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Sequences of ICE-closed Subcategories via Preordered (tau ^{-1})-rigid Modules 通过预购(tau ^{-1}) -刚性模块的ice闭合子类别序列
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2025-07-14 DOI: 10.1007/s10468-025-10347-4
Eric J. Hanson
{"title":"Sequences of ICE-closed Subcategories via Preordered (tau ^{-1})-rigid Modules","authors":"Eric J. Hanson","doi":"10.1007/s10468-025-10347-4","DOIUrl":"10.1007/s10468-025-10347-4","url":null,"abstract":"<div><p>Let <span>(Lambda )</span> be a finite-dimensional basic algebra. Sakai recently used certain sequences of image-cokernel-extension-closed (ICE-closed) subcategories of finitely generated <span>(Lambda )</span>-modules to classify certain (generalized) intermediate <i>t</i>-structures in the bounded derived category. We classify these “contravariantly finite ICE-sequences” using concepts from <span>(tau )</span>-tilting theory. More precisely, we introduce “cogen-preordered <span>(tau ^{-1})</span>-rigid modules” as a generalization of (the dual of) the “TF-ordered <span>(tau )</span>-rigid modules” of Mendoza and Treffinger. We then establish a bijection between the set of cogen-preordered <span>(tau ^{-1})</span>-rigid modules and certain sequences of intervals of torsion-free classes. Combined with the results of Sakai, this yields a bijection with the set of contravariantly finite ICE-sequences (of finite length), and thus also with the set of <span>((m+1))</span>-intermediate <i>t</i>-structures whose aisles are homology-determined.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 4","pages":"997 - 1014"},"PeriodicalIF":0.6,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10347-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144929284","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
Invariants that are Covering Spaces and their Hopf Algebras 覆盖空间的不变量及其Hopf代数
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2025-07-08 DOI: 10.1007/s10468-025-10343-8
Ehud Meir
{"title":"Invariants that are Covering Spaces and their Hopf Algebras","authors":"Ehud Meir","doi":"10.1007/s10468-025-10343-8","DOIUrl":"10.1007/s10468-025-10343-8","url":null,"abstract":"<div><p>In a previous paper by the author a universal ring of invariants for algebraic structures of a given type was constructed. This ring is a polynomial algebra that is generated by certain trace diagrams. It was shown that this ring admits the structure of a rational positive self adjoint Hopf algebra (abbreviated rational PSH-algebra), and was conjectured that it always admits a lattice that is a PSH-algebra, a structure that was introduced by Zelevinsky. In this paper we solve this conjecture, showing that the universal ring of invariants splits as the tensor product of rational PSH-algebras that are either polynomial algebras in a single variable, or admit a lattice that is a PSH-algebra. We do so by considering diagrams as topological spaces, and using tools from the theory of covering spaces. As an application we derive a formula that connects Kronecker coefficients with finite index subgroups of free groups and representations of their Weyl groups, and a formula for the number of conjugacy classes of finite index subgroup in a finitely generated group that admits a surjective homomorphism onto the group of integers.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 4","pages":"883 - 920"},"PeriodicalIF":0.6,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10343-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144929288","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
Demazure Filtration of Tensor Product Modules of Current Lie Algebra of Type (A_1) 一类当前李代数张量积模的变形滤波 (A_1)
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-04-29 DOI: 10.1007/s10468-025-10334-9
Divya Setia, Tanusree Khandai
{"title":"Demazure Filtration of Tensor Product Modules of Current Lie Algebra of Type (A_1)","authors":"Divya Setia,&nbsp;Tanusree Khandai","doi":"10.1007/s10468-025-10334-9","DOIUrl":"10.1007/s10468-025-10334-9","url":null,"abstract":"<div><p>In this paper we study the structure of finite-dimensional representations of the current Lie algebra of type <span>(A_1)</span>, <span>(mathfrak {sl}_2[t])</span>, which are obtained by taking tensor products of local Weyl modules with Demazure modules. We show that such a representation admits a Demazure flag and obtain a closed formula for the graded multiplicities of the level 2 Demazure modules in the filtration of the tensor product of two local Weyl modules for <span>(mathfrak {sl}_2[t])</span>. Furthermore, we show that the tensor product of a local Weyl module with an irreducible <span>(mathfrak {sl}_2[t])</span> module admits a Demazure filtration and derive the graded character of such tensor product modules. In conjunction with the results of Chari et al. (SIGMA Symmetry Integrability Geom. Methods Appl. <b>10</b>(032), 2014), our findings provide evidence for the conjecture in Blanton (2017) that the tensor product of Demazure modules of levels m and n respectively has a filtration by Demazure modules of level m + n.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"679 - 707"},"PeriodicalIF":0.5,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100238","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
