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Entwined Modules Over Representations of Categories 类别表示之上的缠绕模块
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-05-22 DOI: 10.1007/s10468-023-10203-3
Abhishek Banerjee
{"title":"Entwined Modules Over Representations of Categories","authors":"Abhishek Banerjee","doi":"10.1007/s10468-023-10203-3","DOIUrl":"10.1007/s10468-023-10203-3","url":null,"abstract":"<div><p>We introduce a theory of modules over a representation of a small category taking values in entwining structures over a semiperfect coalgebra. This takes forward the aim of developing categories of entwined modules to the same extent as that of module categories as well as the philosophy of Mitchell of working with rings with several objects. The representations are motivated by work of Estrada and Virili, who developed a theory of modules over a representation taking values in small preadditive categories, which were then studied in the same spirit as sheaves of modules over a scheme. We also describe, by means of Frobenius and separable functors, how our theory relates to that of modules over the underlying representation taking values in small <i>K</i>-linear categories.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52368919","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
Invariants of Weyl Group Action and q-characters of Quantum Affine Algebras Weyl群作用的不变量与量子仿射代数的q-特征
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-05-09 DOI: 10.1007/s10468-023-10205-1
Rei Inoue, Takao Yamazaki
{"title":"Invariants of Weyl Group Action and q-characters of Quantum Affine Algebras","authors":"Rei Inoue,&nbsp;Takao Yamazaki","doi":"10.1007/s10468-023-10205-1","DOIUrl":"10.1007/s10468-023-10205-1","url":null,"abstract":"<div><p>Let <i>W</i> be the Weyl group corresponding to a finite dimensional simple Lie algebra <span>(mathfrak {g})</span> of rank <span>(ell )</span> and let <span>(m&gt;1)</span> be an integer. In Inoue (Lett. Math. Phys. 111(1):32, 2021), by applying cluster mutations, a <i>W</i>-action on <span>(mathcal {Y}_m)</span> was constructed. Here <span>(mathcal {Y}_m)</span> is the rational function field on <span>(cmell )</span> commuting variables, where <span>(c in { 1, 2, 3 })</span> depends on <span>(mathfrak {g})</span>. This was motivated by the <i>q</i>-character map <span>(chi _q)</span> of the category of finite dimensional representations of quantum affine algebra <span>(U_q(hat{mathfrak {g}}))</span>. We showed in Inoue (Lett. Math. Phys. 111(1):32, 2021) that when <i>q</i> is a root of unity, <span>(textrm{Im} chi _q)</span> is a subring of the <i>W</i>-invariant subfield <span>(mathcal {Y}_m^W)</span> of <span>(mathcal {Y}_m)</span>. In this paper, we give more detailed study on <span>(mathcal {Y}_m^W)</span>; for each reflection <span>(r_i in W)</span> associated to the <i>i</i>th simple root, we describe the <span>(r_i)</span>-invariant subfield <span>(mathcal {Y}_m^{r_i})</span> of <span>(mathcal {Y}_m)</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45353336","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
Littlewood-Richardson rule for generalized Schur Q-functions 广义Schur q函数的Littlewood-Richardson规则
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-04-03 DOI: 10.1007/s10468-023-10204-2
Fang Huang, Yanjun Chu, Chuanzhong Li
{"title":"Littlewood-Richardson rule for generalized Schur Q-functions","authors":"Fang Huang,&nbsp;Yanjun Chu,&nbsp;Chuanzhong Li","doi":"10.1007/s10468-023-10204-2","DOIUrl":"10.1007/s10468-023-10204-2","url":null,"abstract":"<div><p>Littlewood-Richardson rule gives the expansion formula for decomposing a product of two Schur functions as a linear sum of Schur functions, while the decomposition formula for the multiplication of two Schur Q-functions is also given as the combinatorial model by using the shifted tableaux. In this paper, we firstly use the shifted Littlewood-Richardson coefficients to give the coefficients of generalized Schur Q-function expanded as a sum of Schur Q-functions and the structure constants for the multiplication of two generalized Schur Q-functions, respectively. Then we will combine the vertex operator realizations of generalized Schur Q-functions and raising operators to construct the algebraic forms for the multiplication of generalized Schur Q-functions.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47815158","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
Representation Stability and Finite Orthogonal Groups 表示稳定性与有限正交群
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-03-29 DOI: 10.1007/s10468-023-10202-4
Arun S. Kannan, Zifan Wang
{"title":"Representation Stability and Finite Orthogonal Groups","authors":"Arun S. Kannan,&nbsp;Zifan Wang","doi":"10.1007/s10468-023-10202-4","DOIUrl":"10.1007/s10468-023-10202-4","url":null,"abstract":"<div><p>In this paper, we prove homological stability results about orthogonal groups over finite commutative rings where 2 is a unit. Inspired by Putman and Sam (2017), we construct a category <b>OrI</b>(<i>R</i>) and prove a local Noetherianity theorem for the category of <b>OrI</b>(<i>R</i>)-modules. This implies an asymptotic structure theorem for orthogonal groups. In addition, we show general homological stability theorems for orthogonal groups, with both untwisted and twisted coefficients.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-023-10202-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46047712","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 Quasi Steinberg Characters of Complex Reflection Groups 关于复反射群的拟Steinberg性质
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-02-27 DOI: 10.1007/s10468-023-10201-5
Ashish Mishra, Digjoy Paul, Pooja Singla
