Algebras and Representation Theory最新文献

{"title":"On the Fundamental Group of a Spherical Variety","authors":"Friedrich Knop","doi":"10.1007/s10468-025-10330-z","DOIUrl":"10.1007/s10468-025-10330-z","url":null,"abstract":"<div><p>Let <span>(X=G/H)</span> be a homogeneous spherical variety over an algebraically closed field of characteristic <span>(pge 0)</span>. We compute the <span>(p')</span>-parts of <span>(pi _0(H))</span> and <span>(pi _1(X))</span> from the spherical system of <i>X</i>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 6","pages":"1431 - 1437"},"PeriodicalIF":0.6,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146027188","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":"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}

{"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}

{"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}

{"title":"Projections of Nilpotent Orbits in a Simple Lie Algebra and Shared Orbits","authors":"Dmitri I. Panyushev","doi":"10.1007/s10468-025-10322-z","DOIUrl":"10.1007/s10468-025-10322-z","url":null,"abstract":"<div><p>Let <i>G</i> be a simple algebraic group and <span>(mathcal {O}subset {mathfrak g}={mathrm {Lie,}}G)</span> a nilpotent orbit. If <i>H</i> is a reductive subgroup of <i>G</i> with <span>({mathfrak h}={mathrm {Lie,}}H)</span>, then <span>({mathfrak g}={mathfrak h}oplus {mathfrak m})</span>, where <span>({mathfrak m}={mathfrak h}^perp )</span>. We consider the natural projections <span>(varvec{varphi }: overline{mathcal {O}}rightarrow mathfrak {h})</span> and <span>(varvec{psi }: overline{mathcal {O}}rightarrow mathfrak {m})</span> and two related properties of <span>((H, mathcal {O}))</span>: </p><div><div><span>$$ (mathcal {P}_1): overline{mathcal {O}}cap {mathfrak m}={0}; qquad (mathcal {P}_2): H text { has a dense orbit in } mathcal {O}. $$</span></div></div><p>It is shown that either of these properties implies that <i>H</i> is semisimple. We prove that <span>((mathcal {P}_1))</span> implies <span>((mathcal {P}_2))</span> for all <span>(mathcal {O})</span> and the converse holds for <span>(mathcal {O}_textsf{min})</span>, the minimal nilpotent orbit. If <span>((mathcal {P}_1))</span> holds, then <span>(varvec{varphi })</span> is finite and <span>([varvec{varphi }(e),varvec{psi }(e)]=0)</span> for all <span>(ein mathcal {O})</span>. Then <span>(overline{varvec{varphi }(mathcal {O})})</span> is the closure of a nilpotent <i>H</i>-orbit <span>(mathcal {O}')</span>. The orbit <span>(mathcal {O}')</span> is “shared” in the sense of Brylinski–Kostant (J. Am. Math. Soc. <b>7</b>(2), 269–298 1994). We obtain a classification of all pairs <span>((H,mathcal {O}))</span> with property <span>((mathcal {P}_1))</span> and discuss various relations between <span>(mathcal {O})</span> and <span>(mathcal {O}')</span>. In particular, we detect an omission in the list of pairs of simple groups (<i>H</i>, <i>G</i>) having a shared orbit that was given by Brylinski and Kostant. It is also proved that <span>((mathcal {P}_1))</span> for <span>((H,mathcal {O}_textsf{min}))</span> implies that <span>(overline{G{cdot }varvec{varphi }(mathcal {O}_textsf{min})}=overline{G{cdot }varvec{psi }(mathcal {O}_textsf{min})})</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"423 - 444"},"PeriodicalIF":0.5,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100279","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":"Scalar Extensions of Quiver Representations Over (mathbb {F}_1)","authors":"Markus Kleinau","doi":"10.1007/s10468-025-10326-9","DOIUrl":"10.1007/s10468-025-10326-9","url":null,"abstract":"<div><p>Let <i>V</i> and <i>W</i> be quiver representations over <span>(mathbb {F}_1)</span> and let <i>K</i> be a field. The scalar extensions <span>(V^K)</span> and <span>(W^K)</span> are quiver representations over <i>K</i> with a distinguished, very well-behaved basis. We construct a basis of <span>({{,textrm{Hom},}}_{KQ}(V^K,W^K))</span> generalising the well-known basis of the morphism spaces between string and tree modules. We use this basis to give a combinatorial characterisation of absolutely indecomposable representations. Furthermore, we show that indecomposable representations with finite nice length are absolutely indecomposable. This answers a question of Jun and Sistko.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"531 - 548"},"PeriodicalIF":0.5,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10326-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144100201","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":"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}

{"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}

{"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}

{"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}