The Defining Characteristic Case of the Representations of (textrm{GL}_{n}) and (textrm{SL}_{n}) Over Principal Ideal Local Rings 主理想局部环上(textrm{GL}_{n})和(textrm{SL}_{n})表示的定义特征情形
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-04-21 DOI: 10.1007/s10468-025-10333-w
Nariel Monteiro
{"title":"The Defining Characteristic Case of the Representations of (textrm{GL}_{n}) and (textrm{SL}_{n}) Over Principal Ideal Local Rings","authors":"Nariel Monteiro","doi":"10.1007/s10468-025-10333-w","DOIUrl":"10.1007/s10468-025-10333-w","url":null,"abstract":"<div><p>Let <span>(W_{r}(mathbb {F}_{q}))</span> be the ring of Witt vectors of length <i>r</i> with residue field <span>(mathbb {F}_{q})</span> of characteristic <i>p</i>. In this paper, we study the defining characteristic case of the representations of <span>(textrm{GL}_{n})</span> and <span>(textrm{SL}_{n})</span> over the principal ideal local rings <span>(W_{r}(mathbb {F}_{q}))</span> and <span>(mathbb {F}_{q}[t]/t^{r})</span>. Let <span>({textbf{G}})</span> be either <span>(textrm{GL}_{n})</span> or <span>(textrm{SL}_{n})</span> and <i>F</i> a perfect field of characteristic <i>p</i>, we prove that for most <i>p</i> the group algebras <span>(F[{textbf{G}}(W_{r}(mathbb {F}_{q}))])</span> and <span>(F[{textbf{G}}(mathbb {F}_{q}[t]/t^{r})])</span> are not stably equivalent of Morita type. Thus, the group algebras <span>(F[{textbf{G}}(W_{r}(mathbb {F}_{q}))])</span> and <span>(F[{textbf{G}}(mathbb {F}_{q}[t]/t^{r})])</span> are not isomorphic in the defining characteristic case.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"669 - 677"},"PeriodicalIF":0.5,"publicationDate":"2025-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100146","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
Involutions in Coxeter groups 考克斯特群体的内讧
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-04-08 DOI: 10.1007/s10468-025-10332-x
Anna Reimann, Yuri Santos Rego, Petra Schwer, Olga Varghese
{"title":"Involutions in Coxeter groups","authors":"Anna Reimann,&nbsp;Yuri Santos Rego,&nbsp;Petra Schwer,&nbsp;Olga Varghese","doi":"10.1007/s10468-025-10332-x","DOIUrl":"10.1007/s10468-025-10332-x","url":null,"abstract":"<div><p>We combinatorially characterize the number <span>(textrm{cc}_2)</span> of conjugacy classes of involutions in any Coxeter group in terms of higher rank odd graphs. This notion naturally generalizes the concept of odd graphs, used previously to count the number of conjugacy classes of reflections. Moreover, we provide formulae for finite and affine types, besides computing <span>(textrm{cc}_2)</span> for all triangle groups and RACGs.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"647 - 667"},"PeriodicalIF":0.5,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10332-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100240","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 Interpolation Categories for the Hyperoctahedral Group 关于高八面体群的插值范畴
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-04-01 DOI: 10.1007/s10468-025-10331-y
Th. Heidersdorf, G. Tyriard
{"title":"On Interpolation Categories for the Hyperoctahedral Group","authors":"Th. Heidersdorf,&nbsp;G. Tyriard","doi":"10.1007/s10468-025-10331-y","DOIUrl":"10.1007/s10468-025-10331-y","url":null,"abstract":"<div><p>Two different types of Deligne categories have been defined to interpolate the finite dimensional complex representations of the hyperoctahedral group. The first one, initially defined by Knop and then further studied by Likeng and Savage, uses a categorical analogue of the permutation representation as a tensor generator. The second one, due to Flake and Maassen, is tensor generated by a categorical analogue of the reflection representation. We construct a symmetric monoidal functor between the two and show that it is an equivalence of symmetric monoidal categories.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"613 - 646"},"PeriodicalIF":0.5,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10331-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100167","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
A Partial Classification of Simple Regular Representations of Bimodules Type ((2,,2)) Over the Field of Laurent Series 双模型简单正则表示((2,,2))在Laurent级数域上的部分分类
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-03-26 DOI: 10.1007/s10468-025-10327-8
Hernán Giraldo, David Reynoso-Mercado, Pedro Rizzo
