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Stability of $$imath $$ canonical Bases of Locally Finite Type 局部有限类型的 $$imath $$ 规范基础的稳定性
IF 0.7 3区 数学
Transformation Groups Pub Date : 2024-09-14 DOI: 10.1007/s00031-024-09876-x
Hideya Watanabe
{"title":"Stability of $$imath $$ canonical Bases of Locally Finite Type","authors":"Hideya Watanabe","doi":"10.1007/s00031-024-09876-x","DOIUrl":"https://doi.org/10.1007/s00031-024-09876-x","url":null,"abstract":"<p>We prove the stability conjecture of <span>(imath )</span>canonical bases, which was raised by Huanchen Bao and Weiqiang Wang in 2016, for all locally finite types. To this end, we characterize the trivial module over the <span>(imath )</span>quantum groups of such type at <span>(q = infty )</span>. This result can be seen as a very restrictive version of the <span>(imath )</span>crystal base theory for locally finite types.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142257865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Counting Parabolic Principal G-Bundles with Nilpotent Sections Over $$mathbb {P}^{1}$$ 在 $$mathbb {P}^{1}$ 上数抛物线主 G 束带的无热点部分
IF 0.7 3区 数学
Transformation Groups Pub Date : 2024-09-13 DOI: 10.1007/s00031-024-09877-w
Rahul Singh
{"title":"Counting Parabolic Principal G-Bundles with Nilpotent Sections Over $$mathbb {P}^{1}$$","authors":"Rahul Singh","doi":"10.1007/s00031-024-09877-w","DOIUrl":"https://doi.org/10.1007/s00031-024-09877-w","url":null,"abstract":"<p>Let <i>G</i> be a split connected reductive group over <span>(mathbb {F}_q)</span> and let <span>(mathbb {P}^1)</span> be the projective line over <span>(mathbb {F}_q)</span>. Firstly, we give an explicit formula for the number of <span>(mathbb {F}_{q})</span>-rational points of generalized Steinberg varieties of <i>G</i>. Secondly, for each principal <i>G</i>-bundle over <span>(mathbb {P}^1)</span>, we give an explicit formula counting the number of triples consisting of parabolic structures at 0 and <span>(infty )</span> and a compatible nilpotent section of the associated adjoint bundle. In the case of <span>(GL_{n})</span> we calculate a generating function of such volumes re-deriving a result of Mellit.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"71 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142213012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Regularity of Unipotent Elements in Total Positivity 全正中单能元素的规则性
IF 0.7 3区 数学
Transformation Groups Pub Date : 2024-09-05 DOI: 10.1007/s00031-024-09871-2
Haiyu Chen, Kaitao Xie
{"title":"Regularity of Unipotent Elements in Total Positivity","authors":"Haiyu Chen, Kaitao Xie","doi":"10.1007/s00031-024-09871-2","DOIUrl":"https://doi.org/10.1007/s00031-024-09871-2","url":null,"abstract":"<p>Let <i>G</i> be a connected reductive group split over <span>(mathbb R)</span>. We show that every unipotent element in the totally nonnegative monoid of <i>G</i> is regular in some Levi subgroups, confirming a conjecture of Lusztig.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"64 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142213010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Rational Singularities for Moment Maps of Totally Negative Quivers 完全负引信矩图的有理奇异性
IF 0.7 3区 数学
Transformation Groups Pub Date : 2024-08-09 DOI: 10.1007/s00031-024-09873-0
Tanguy Vernet
{"title":"Rational Singularities for Moment Maps of Totally Negative Quivers","authors":"Tanguy Vernet","doi":"10.1007/s00031-024-09873-0","DOIUrl":"https://doi.org/10.1007/s00031-024-09873-0","url":null,"abstract":"<p>We prove that the zero-fiber of the moment map of a totally negative quiver has rational singularities. Our proof consists in generalizing dimension bounds on jet spaces of this fiber, which were introduced by Budur. We also transfer the rational singularities property to other moduli spaces of objects in 2-Calabi-Yau categories, based on recent work of Davison. This has interesting arithmetic applications on quiver moment maps and moduli spaces of objects in 2-Calabi-Yau categories. First, we generalize results of Wyss on the asymptotic behaviour of counts of jets of quiver moment maps over finite fields. Moreover, we interpret the limit of counts of jets on a given moduli space as its <i>p</i>-adic volume under a canonical measure analogous to the measure built by Carocci, Orecchia and Wyss on certain moduli spaces of coherent sheaves.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"16 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Filtered Fiber Functors Over a General Base 一般基上的滤波纤维函数
