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Clebsch-Gordan Coefficients for Macdonald Polynomials 麦克唐纳多项式的Clebsch-Gordan系数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-12-07 DOI: 10.1007/s10468-024-10303-8
Aritra Bhattacharya, Arun Ram
{"title":"Clebsch-Gordan Coefficients for Macdonald Polynomials","authors":"Aritra Bhattacharya,&nbsp;Arun Ram","doi":"10.1007/s10468-024-10303-8","DOIUrl":"10.1007/s10468-024-10303-8","url":null,"abstract":"<div><p>In this paper we use the double affine Hecke algebra to compute the Macdonald polynomial products <span>(E_ell P_m)</span> and <span>(P_ell P_m)</span> for type <span>(SL_2)</span> and type <span>(GL_2)</span> Macdonald polynomials. Our method follows the ideas of Martha Yip but executes a compression to reduce the sum from <span>(2cdot 3^{ell -1})</span> signed terms to <span>(2ell )</span> positive terms. We show that our rule for <span>(P_ell P_m)</span> is equivalent to a special case of the Pieri rule of Macdonald. Our method shows that computing <span>(E_ell {textbf {1}}_0)</span> and <span>({textbf {1}}_0 E_ell {textbf {1}}_0)</span> in terms of a special basis of the double affine Hecke algebra provides universal compressed formulas for multiplication by <span>(E_ell )</span> and <span>(P_ell )</span>. The formulas for a specific products <span>(E_ell P_m)</span> and <span>(P_ell P_m)</span> are obtained by evaluating the universal formulas at <span>(t^{-frac{1}{2}}q^{-frac{m}{2}})</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2423 - 2464"},"PeriodicalIF":0.5,"publicationDate":"2024-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10303-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994514","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
Isomorphism Problems and Groups of Automorphisms for Ore Extensions (K[x][y; ffrac{d}{dx} ]) (Prime Characteristic) 矿扩展的同构问题和自同构群(K[x][y; ffrac{d}{dx} ])（素特征）
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-12-04 DOI: 10.1007/s10468-024-10301-w
V. V. Bavula
{"title":"Isomorphism Problems and Groups of Automorphisms for Ore Extensions (K[x][y; ffrac{d}{dx} ]) (Prime Characteristic)","authors":"V. V. Bavula","doi":"10.1007/s10468-024-10301-w","DOIUrl":"10.1007/s10468-024-10301-w","url":null,"abstract":"<div><p>Let <span>(Lambda (f) = K[x][y; ffrac{d}{dx} ])</span> be an Ore extension of a polynomial algebra <i>K</i>[<i>x</i>] over an arbitrary field <i>K</i> of characteristic <span>(p&gt;0)</span> where <span>(fin K[x])</span>. For each polynomial <i>f</i>, the automorphism group of the algebras <span>(Lambda (f))</span> is explicitly described. The automorphism group <span>(textrm{Aut}_K(Lambda (f))=mathbb {S}rtimes G_f)</span> is a semidirect product of two explicit groups where <span>(G_f)</span> is the <i>eigengroup</i> of the polynomial <i>f</i> (the set of all automorphisms of <i>K</i>[<i>x</i>] such that <i>f</i> is their common eigenvector). For each polynomial <i>f</i>, the eigengroup <span>(G_f)</span> is explicitly described. It is proven that every subgroup of <span>(textrm{Aut}_K(K[x]))</span> is the eigengroup of a polynomial. It is proven that the Krull and global dimensions of the algebra <span>(Lambda (f))</span> are 2. The prime, completely prime, primitive and maximal ideals of the algebra <span>(Lambda (f))</span> are classified.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2389 - 2422"},"PeriodicalIF":0.5,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10301-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994558","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
Hopf Algebra (Co)actions on Rational Functions 有理函数上的Hopf代数（Co）作用
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-11-23 DOI: 10.1007/s10468-024-10294-6
Ulrich Krähmer, Blessing Bisola Oni
{"title":"Hopf Algebra (Co)actions on Rational Functions","authors":"Ulrich Krähmer,&nbsp;Blessing Bisola Oni","doi":"10.1007/s10468-024-10294-6","DOIUrl":"10.1007/s10468-024-10294-6","url":null,"abstract":"<div><p>In the theory of quantum automorphism groups, one constructs Hopf algebras acting on an algebra <i>K</i> from certain algebra morphisms <span>( sigma :K rightarrow textrm{M}_n(K))</span>. This approach is applied to the field <span>(K=k(t))</span> of rational functions, and it is investigated when these actions restrict to actions on the coordinate ring <span>(B=k[t^2,t^3])</span> of the cusp. An explicit example is described in detail and shown to define a new quantum homogeneous space structure on the cusp.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2187 - 2216"},"PeriodicalIF":0.5,"publicationDate":"2024-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10294-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995681","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
