Vinit Sinha, Amit Kuber, Annoy Sengupta, Bhargav Kale
{"title":"Hammocks for Non-Domestic String Algebras","authors":"Vinit Sinha, Amit Kuber, Annoy Sengupta, Bhargav Kale","doi":"10.1007/s10468-024-10285-7","DOIUrl":"10.1007/s10468-024-10285-7","url":null,"abstract":"<div><p>We show that the order type of the simplest version of a hammock for string algebras lies in the class of <i>finite description</i> linear orders–the smallest class of linear orders containing <span>(textbf{0})</span>, <span>(textbf{1})</span>, and that is closed under isomorphisms, finite order sum, anti-lexicographic product with <span>(omega )</span> and <span>(omega ^*)</span>, and shuffle of finite subsets–using condensation (localization) of linear orders as a tool. We also introduce two finite subsets of the set of bands and use them to describe the location of left <span>(mathbb {N})</span>-strings in the completion of hammocks.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 5","pages":"1869 - 1908"},"PeriodicalIF":0.5,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204751","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}
{"title":"Translation Hopf Algebras and Hopf Heaps","authors":"Tomasz Brzeziński, Małgorzata Hryniewicka","doi":"10.1007/s10468-024-10283-9","DOIUrl":"10.1007/s10468-024-10283-9","url":null,"abstract":"<div><p>To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural structure of a Hopf heap with the translation Hopf algebra isomorphic to the acting Hopf algebra. It is then shown that this assignment establishes an equivalence between categories of Hopf heaps and Hopf-Galois co-objects.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 5","pages":"1805 - 1819"},"PeriodicalIF":0.5,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10283-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204752","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}
{"title":"Hopf Algebras with the Dual Chevalley Property of Finite Corepresentation Type","authors":"Jing Yu, Kangqiao Li, Gongxiang Liu","doi":"10.1007/s10468-024-10284-8","DOIUrl":"10.1007/s10468-024-10284-8","url":null,"abstract":"<div><p>Let <i>H</i> be a finite-dimensional Hopf algebra over an algebraically closed field <span>(Bbbk )</span> with the dual Chevalley property. We prove that <i>H</i> is of finite corepresentation type if and only if it is coNakayama, if and only if the link quiver <span>(textrm{Q}(H))</span> of <i>H</i> is a disjoint union of basic cycles, if and only if the link-indecomposable component <span>(H_{(1)})</span> containing <span>(Bbbk 1)</span> is a pointed Hopf algebra and the link quiver of <span>(H_{(1)})</span> is a basic cycle.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 5","pages":"1821 - 1867"},"PeriodicalIF":0.5,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226101","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}
{"title":"Quantum Frobenius Splittings and Cluster Structures","authors":"Jinfeng Song","doi":"10.1007/s10468-024-10281-x","DOIUrl":"10.1007/s10468-024-10281-x","url":null,"abstract":"<div><p>We prove that the duals of the quantum Frobenius morphisms and their splittings by Lusztig are compatible with quantum cluster monomials. After specialization, we deduce that the canonical Frobenius splittings on flag varieties are compatible with cluster algebra structures on Schubert cells.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 5","pages":"1773 - 1797"},"PeriodicalIF":0.5,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141948726","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}
{"title":"Representations of the Restricted Lie Superalgebra p(n)","authors":"Chaowen Zhang","doi":"10.1007/s10468-024-10280-y","DOIUrl":"10.1007/s10468-024-10280-y","url":null,"abstract":"<div><p>In this paper, we first study the center of a quotient of the reduced enveloping algebra of <i>p</i>(<i>n</i>), then we use the obtained results to investigate the restricted representation of <i>p</i>(<i>n</i>).</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 4","pages":"1709 - 1733"},"PeriodicalIF":0.5,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885878","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}
{"title":"A Note on the Codegree of Finite Groups","authors":"Mark L. Lewis, Quanfu Yan","doi":"10.1007/s10468-024-10282-w","DOIUrl":"10.1007/s10468-024-10282-w","url":null,"abstract":"<div><p>Let <span>(chi )</span> be an irreducible character of a group <i>G</i>, and <span>(S_c(G)=sum _{chi in textrm{Irr}(G)}textrm{cod}(chi ))</span> be the sum of the codegrees of the irreducible characters of <i>G</i>. Write <span>(textrm{fcod} (G)=frac{S_c(G)}{|G|}.)</span> We aim to explore the structure of finite groups in terms of <span>(textrm{fcod} (G).)</span> On the other hand, we determine the lower bound of <span>(S_c(G))</span> for nonsolvable groups and prove that if <i>G</i> is nonsolvable, then <span>(S_c(G)geqslant S_c(A_5)=68,)</span> with equality if and only if <span>(Gcong A_5.)</span> Additionally, we show that there is a solvable group so that it has the codegree sum as <span>(A_5.)</span></p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 5","pages":"1799 - 1804"},"PeriodicalIF":0.5,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10282-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885879","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}