IF 0.7 3区 数学
Transformation Groups Pub Date : 2024-08-08 DOI: 10.1007/s00031-024-09875-y
Paul Ziegler
{"title":"Filtered Fiber Functors Over a General Base","authors":"Paul Ziegler","doi":"10.1007/s00031-024-09875-y","DOIUrl":"https://doi.org/10.1007/s00031-024-09875-y","url":null,"abstract":"<p>We prove that every filtered fiber functor on the category of dualizable representations of a smooth affine group scheme with enough dualizable representations comes from a graded fiber functor.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"60 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Lattices in $$mathbb {R}^nrtimes textrm{SL}_2(mathbb {R})$$ 在 $$mathbb {R}^nrtimes textrm{SL}_2(mathbb {R})$$ 中的网格
IF 0.7 3区 数学
Transformation Groups Pub Date : 2024-08-07 DOI: 10.1007/s00031-024-09874-z
M. M. Radhika, Sandip Singh
{"title":"Lattices in $$mathbb {R}^nrtimes textrm{SL}_2(mathbb {R})$$","authors":"M. M. Radhika, Sandip Singh","doi":"10.1007/s00031-024-09874-z","DOIUrl":"https://doi.org/10.1007/s00031-024-09874-z","url":null,"abstract":"<p>We determine the existence of cocompact lattices in groups of the form <span>(textrm{V}rtimes textrm{SL}_2(mathbb {R}))</span>, where <span>(textrm{V})</span> is a finite dimensional real representation of <span>(textrm{SL}_2(mathbb {R}))</span>. It turns out that the answer depends on the parity of <span>(dim (textrm{V}))</span> when the representation is irreducible.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"45 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Classification and Double Commutant Property for Dual Pairs in an Orthosymplectic Lie Supergroup 正交李超群中双对的分类和双换向性质
IF 0.7 3区 数学
Transformation Groups Pub Date : 2024-08-07 DOI: 10.1007/s00031-024-09868-x
Allan Merino, Hadi Salmasian
{"title":"Classification and Double Commutant Property for Dual Pairs in an Orthosymplectic Lie Supergroup","authors":"Allan Merino, Hadi Salmasian","doi":"10.1007/s00031-024-09868-x","DOIUrl":"https://doi.org/10.1007/s00031-024-09868-x","url":null,"abstract":"<p>Let <span>(textrm{E}=textrm{E}_{bar{0}}oplus textrm{E}_{bar{1}})</span> be a real or complex <span>(mathbb {Z}_2)</span>-graded vector space equipped with an even supersymmetric bilinear form that restricts to a symplectic form on <span>(textrm{E}_{bar{0}})</span> and an orthogonal form on <span>(textrm{E}_{bar{1}})</span>. We obtain a full classification of reductive dual pairs in the (real or complex) orthosymplectic Lie superalgebra <span>(mathfrak {spo})</span>(E) and its associated Lie supergroup <span>({textbf {SpO}}(textrm{E}))</span>. Similar to the purely even case, dual pairs are divided into two subclasses: Type I and Type II. The main difference with the purely even case occurs in the characterization of (super)hermitian forms on modules over division superalgebras. We then use this classification to prove that for a reductive dual pair <span>((mathscr {G},, mathscr {G}') = ((textrm{G},, mathfrak {g}),, (textrm{G}',, mathfrak {g}')))</span> in <span>({textbf {SpO}}(textrm{E}))</span>, the superalgebra <span>({textbf {WC}}(textrm{E})^{mathscr {G}})</span> that consists of <span>(mathscr {G})</span>-invariant elements in the Weyl-Clifford algebra <span>({textbf {WC}}(textrm{E}))</span>, when it is equipped with the natural action of the orthosymplectic Lie supergroup <span>({textbf {SpO}}(textrm{E}))</span>, is generated by the Lie superalgebra <span>(mathfrak {g}')</span>. As an application of the latter double commutant property, we prove that Howe duality holds for the dual pairs <span>(( {{textbf {SpO}}}(2n|1),, {{textbf {OSp}}}(2k|2l)) subseteq {{textbf {SpO}}}(mathbb {C}^{2k|2l} otimes mathbb {C}^{2n|1}))</span>.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"4 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141942976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 Non-Normal Subvarieties of the Moduli Space of Riemann Surfaces 论黎曼曲面模空间的非正态子变量
