Algebras and Representation Theory最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}