{"title":"A Partial Classification of Simple Regular Representations of Bimodules Type ((2,,2)) Over the Field of Laurent Series","authors":"Hernán Giraldo,&nbsp;David Reynoso-Mercado,&nbsp;Pedro Rizzo","doi":"10.1007/s10468-025-10327-8","DOIUrl":"10.1007/s10468-025-10327-8","url":null,"abstract":"<div><p>In this paper, we use Galois descent techniques to find suitable representatives of the regular simple representations of the species of type (2, 2) over <span>(k_n:= k[varepsilon ^{1/n}])</span>, where <i>n</i> is a positive integer and <span>(k:=mathbb {C}(!(varepsilon )!))</span> is the field of Laurent series over the complexes. These regular representations are essential for the definition of canonical algebras. Our work is inspired by the work done for species of type (1, 4) on <i>k</i> in Geiss and Reynoso-Mercado (Bol. Soc. Mat. Mex. <b>30</b>(3):87, 2024). We presents all the regular simple representations on the <i>n</i>-crown quiver, and from these, we establish a partial classification of regular simple representations of bimodules type (2, 2).</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"549 - 577"},"PeriodicalIF":0.5,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10327-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100236","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
PBW-deformations of Graded Algebras with Braiding Relations 具有编织关系的梯度代数的pbw变形
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-03-18 DOI: 10.1007/s10468-025-10328-7
Yujie Gao, Shilin Yang
{"title":"PBW-deformations of Graded Algebras with Braiding Relations","authors":"Yujie Gao,&nbsp;Shilin Yang","doi":"10.1007/s10468-025-10328-7","DOIUrl":"10.1007/s10468-025-10328-7","url":null,"abstract":"<div><p>The aim of this paper is to describe all PBW-deformations of the connected graded <span>({mathbb {K}})</span>-algebra <span>(mathcal {A})</span> generated by <span>(x_i, 1le ile n,)</span> with the braiding relations: </p><div><div><span>$$begin{aligned} left{ begin{array}{ll} x_i^2=0, 1le ile n, x_ix_j=x_jx_i, {|j-i|} &gt;1, x_ix_{i+1}x_i=x_{i+1}x_ix_{i+1}, 1le ile n-1. end{array}right. end{aligned}$$</span></div></div><p>Firstly, the complexity <span>(mathcal {C}({mathcal {A}}))</span> of the algebra <span>({mathcal {A}})</span> is computed. Then all PBW-deformations of <span>(mathcal {A})</span> when <span>(nge 2)</span> are given explicitly with the help of the general PBW-deformation theory introduced by Cassidy and Shelton. Finally, it is shown that each non-trivial PBW-deformation of <span>(mathcal {A})</span> is isomorphic to a Iwahori-Hecke algebra <span>(H_q(n+1))</span> (of type <i>A</i>) with <i>n</i> generators and an appropriate parameter <i>q</i>. Here, trivial PBW-deformations of <span>({mathcal {A}})</span> mean that those PBW-deformations that are isomorphic to <span>({mathcal {A}}.)</span></p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"579 - 611"},"PeriodicalIF":0.5,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100229","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
(D_7^{(1)})- Geometric Crystal at the Spin Node (D_7^{(1)})-旋转节点的几何晶体
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-03-12 DOI: 10.1007/s10468-025-10325-w
Kailash C. Misra, Toshiki Nakashima, Suchada Pongprasert
{"title":"(D_7^{(1)})- Geometric Crystal at the Spin Node","authors":"Kailash C. Misra,&nbsp;Toshiki Nakashima,&nbsp;Suchada Pongprasert","doi":"10.1007/s10468-025-10325-w","DOIUrl":"10.1007/s10468-025-10325-w","url":null,"abstract":"<div><p>Let <span>(mathfrak {g})</span> be an affine Lie algebra with index set <span>(varvec{I})</span> = {<b>0, 1, 2,</b> <span>(ldots , varvec{n}})</span>. It is conjectured that for each Dynkin node <span>(varvec{k} in varvec{I} setminus {{textbf {0}}})</span> the affine Lie algebra <span>(mathfrak {g})</span> has a positive geometric crystal. In this paper, we construct a positive geometric crystal for the affine Lie algebra <span>(varvec{D}_{textbf {7}}^{{textbf {(1)}}})</span> corresponding to the Dynkin spin node <span>(varvec{k}= {textbf {7}})</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"509 - 530"},"PeriodicalIF":0.5,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100225","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
Primed Decomposition Tableaux and Extended Queer Crystals 启动分解表和扩展酷儿晶体
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-03-11 DOI: 10.1007/s10468-025-10323-y
Eric Marberg, Kam Hung Tong
{"title":"Primed Decomposition Tableaux and Extended Queer Crystals","authors":"Eric Marberg,&nbsp;Kam Hung Tong","doi":"10.1007/s10468-025-10323-y","DOIUrl":"10.1007/s10468-025-10323-y","url":null,"abstract":"<div><p>Our previous work introduced a category of extended queer crystals, whose connected normal objects have unique highest weight elements and characters that are Schur <i>Q</i>-polynomials. The initial models for such crystals were based on semistandard shifted tableaux. Here, we introduce a simpler construction using certain “primed” decomposition tableaux, which slightly generalize the decomposition tableaux used in work of Grantcharov et al. This leads to a new, shorter proof of the highest weight properties of the normal subcategory of extended queer crystals. Along the way, we analyze a primed extension of Grantcharov et al.’s insertion scheme for decomposition tableaux.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"445 - 482"},"PeriodicalIF":0.5,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10323-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100278","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
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