3-Preprojective Algebras of Type D D型的3-预投影代数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-11-16 DOI: 10.1007/s10468-024-10297-3
Jordan Haden
{"title":"3-Preprojective Algebras of Type D","authors":"Jordan Haden","doi":"10.1007/s10468-024-10297-3","DOIUrl":"10.1007/s10468-024-10297-3","url":null,"abstract":"<div><p>We present a family of selfinjective algebras of type D, which arise from the 3-preprojective algebras of type A by taking a <span>(mathbb {Z}_3)</span>-quotient. We show that a subset of these are themselves 3-preprojective algebras, and that the associated 2-representation-finite algebras are fractional Calabi-Yau. In addition, we show our work is connected to modular invariants for SU(3).</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2295 - 2320"},"PeriodicalIF":0.5,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10297-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995278","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
Symmetries of Algebras Captured by Actions of Weak Hopf Algebras 弱Hopf代数作用俘获的代数的对称性
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-11-13 DOI: 10.1007/s10468-024-10295-5
Fabio Calderón, Hongdi Huang, Elizabeth Wicks, Robert Won
{"title":"Symmetries of Algebras Captured by Actions of Weak Hopf Algebras","authors":"Fabio Calderón,&nbsp;Hongdi Huang,&nbsp;Elizabeth Wicks,&nbsp;Robert Won","doi":"10.1007/s10468-024-10295-5","DOIUrl":"10.1007/s10468-024-10295-5","url":null,"abstract":"<div><p>In this paper, we present a generalization of well-established results regarding symmetries of <span>(Bbbk )</span>-algebras, where <span>(Bbbk )</span> is a field. Traditionally, for a <span>(Bbbk )</span>-algebra <i>A</i>, the group of <span>(Bbbk )</span>-algebra automorphisms of <i>A</i> captures the symmetries of <i>A</i> via group actions. Similarly, the Lie algebra of derivations of <i>A</i> captures the symmetries of <i>A</i> via Lie algebra actions. In this paper, given a category <span>(mathcal {C})</span> whose objects possess <span>(Bbbk )</span>-linear monoidal categories of modules, we introduce an objec <span>(operatorname {Sym}_{mathcal {C}}(A))</span> that captures the symmetries of <i>A</i> via actions of objects in <span>(mathcal {C})</span>. Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected <span>(Bbbk )</span>-algebra <i>A</i>, some of its symmetries are naturally captured within the weak Hopf framework.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2217 - 2266"},"PeriodicalIF":0.5,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994464","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
Equational Quantum Quasigroups 方程量子拟群
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-11-11 DOI: 10.1007/s10468-024-10300-x
Jonathan D. H. Smith
{"title":"Equational Quantum Quasigroups","authors":"Jonathan D. H. Smith","doi":"10.1007/s10468-024-10300-x","DOIUrl":"10.1007/s10468-024-10300-x","url":null,"abstract":"<div><p>As a self-dual framework to unify the study of quasigroups and Hopf algebras, quantum quasigroups are defined using a quantum analogue of the combinatorial approach to classical quasigroups, merely requiring invertibility of the left and right composites. In this paper, quantum quasigroups are redefined with a quantum analogue of the equational approach to classical quasigroups. Here, the left and right composites of auxiliary quantum quasigroups participate in diagrams whose commutativity witnesses the required invertibilities. Whenever the original and two auxiliary quantum quasigroups appear on an equal footing, the triality symmetry of the language of equational quasigroups is replicated. In particular, the problem arises as to when this triality emerges in the Hopf algebra context.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2355 - 2387"},"PeriodicalIF":0.5,"publicationDate":"2024-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994358","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 J-folded Alcove Paths and Combinatorial Representations of Affine Hecke Algebras 仿射Hecke代数的j折叠凹形路径与组合表示
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-11-05 DOI: 10.1007/s10468-024-10293-7
Jérémie Guilhot, Eloise Little, James Parkinson
{"title":"On J-folded Alcove Paths and Combinatorial Representations of Affine Hecke Algebras","authors":"Jérémie Guilhot,&nbsp;Eloise Little,&nbsp;James Parkinson","doi":"10.1007/s10468-024-10293-7","DOIUrl":"10.1007/s10468-024-10293-7","url":null,"abstract":"<div><p>We introduce the combinatorial model of <i>J</i>-folded alcove paths in an affine Weyl group and construct representations of affine Hecke algebras using this model. We study boundedness of these representations, and we state conjectures linking our combinatorial formulae to Kazhdan-Lusztig theory and Opdam’s Plancherel Theorem.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2131 - 2185"},"PeriodicalIF":0.5,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10293-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994378","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
