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The Parity of Lusztig’s Restriction Functor and Green’s Formula for a Quiver with Automorphism 自同构颤振的Lusztig限制函子和Green公式的宇称性
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-03-04 DOI: 10.1007/s10468-025-10324-x
Jiepeng Fang, Yixin Lan, Yumeng Wu
{"title":"The Parity of Lusztig’s Restriction Functor and Green’s Formula for a Quiver with Automorphism","authors":"Jiepeng Fang,&nbsp;Yixin Lan,&nbsp;Yumeng Wu","doi":"10.1007/s10468-025-10324-x","DOIUrl":"10.1007/s10468-025-10324-x","url":null,"abstract":"<div><p>In Fang et al. (J. Algebra <b>618</b>, 67–95 2023), Fang-Lan-Xiao proved a formula about Lusztig’s induction and restriction functors which can induce Green’s formula for the path algebra of a quiver over a finite field via the trace map. In this paper, we generalize their formula to that for the mixed semisimple perverse sheaves for a quiver with an automorphism. By applying the trace map, we obtain Green’s formula for any finite-dimensional hereditary algebra over a finite field.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"483 - 508"},"PeriodicalIF":0.5,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100198","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
Torsion Pairs and t-Structures in (textrm{D}^{b}(text {coh}(mathbb {X}))) 扭力副与t型结构 (textrm{D}^{b}(text {coh}(mathbb {X})))
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-25 DOI: 10.1007/s10468-025-10321-0
Edson Ribeiro Alvares, D. D. Silva
{"title":"Torsion Pairs and t-Structures in (textrm{D}^{b}(text {coh}(mathbb {X})))","authors":"Edson Ribeiro Alvares,&nbsp;D. D. Silva","doi":"10.1007/s10468-025-10321-0","DOIUrl":"10.1007/s10468-025-10321-0","url":null,"abstract":"<div><p>We present a bijection between torsion pairs in <span>(text {coh}(mathbb ))</span> with corresponding <i>t</i>-structures in <span>(textrm{D}^{b}(text {coh}(mathbb {X})))</span> where <span>(mathbb {X})</span> represents a weighted projective line. When focusing on the split case, we derive a bijection between this class and corresponding torsion pairs in the module category of a concealed canonical algebra. Additionally, we demonstrate that if the aisle of a split t-structure in the derived category of a hereditary category contains an Ext-projective object, then it admits a tilting complex. Finally, we use the structure of the Auslander-Reiten quiver of <span>(textrm{D}^{b}(text {coh}(mathbb {X})))</span> in order to classify split t-structures in <span>(textrm{D}^{b}(text {coh}(mathbb {X})))</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"407 - 421"},"PeriodicalIF":0.5,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100233","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 Common Zeros of Characters of Finite Groups 关于有限群的公零
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-20 DOI: 10.1007/s10468-025-10320-1
Mark L. Lewis, Lucia Morotti, Emanuele Pacifici, Lucia Sanus, Hung P. Tong-Viet
{"title":"On Common Zeros of Characters of Finite Groups","authors":"Mark L. Lewis,&nbsp;Lucia Morotti,&nbsp;Emanuele Pacifici,&nbsp;Lucia Sanus,&nbsp;Hung P. Tong-Viet","doi":"10.1007/s10468-025-10320-1","DOIUrl":"10.1007/s10468-025-10320-1","url":null,"abstract":"<div><p>Let <i>G</i> be a finite group, and let <span>(textrm{Irr}(G))</span> denote the set of the irreducible complex characters of <i>G</i>. An element <span>(gin G)</span> is called a <i>vanishing element</i> of <i>G</i> if there exists <span>(chi in textrm{Irr}(G))</span> such that <span>(chi (g)=0)</span> (i.e., <i>g</i> is a <i>zero</i> of <span>(chi )</span>) and, in this case, the conjugacy class <span>(g^G)</span> of <i>g</i> in <i>G</i> is called a <i>vanishing conjugacy class</i>. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group <i>G</i> such that every non-linear <span>(chi in textrm{Irr}(G))</span> vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"395 - 406"},"PeriodicalIF":0.5,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10320-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100322","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
Mickelsson Algebras via Hasse Diagrams 通过Hasse图的Mickelsson代数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-12 DOI: 10.1007/s10468-024-10311-8
Andrey Mudrov, Vladimir Stukopin