IF 0.7 3区 数学
Transformation Groups Pub Date : 2024-07-27 DOI: 10.1007/s00031-024-09870-3
Rubén A. Hidalgo, Jennifer Paulhus, Sebastián Reyes-Carocca, Anita M. Rojas
{"title":"On Non-Normal Subvarieties of the Moduli Space of Riemann Surfaces","authors":"Rubén A. Hidalgo, Jennifer Paulhus, Sebastián Reyes-Carocca, Anita M. Rojas","doi":"10.1007/s00031-024-09870-3","DOIUrl":"https://doi.org/10.1007/s00031-024-09870-3","url":null,"abstract":"<p>In this article, we consider certain irreducible subvarieties of the moduli space of compact Riemann surfaces determined by the specification of actions of finite groups. We address the general problem of determining which among them are non-normal subvarieties of the moduli space. We obtain several new examples of subvarieties with this property.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"39 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141778228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Extensions of Deformed W-algebras via qq-characters 通过 qq 字符的变形 W 后缀扩展
IF 0.7 3区 数学
Transformation Groups Pub Date : 2024-07-23 DOI: 10.1007/s00031-024-09869-w
B. Feigin, M. Jimbo, E. Mukhin
{"title":"Extensions of Deformed W-algebras via qq-characters","authors":"B. Feigin, M. Jimbo, E. Mukhin","doi":"10.1007/s00031-024-09869-w","DOIUrl":"https://doi.org/10.1007/s00031-024-09869-w","url":null,"abstract":"<p>We use combinatorics of <i>qq</i>-characters to study extensions of deformed <i>W</i>-algebras. We describe additional currents and part of the relations in the cases of <span>(mathfrak {gl}(n|m))</span> and <span>(mathfrak {osp}(2|2n))</span>.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"52 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141778229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 the Canonical Bundle of Complex Solvmanifolds and Applications to Hypercomplex Geometry 论复数 Solvmanifolds 的典范束及其在超复数几何中的应用
IF 0.7 3区 数学
Transformation Groups Pub Date : 2024-07-10 DOI: 10.1007/s00031-024-09866-z
Adrián Andrada, Alejandro Tolcachier
{"title":"On the Canonical Bundle of Complex Solvmanifolds and Applications to Hypercomplex Geometry","authors":"Adrián Andrada, Alejandro Tolcachier","doi":"10.1007/s00031-024-09866-z","DOIUrl":"https://doi.org/10.1007/s00031-024-09866-z","url":null,"abstract":"<p>We study complex solvmanifolds <span>(Gamma backslash G)</span> with holomorphically trivial canonical bundle. We show that the trivializing section of this bundle can be either invariant or non-invariant by the action of <i>G</i>. First we characterize the existence of invariant trivializing sections in terms of the Koszul 1-form <span>(psi )</span> canonically associated to <span>((mathfrak {g},J))</span>, where <span>(mathfrak {g})</span> is the Lie algebra of <i>G</i>, and we use this characterization to produce new examples of complex solvmanifolds with trivial canonical bundle. Moreover, we provide an algebraic obstruction, also in terms of <span>(psi )</span>, for a complex solvmanifold to have trivial (or more generally holomorphically torsion) canonical bundle. Finally, we exhibit a compact hypercomplex solvmanifold <span>((M^{4n},{J_1,J_2,J_3}))</span> such that the canonical bundle of <span>((M,J_{alpha }))</span> is trivial only for <span>(alpha =1)</span>, so that <i>M</i> is not an <span>({text {SL}}(n,mathbb {H}))</span>-manifold.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"24 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141587078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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