{"title":"Categorical Idempotents Via Shifted 0-Affine Algebras","authors":"You-Hung Hsu","doi":"10.1007/s10468-024-10279-5","DOIUrl":"10.1007/s10468-024-10279-5","url":null,"abstract":"<div><p>We show that a categorical action of shifted 0-affine algebra naturally gives two families of pairs of complementary idempotents in the triangulated monoidal category of triangulated endofunctors for each weight category. Consequently, we obtain two families of pairs of complementary idempotents in the triangulated monoidal category <span>({textrm{D}}^btextrm{Coh}(G/P times G/P))</span>. As an application, this provides examples where the projection functors of a semiorthogonal decomposition are kernel functors, and we determine the generators of the component categories in the Grassmannians case.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 4","pages":"1735 - 1772"},"PeriodicalIF":0.5,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872896","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}
{"title":"Some Universal Constructions in Representation Theory","authors":"Claudio Procesi","doi":"10.1007/s10468-024-10278-6","DOIUrl":"10.1007/s10468-024-10278-6","url":null,"abstract":"<div><p>We add some further constructions to the general Theory of Cayley Hamilton algebras developed in the papers [Procesi, C.: J. Algebra <b>107</b>, 63–74 \u0000(1987), Procesi, C.: Naz.LinceiRend. Lincei Mat.Appl.<b>32</b>(1), 23–61 \u0000(2021), Procesi, C.: Indag. Math. (N.S.) <b>32</b>(6), 1190–1228 \u0000(2021) ].</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2001 - 2017"},"PeriodicalIF":0.5,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141864635","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}
{"title":"Weak Silting Modules","authors":"Qianqian Yuan, Hailou Yao","doi":"10.1007/s10468-024-10276-8","DOIUrl":"10.1007/s10468-024-10276-8","url":null,"abstract":"<div><p>It is well-established that weak <i>n</i>-tilting modules serve as generalizations of both <i>n</i>-tilting and <i>n</i>-cotilting modules. The primary objective of this paper is to delineate the characterizations of weak <i>n</i>-silting modules and elaborate on their applications. Specifically, we aim to establish the \"triangular relation\" within the framework of silting theory in a module category, and provide novel characterizations of weak <i>n</i>-tilting modules. Furthermore, we delve into the properties of <i>n</i>-(co)silting modules and their interrelations with some other types of modules. Additionally, we explore the conditions under which a weak <i>n</i>-silting module can be classified as partial <i>n</i>-silting, weak <i>n</i>-tilting, or partial <i>n</i>-tilting. Notably, we establish and prove that Bazzoni’s renowned characterization of pure-injectivity for cotilting modules remains valid for weak <i>n</i>-silting modules with respect to <span>(mathcal {F}_{mathbb {T}})</span>. Lastly, we investigate weak <i>n</i>-silting and weak <i>n</i>-tilting objects in a morphism category.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 4","pages":"1681 - 1707"},"PeriodicalIF":0.5,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780754","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}
{"title":"Butler’s Method Applied to (mathbb {Z}_p[C_ptimes C_p])-Permutation Modules","authors":"John W. MacQuarrie, Marlon Estanislau","doi":"10.1007/s10468-024-10277-7","DOIUrl":"10.1007/s10468-024-10277-7","url":null,"abstract":"<div><p>Let <i>G</i> be a finite <i>p</i>-group with normal subgroup <span>(varvec{N})</span> of order <span>(varvec{p})</span>. The first author and Zalesskii have previously shown that a <span>(mathbb {Z}_p)</span> <span>(varvec{G})</span>-lattice is a permutation module if, and only if, its <span>(varvec{N})</span>-invariants, its <span>(varvec{N})</span>-coinvariants, and a third module are all <i>G</i>/<i>N</i> permutation modules over <span>(mathbb {Z}_p, mathbb {Z}_p)</span> and <span>(mathbb {Z}_p)</span> respectively. The necessity of the first two conditions is easily shown but the necessity of the third was not known. We apply a correspondence due to Butler, which associates to a <span>(mathbb {Z}_p)</span> <span>(varvec{G})</span>-lattice for an abelian <span>(varvec{p})</span>-group a set of simple combinatorial data, to demonstrate the necessity of the conditions, using the correspondence to construct highly non-trivial counterexamples to the claim that if both the <span>(varvec{N})</span>-invariants and the <span>(varvec{N})</span>-coinvariants of a given lattice <span>(varvec{U})</span> are permutation modules, then so is <span>(varvec{U})</span>. Our approach, which is new, is to translate the desired properties to the combinatorial side, find the counterexample there, and translate it back to a lattice.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 4","pages":"1671 - 1680"},"PeriodicalIF":0.5,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141742556","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}
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