Generalised Lat-Igusa-Todorov Algebras and Morita Contexts 广义latu - igusa - todorov代数与Morita上下文
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-10-14 DOI: 10.1007/s10468-024-10289-3
Marcelo Lanzilotta, José Vivero
{"title":"Generalised Lat-Igusa-Todorov Algebras and Morita Contexts","authors":"Marcelo Lanzilotta,&nbsp;José Vivero","doi":"10.1007/s10468-024-10289-3","DOIUrl":"10.1007/s10468-024-10289-3","url":null,"abstract":"<div><p>In this paper we define (special) GLIT classes and (special) GLIT algebras. We prove that GLIT algebras, which generalise Lat-Igusa-Todorov algebras, satisfy the finitistic dimension conjecture and give several properties and examples. In addition we show that special GLIT algebras are exactly those that have finite finitistic dimension. Lastly we study Morita algebras arising from a Morita context and give conditions for them to be (special) GLIT in terms of the algebras and bimodules used in their definition. As a consequence we obtain simple conditions for a triangular matrix algebra to be (special) GLIT and also prove that the tensor product of a GLIT <span>(mathbb {K})</span>-algebra with a path algebra of a finite quiver without oriented cycles is GLIT.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2045 - 2066"},"PeriodicalIF":0.5,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10289-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994470","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
Highest Weight Modules for Affine and Loop Superalgebras of (mathfrak {osp}_{1|2}(mathbb C)) 的仿射超代数和环超代数的最大权模 (mathfrak {osp}_{1|2}(mathbb C))
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-09-28 DOI: 10.1007/s10468-024-10292-8
Fulin Chen, Xin Huang, Shaobin Tan
{"title":"Highest Weight Modules for Affine and Loop Superalgebras of (mathfrak {osp}_{1|2}(mathbb C))","authors":"Fulin Chen,&nbsp;Xin Huang,&nbsp;Shaobin Tan","doi":"10.1007/s10468-024-10292-8","DOIUrl":"10.1007/s10468-024-10292-8","url":null,"abstract":"<div><p>This paper is about the highest weight module theory for affine superalgebra <span>(widetilde{mathfrak g})</span> of <span>({mathfrak g}={mathfrak {osp}_{1|2}(mathbb C)})</span> and loop superalgebra <span>({mathfrak g}{otimes }{mathbb {C}}[t,t^{-1}])</span>. Among the main results, we obtain (i) a necessary and sufficient condition for Verma type <span>(ell )</span>-highest weight <span>(widetilde{mathfrak g})</span>-modules to be irreducible; (ii) a free field(-like) realization of all irreducible <span>(ell )</span>-highest weight <span>(widetilde{mathfrak g})</span>-modules; (iii) a character formula for all irreducible <span>(ell )</span>-highest weight <span>(widetilde{mathfrak g})</span>-modules with finite dimensional weight spaces. We also obtain three similar results for highest weight <span>({mathfrak g}{otimes }{mathbb {C}}[t,t^{-1}])</span>-modules.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2099 - 2130"},"PeriodicalIF":0.5,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995695","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
Automorphisms of Quantum Toroidal Algebras from an Action of the Extended Double Affine Braid Group 扩展双仿射编织群作用下量子环面代数的自同构
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-09-24 DOI: 10.1007/s10468-024-10291-9
Duncan Laurie
{"title":"Automorphisms of Quantum Toroidal Algebras from an Action of the Extended Double Affine Braid Group","authors":"Duncan Laurie","doi":"10.1007/s10468-024-10291-9","DOIUrl":"10.1007/s10468-024-10291-9","url":null,"abstract":"<div><p>We first construct an action of the extended double affine braid group <span>(mathcal {ddot{B}})</span> on the quantum toroidal algebra <span>(U_{q}(mathfrak {g}_{textrm{tor}}))</span> in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of <span>(mathcal {ddot{B}})</span> we produce automorphisms and anti-involutions of <span>(U_{q}(mathfrak {g}_{textrm{tor}}))</span> which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements <i>C</i> and <span>(k_{0}^{a_{0}}dots k_{n}^{a_{n}})</span> up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type <i>A</i> due to Miki which have been instrumental in the study of the structure and representation theory of <span>(U_{q}(mathfrak {sl}_{n+1,textrm{tor}}))</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2067 - 2097"},"PeriodicalIF":0.5,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10291-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995845","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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