Transformation Groups最新文献

{"title":"Subgroup Proximity in Banach Lie Groups","authors":"Alexandru Chirvasitu","doi":"10.1007/s00031-024-09859-y","DOIUrl":"https://doi.org/10.1007/s00031-024-09859-y","url":null,"abstract":"<p>Let <i>U</i> be a Banach Lie group and <span>(Gle U)</span> a compact subgroup. We show that closed Lie subgroups of <i>U</i> contained in sufficiently small neighborhoods <span>(Vsupseteq G)</span> are compact, and conjugate to subgroups of <i>G</i> by elements close to <span>(1in U)</span>; this generalizes a well-known result of Montgomery and Zippin’s from finite- to infinite-dimensional Lie groups. Along the way, we also prove an approximate counterpart to Jordan’s theorem on finite subgroups of general linear groups: finite subgroups of <i>U</i> contained in sufficiently small neighborhoods <span>(Vsupseteq G)</span> have normal abelian subgroups of index bounded in terms of <span>(Gle U)</span> alone. Additionally, various spaces of compact subgroups of <i>U</i>, equipped with the Hausdorff metric attached to a complete metric on <i>U</i>, are shown to be analytic Banach manifolds; this is the case for both (a) compact groups of a given, fixed dimension, or (b) compact (possibly disconnected) semisimple subgroups. Finally, we also prove that the operation of taking the centralizer (or normalizer) of a compact subgroup of <i>U</i> is continuous (respectively upper semicontinuous) in the appropriate sense.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140803285","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}

{"title":"Discrete Degree of Symmetry of Manifolds","authors":"Ignasi Mundet i Riera","doi":"10.1007/s00031-024-09858-z","DOIUrl":"https://doi.org/10.1007/s00031-024-09858-z","url":null,"abstract":"<p>We define the discrete degree of symmetry disc-sym(<i>X</i>) of a closed <i>n</i>-manifold <i>X</i> as the biggest <span>(mge 0)</span> such that <i>X</i> supports an effective action of <span>((mathbb {Z}/r)^m)</span> for arbitrarily big values of <i>r</i>. We prove that if <i>X</i> is connected then disc-sym<span>((X)le 3n/2)</span>. We propose the question of whether for every closed connected <i>n</i>-manifold <i>X</i> the inequality disc-sym<span>((X)le n)</span> holds true, and whether the only closed connected <i>n</i>-manifold <i>X</i> for which disc-sym(X)<span>(=n)</span> is the torus <span>(T^n)</span>. We prove partial results providing evidence for an affirmative answer to this question.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140630888","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}

{"title":"Deligne Categories and Representations of the Finite General Linear Group, Part 1: Universal Property","authors":"Inna Entova-Aizenbud, Thorsten Heidersdorf","doi":"10.1007/s00031-023-09840-1","DOIUrl":"https://doi.org/10.1007/s00031-023-09840-1","url":null,"abstract":"<p>We study the Deligne interpolation categories <span>(underline{textrm{Rep}}(GL_{t}({mathbb F}_q)))</span> for <span>(tin mathbb {C})</span>, first introduced by F. Knop. These categories interpolate the categories of finite-dimensional complex representations of the finite general linear group <span>(GL_n(mathbb {F}_q))</span>. We describe the morphism spaces in this category via generators and relations. We show that the generating object of this category (an analogue of the representation <span>({mathbb C}{mathbb F}_q^n)</span> of <span>(GL_n(mathbb {F}_q))</span>) carries the structure of a Frobenius algebra with a compatible <span>({mathbb F}_q)</span>-linear structure; we call such objects <span>(mathbb {F}_q)</span>-linear Frobenius spaces and show that <span>(underline{textrm{Rep}}(GL_{t}({mathbb F}_q)))</span> is the universal symmetric monoidal category generated by such an <span>(mathbb {F}_q)</span>-linear Frobenius space of categorical dimension <i>t</i>. In the second part of the paper, we prove a similar universal property for a category of representations of <span>(GL_{infty }(mathbb {F}_q))</span>.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140616251","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}