{"title":"Mickelsson Algebras via Hasse Diagrams","authors":"Andrey Mudrov,&nbsp;Vladimir Stukopin","doi":"10.1007/s10468-024-10311-8","DOIUrl":"10.1007/s10468-024-10311-8","url":null,"abstract":"<div><p>Let <span>(mathcal {A})</span> be an associative algebra containing either classical or quantum universal enveloping algebra of a semi-simple complex Lie algebra <span>(mathfrak {g})</span>. We present a construction of the Mickelsson algebra <span>(Z(mathcal {A},mathfrak {g}))</span> relative to the left ideal in <span>(mathcal {A})</span> generated by positive root vectors. Our method employs a calculus on Hasse diagrams associated with classical or quantum <span>(mathfrak {g})</span>-modules. We give an explicit expression for a PBW basis in <span>(Z(mathcal {A},mathfrak {g}))</span> in the case when <span>(mathcal {A}=U(mathfrak {a}))</span> of a finite-dimensional Lie algebra <span>(mathfrak {a}supset mathfrak {g})</span>. For <span>(mathcal {A}=U_q(mathfrak {a}))</span> and <span>(mathfrak {g})</span> the commutant of a Levi subalgebra in <span>(mathfrak {a})</span>, we construct a PBW basis in terms of quantum Lax operators, upon extension of the ground ring of scalars to <span>(mathbb {C}[[hbar ]])</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"353 - 367"},"PeriodicalIF":0.5,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100223","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
Dirac Cohomology and (Theta )-correspondence for Complex Dual Pairs 复对偶对的狄拉克上同调和(Theta ) -对应
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-08 DOI: 10.1007/s10468-025-10319-8
S. Afentoulidis-Almpanis, G. Liu, S. Mehdi
{"title":"Dirac Cohomology and (Theta )-correspondence for Complex Dual Pairs","authors":"S. Afentoulidis-Almpanis,&nbsp;G. Liu,&nbsp;S. Mehdi","doi":"10.1007/s10468-025-10319-8","DOIUrl":"10.1007/s10468-025-10319-8","url":null,"abstract":"<div><p>We study the behavior of Dirac cohomology under Howe’s <span>(Theta )</span>-correspondence in the case of complex reductive dual pairs. More precisely, if <span>((G_1,G_2))</span> is a complex reductive dual pair with <span>(G_1)</span> and <span>(G_2)</span> viewed as real groups, we describe those Harish-Chandra modules <span>(pi _1)</span> of <span>(G_1)</span> with nonzero Dirac cohomology whose <span>(Theta )</span>-liftings <span>(Theta (pi _1))</span> still have nonzero Dirac cohomology. In this case, we compute explicitly the Dirac cohomology of <span>(Theta (pi _1))</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"337 - 352"},"PeriodicalIF":0.5,"publicationDate":"2025-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10319-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553874","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
Endomorphism Algebras of Equivariant Exceptional Collections 等变异常集合的自同态代数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-07 DOI: 10.1007/s10468-025-10313-0
Andreas Krug, Erik Nikolov
{"title":"Endomorphism Algebras of Equivariant Exceptional Collections","authors":"Andreas Krug,&nbsp;Erik Nikolov","doi":"10.1007/s10468-025-10313-0","DOIUrl":"10.1007/s10468-025-10313-0","url":null,"abstract":"<div><p>Given an action of a finite group on a triangulated category with a suitable strong exceptional collection, a construction of Elagin produces an associated strong exceptional collection on the equivariant category. In the untwisted case, we prove that the endomorphism algebra of the induced exceptional collection is the basic reduction of the skew group algebra of the endomorphism algebra of the original exceptional collection.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"193 - 210"},"PeriodicalIF":0.5,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10313-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553871","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
Exhaustion of Supercuspidal Representations of p-adic Spin Groups: Semisimple Characters p-adic Spin Groups 的 Supercuspidal Representations 的穷尽：半简单字符
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-04 DOI: 10.1007/s10468-025-10318-9
Ngô Văn Định
{"title":"Exhaustion of Supercuspidal Representations of p-adic Spin Groups: Semisimple Characters","authors":"Ngô Văn Định","doi":"10.1007/s10468-025-10318-9","DOIUrl":"10.1007/s10468-025-10318-9","url":null,"abstract":"<div><p>Let <span>(varvec{widehat{G}})</span> be a spin group over a locally compact non-archimedean local field <span>(varvec{F})</span> of odd residual characteristic. We defined lifted self-dual semisimple characters for <span>(varvec{widehat{G}})</span> and from them, we constructed a large class of supercuspidal representations of <span>(varvec{widehat{G}})</span> when <span>(varvec{F})</span> is of characteristic zero. In this paper, we show that any positive level supercuspidal representation of <span>(varvec{widehat{G}})</span> contains such a character.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"315 - 335"},"PeriodicalIF":0.5,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553795","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 Mirković–Vilonen Polytopes