{"title":"On Quasi Steinberg Characters of Complex Reflection Groups","authors":"Ashish Mishra,&nbsp;Digjoy Paul,&nbsp;Pooja Singla","doi":"10.1007/s10468-023-10201-5","DOIUrl":"10.1007/s10468-023-10201-5","url":null,"abstract":"<div><p>Let <i>G</i> be a finite group and <i>p</i> be a prime number dividing the order of <i>G</i>. An irreducible character <i>χ</i> of <i>G</i> is called a quasi <i>p</i>-Steinberg character if <i>χ</i>(<i>g</i>) is nonzero for every <i>p</i>-regular element <i>g</i> in <i>G</i>. In this paper, we classify the quasi <i>p</i>-Steinberg characters of complex reflection groups <i>G</i>(<i>r</i>,<i>q</i>,<i>n</i>) and exceptional complex reflection groups. In particular, we obtain this classification for Weyl groups of type <i>B</i><sub><i>n</i></sub> and type <i>D</i><sub><i>n</i></sub>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42671291","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
Whittaker Modules Over Some Generalized Weyl Algebras 广义Weyl代数上的Whittaker模
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-02-15 DOI: 10.1007/s10468-023-10200-6
Hongjia Chen, Longhui Wang
{"title":"Whittaker Modules Over Some Generalized Weyl Algebras","authors":"Hongjia Chen,&nbsp;Longhui Wang","doi":"10.1007/s10468-023-10200-6","DOIUrl":"10.1007/s10468-023-10200-6","url":null,"abstract":"<div><p>In Benkart and Ondrus (Represent. Theory <b>13</b>, 141–164 2009), Benkart and Ondrus investigated Whittaker modules for generalized Weyl algebras. Following the results of Benkart and Ondrus, we study Whittaker modules for three special kinds of generalized Weyl algebras in this note, including Rueda’s algebras, the algebras <i>U</i><sub><i>q</i></sub>(<i>f</i>(<i>K</i>)) and <i>U</i><sub><i>q</i></sub>(<i>f</i>(<i>K</i>,<i>H</i>)). In particular, we acquire the centers of the last two classes of algebras before giving an explicit description of their simple Whittaker modules.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43416811","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
Local Characterizations for Decomposability of 2-Parameter Persistence Modules 2-参数持久模可分解性的局部刻画
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-02-14 DOI: 10.1007/s10468-022-10189-4
Magnus B. Botnan, Vadim Lebovici, Steve Oudot
{"title":"Local Characterizations for Decomposability of 2-Parameter Persistence Modules","authors":"Magnus B. Botnan,&nbsp;Vadim Lebovici,&nbsp;Steve Oudot","doi":"10.1007/s10468-022-10189-4","DOIUrl":"10.1007/s10468-022-10189-4","url":null,"abstract":"<div><p>We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corresponding to the setting of 2-parameter persistence in topological data analysis. Our indecomposables of interest belong to the so-called interval modules, which by definition are indicator representations of intervals in the poset. While the whole class of interval modules does not admit such a local characterization, we show that the subclass of rectangle modules does admit one and that it is, in some precise sense, the largest subclass to do so.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48067216","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
Free Rota-Baxter Family Algebras and Free (tri)dendriform Family Algebras 自由Rota-Baxter族代数和自由(三)树形族代数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-02-09 DOI: 10.1007/s10468-022-10198-3
Yuanyuan Zhang, Xing Gao, Dominique Manchon
{"title":"Free Rota-Baxter Family Algebras and Free (tri)dendriform Family Algebras","authors":"Yuanyuan Zhang,&nbsp;Xing Gao,&nbsp;Dominique Manchon","doi":"10.1007/s10468-022-10198-3","DOIUrl":"10.1007/s10468-022-10198-3","url":null,"abstract":"<div><p>In this paper, we first construct the free Rota-Baxter family algebra generated by some set <i>X</i> in terms of typed angularly <i>X</i>-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free (tri)dendriform family algebra.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46268700","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
Defining an Affine Partition Algebra 定义仿射划分代数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-02-02 DOI: 10.1007/s10468-022-10196-5
Samuel Creedon, Maud De Visscher
{"title":"Defining an Affine Partition Algebra","authors":"Samuel Creedon,&nbsp;Maud De Visscher","doi":"10.1007/s10468-022-10196-5","DOIUrl":"10.1007/s10468-022-10196-5","url":null,"abstract":"<div><p>We define an affine partition algebra by generators and relations and prove a variety of basic results regarding this new algebra analogous to those of other affine diagram algebras. In particular we show that it extends the Schur-Weyl duality between the symmetric group and the partition algebra. We also relate it to the affine partition category recently defined by J. Brundan and M. Vargas. Moreover, we show that this affine partition category is a <i>full</i> monoidal subcategory of the Heisenberg category.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-022-10196-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42806383","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
Morphisms represented by monomorphisms with n-torsionfree cokernel 由具有n-或无离子cokernel的单态射表示的态射
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-01-28 DOI: 10.1007/s10468-022-10192-9
Yuya Otake
{"title":"Morphisms represented by monomorphisms with n-torsionfree cokernel","authors":"Yuya Otake","doi":"10.1007/s10468-022-10192-9","DOIUrl":"10.1007/s10468-022-10192-9","url":null,"abstract":"<div><p>We introduce and study a new class of morphisms which includes morphisms represented by monomorphisms in the sense of Auslander and Bridger. As an application, we give not only an extension of Kato’s theorem on morphisms represented by monomorphisms, but also a common generalization of several results due to Auslander and Bridger that describe relationships between torsionfreeness and the grades of Ext modules.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43396482","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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