories","authors":"Yu Liu,&nbsp;Panyue Zhou,&nbsp;Yu Zhou,&nbsp;Bin Zhu","doi":"10.1007/s10468-023-10238-6","DOIUrl":"10.1007/s10468-023-10238-6","url":null,"abstract":"<div><p>Presilting and silting subcategories in extriangulated categories were introduced by Adachi and Tsukamoto recently, which are generalizations of those concepts in triangulated categories. Exact categories and triangulated categories are extriangulated categories. In this paper, we prove that the Gabriel-Zisman localization <span>(mathcal {B}/(textsf{thick}hspace{.01in}mathcal W))</span> of an exact category <span>(mathcal {B})</span> with respect to a presilting subcategory <span>(mathcal W)</span> satisfying certain condition can be realized as a subfactor category of <span>(mathcal {B})</span>. Afterwards, we discuss the relation between silting subcategories and tilting subcategories in exact categories, which gives us a kind of important examples of our results. In particular, for a finite dimensional Gorenstein algebra, we get the relative version of the description of the singularity category due to Happel and Chen-Zhang.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"847 - 876"},"PeriodicalIF":0.5,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136232927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Nakayama Algebras and Fuchsian Singularities 中山代数和富奇异性
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-10-20 DOI: 10.1007/s10468-023-10236-8
Helmut Lenzing, Hagen Meltzer, Shiquan Ruan
{"title":"Nakayama Algebras and Fuchsian Singularities","authors":"Helmut Lenzing,&nbsp;Hagen Meltzer,&nbsp;Shiquan Ruan","doi":"10.1007/s10468-023-10236-8","DOIUrl":"10.1007/s10468-023-10236-8","url":null,"abstract":"<div><p>This present paper is devoted to the study of a class of Nakayama algebras <span>(N_n(r))</span> given by the path algebra of the equioriented quiver <span>(mathbb {A}_n)</span> subject to the nilpotency degree <i>r</i> for each sequence of <i>r</i> consecutive arrows. We show that the Nakayama algebras <span>(N_n(r))</span> for certain pairs (<i>n</i>, <i>r</i>) can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras <span>(N_n(r))</span> of Fuchsian type, that is, derived equivalent to the bounded derived categories of extended canonical algebras. We also provide a new way to prove the classification result on Nakayama algebras of piecewise hereditary type, which have been done by Happel–Seidel before.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"815 - 846"},"PeriodicalIF":0.5,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135617429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Kazhdan-Lusztig Algorithm for Whittaker Modules with Arbitrary Infinitesimal Characters 具有任意无穷小字符的惠特克模块的卡兹丹-卢兹蒂算法
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-10-16 DOI: 10.1007/s10468-023-10222-0
Qixian Zhao
{"title":"Kazhdan-Lusztig Algorithm for Whittaker Modules with Arbitrary Infinitesimal Characters","authors":"Qixian Zhao","doi":"10.1007/s10468-023-10222-0","DOIUrl":"10.1007/s10468-023-10222-0","url":null,"abstract":"<div><p>Let <span>(mathfrak {g})</span> be a complex semisimple Lie algebra. We give a description of characters of irreducible Whittaker modules for <span>(mathfrak {g})</span> with any infinitesimal character, along with a Kazhdan-Lusztig algorithm for computing them. This generalizes Miličić-Soergel’s and Romanov’s results for integral infinitesimal characters. As a special case, we recover the non-integral Kazhdan-Lusztig conjecture for Verma modules.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"767 - 814"},"PeriodicalIF":0.5,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136114083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
On the Properties of Acyclic Sign-Skew-Symmetric Cluster Algebras 论无环符号歪斜对称簇代数的性质
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-10-14 DOI: 10.1007/s10468-023-10239-5
Siyang Liu
{"title":"On the Properties of Acyclic Sign-Skew-Symmetric Cluster Algebras","authors":"Siyang Liu","doi":"10.1007/s10468-023-10239-5","DOIUrl":"10.1007/s10468-023-10239-5","url":null,"abstract":"<div><p>We study the tropical dualities and properties of exchange graphs for the totally sign-skew-symmetric cluster algebra under a condition. We prove that the condition always holds for acyclic cluster algebras, then all results hold for the acyclic case.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"745 - 766"},"PeriodicalIF":0.5,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135803539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Dipper Donkin Quantized Matrix Algebra and Reflection Equation Algebra at Root of Unity 统一根的 Dipper Donkin 量化矩阵代数和反射方程代数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-10-10 DOI: 10.1007/s10468-023-10235-9
Sanu Bera, Snehashis Mukherjee
{"title":"Dipper Donkin Quantized Matrix Algebra and Reflection Equation Algebra at Root of Unity","authors":"Sanu Bera,&nbsp;Snehashis Mukherjee","doi":"10.1007/s10468-023-10235-9","DOIUrl":"10.1007/s10468-023-10235-9","url":null,"abstract":"<div><p>In this article the quantized matrix algebras as in the title have been studied at a root of unity. A full classification of simple modules over such quantized matrix algebras of degree 2 along with some finite-dimensional indecomposable modules are explicitly presented.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"723 - 744"},"PeriodicalIF":0.5,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136295889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
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