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-04 DOI: 10.1007/s10468-025-10317-w
Pierre Baumann
{"title":"On Mirković–Vilonen Polytopes","authors":"Pierre Baumann","doi":"10.1007/s10468-025-10317-w","DOIUrl":"10.1007/s10468-025-10317-w","url":null,"abstract":"<div><p>Mirković–Vilonen polytopes encode in a geometrical way the numerical data present in the Kashiwara crystal <span>(B(infty ))</span> of a semisimple group <i>G</i>. We retrieve these polytopes from the coproduct of the Hopf algebra <span>(mathscr {O}(N))</span> of regular functions on a maximal unipotent subgroup <i>N</i> of <i>G</i>. We bring attention to a remarkable behavior that the classical bases (dual canonical, dual semicanonical, Mirković–Vilonen) of <span>(mathscr {O}(N))</span> manifest with respect to the extremal points of these polytopes, which extends the crystal operations. This study leans on a notion of stability for graded bialgebras.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"369 - 393"},"PeriodicalIF":0.5,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100169","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
Subalgebras of Étale Algebras in Braided Fusion Categories 编织融合类中的Étale代数子代数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-02-03 DOI: 10.1007/s10468-025-10314-z
Sebastian Burciu
{"title":"Subalgebras of Étale Algebras in Braided Fusion Categories","authors":"Sebastian Burciu","doi":"10.1007/s10468-025-10314-z","DOIUrl":"10.1007/s10468-025-10314-z","url":null,"abstract":"<div><p>In (Davydov et al. Selecta Mathematica (N.S.) <b>19</b>, 237–269 2013, Rem. 3.4) the authors asked the question if any étale subalgebra of an étale algebra in a braided fusion category is also étale. We give a positive answer to this question if the braided fusion category <span>(mathcal {C})</span> is pseudo-unitary and non-degenerate. In the case of a pseudo-unitary fusion category we also give a new description of the lattice correspondence from (Davydov et al. J. für die reine und angewandte Mathematik. <b>677</b>, 135–177 2013, Theorem 4.10). This new description enables us to describe the two binary operations on the lattice of fusion subcategories.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"211 - 238"},"PeriodicalIF":0.5,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143553841","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
Graded Triangular Bases 渐变三角形底
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2025-01-29 DOI: 10.1007/s10468-025-10315-y
Jonathan Brundan
{"title":"Graded Triangular Bases","authors":"Jonathan Brundan","doi":"10.1007/s10468-025-10315-y","DOIUrl":"10.1007/s10468-025-10315-y","url":null,"abstract":"<div><p>This article develops a practical technique for studying representations of <span>(Bbbk )</span>-linear categories arising in the categorification of quantum groups. We work in terms of locally unital algebras which are <span>(mathbb {Z})</span>-graded with graded pieces that are finite-dimensional and bounded below, developing a theory of <i>graded triangular bases</i> for such algebras. The definition is a graded extension of the notion of triangular basis as formulated in Brundan and Stroppel (Mem. Amer. Math. Soc. <b>293</b>(1459), vii+152 2024). However, in the general graded setting, finitely generated projective modules often fail to be Noetherian, so that existing results from the study of highest weight categories are not directly applicable. Nevertheless, we show that there is still a good theory of <i>standard modules</i>. In motivating examples arising from Kac-Moody 2-categories, these modules categorify the PBW bases for the modified forms of quantum groups constructed by Wang.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"239 - 280"},"PeriodicalIF":0.5,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143554006","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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