{"title":"A Generalization of Kac Polynomials and Tensor Product of Representations of $${{,textrm{GL},}}_n(mathbb {F}_q)$$","authors":"","doi":"10.1007/s00031-024-09854-3","DOIUrl":"https://doi.org/10.1007/s00031-024-09854-3","url":null,"abstract":"<h3>Abstract</h3> <p>Given a <em>generic</em> <em>k</em>-tuple <span> <span>((mathcal {X}_1,dots ,mathcal {X}_k))</span> </span> of split semisimple irreducible characters of <span> <span>(textrm{GL}_n(mathbb {F}_q))</span> </span>, Hausel, Letellier and Rodriguez-Villegas (<em>Adv. Math.</em> 234:85–128, 2013, Theorem 1.4.1) constructed a <em>star-shaped</em> quiver <span> <span>(Q=(I,Omega ))</span> </span> together with a dimension vector <span> <span>(alpha in mathbb {N}^I)</span> </span> and they proved that <span> <EquationNumber>0.0.1</EquationNumber> <span>$$begin{aligned} leftlangle mathcal {X}_1otimes cdots otimes mathcal {X}_k,1rightrangle =a_{Q,alpha }(q) end{aligned}$$</span> </span>where <span> <span>(a_{Q,alpha }(t)in mathbb {Z}[t])</span> </span> is the so-called <em>Kac polynomial</em>, i.e., it is the counting polynomial for the number of isomorphism classes of absolutely indecomposable representations of <em>Q</em> of dimension vector <span> <span>(alpha )</span> </span> over finite fields. Moreover, it was conjectured by Kac (1983) and proved by Hausel-Letellier-Villegas (<em>Ann. of Math. (2)</em> 177(3):1147–1168, 2013) that <span> <span>(a_{Q,alpha }(t))</span> </span> has non-negative integer coefficients. From the above formula together with Kac’s (1983) results, they deduced that <span> <span>(leftlangle mathcal {X}_1otimes cdots otimes mathcal {X}_k,1rightrangle ne 0)</span> </span> if and only if <span> <span>(alpha )</span> </span> is a root of <em>Q</em>; moreover, <span> <span>(leftlangle mathcal {X}_1otimes cdots otimes mathcal {X}_k,1rightrangle =1)</span> </span> exactly when <span> <span>(alpha )</span> </span> is a real root. In this paper, we extend their result to any <em>k</em>-tuple <span> <span>((mathcal {X}_1,dots ,mathcal {X}_k))</span> </span> of split semisimple irreducible characters (which are not necessarily generic). To do that, we introduce a stratification indexed by subsets <span> <span>(Vsubset mathbb {N}^I)</span> </span> on the set of <em>k</em>-tuples of split semisimple irreducible characters of <span> <span>(textrm{GL}_n(mathbb {F}_q))</span> </span>. The part corresponding to <span> <span>(V={alpha })</span> </span> consists of the subset of generic <em>k</em>-tuples <span> <span>((mathcal {X}_1,dots ,mathcal {X}_k))</span> </span>. A <em>k</em>-tuple <span> <span>((mathcal {X}_1,dots ,mathcal {X}_k))</span> </span> in the stratum corresponding to <span> <span>(Vsubset mathbb {N}^I)</span> </span> is said to be of level <em>V</em>. A representation <span> <span>(rho )</span> </span> of <span> <span>((Q,alpha ))</span> </span> is said to be of level at most <span> <span>(Vsubset mathbb {N}^I)</span> </span> if the dimension vectors of the indecomposable components of <span> <span>(rho otimes _{mathbb {F}_q}overline{mathbb {F}}_q)</span> </span> belong to <em>V</em>. Given a <em>k</em>-tuple <span> <span>((mathcal {X}_1,dots ,mathcal {X}_k))</span> </span> of level <em>V</em>, our main theo","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140576733","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}

{"title":"On Algebras of Finite General Representation Type","authors":"Ryan Kinser, Danny Lara","doi":"10.1007/s00031-024-09856-1","DOIUrl":"https://doi.org/10.1007/s00031-024-09856-1","url":null,"abstract":"<p>We introduce the notion of “finite general representation type” for a finite-dimensional algebra, a property related to the “dense orbit property” introduced by Chindris-Kinser-Weyman. We use an interplay of geometric, combinatorial, and algebraic methods to produce a family of algebras of wild representation type but finite general representation type. For completeness, we also give a short proof that the only local algebras of discrete general representation type are already of finite representation type. We end with a Brauer-Thrall style conjecture for general representations of algebras.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313169","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}

{"title":"A Spectral Sequence for Locally Free Isometric Lie Group Actions","authors":"Paweł Raźny","doi":"10.1007/s00031-024-09855-2","DOIUrl":"https://doi.org/10.1007/s00031-024-09855-2","url":null,"abstract":"<p>We present a spectral sequence for free isometric Lie algebra actions (and consequently locally free isometric Lie group actions) which relates the de Rham cohomology of the manifold with the Lie algebra cohomology and basic cohomology (intuitively the cohomology of the orbit space). In the process of developing this sequence, we introduce a new description of the de Rham cohomology of manifolds with such actions which appears to be well suited to this and similar problems. Finally, we provide some simple applications generalizing the Wang long exact sequence to Lie algebra actions of low codimension.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203747","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}

{"title":"$${{,textrm{SO},}}(3)$$ -Homogeneous Decomposition of the Flag Scheme of $${{,textrm{SL},}}_3$$ over $$mathbb {Z}left[ 1/2right] $$","authors":"","doi":"10.1007/s00031-024-09847-2","DOIUrl":"https://doi.org/10.1007/s00031-024-09847-2","url":null,"abstract":"<h3>Abstract</h3> <p>In this paper, we give <span> <span>(mathbb {Z}left[ 1/2right] )</span> </span>-forms of <span> <span>({{,textrm{SO},}}(3,mathbb {C}))</span> </span>-orbits in the flag variety of <span> <span>({{,textrm{SL},}}_3(mathbb {C}))</span> </span>. We also prove that they give a <span> <span>(mathbb {Z}left[ 1/2right] )</span> </span>-form of the <span> <span>({{,textrm{SO},}}(3,mathbb {C}))</span> </span>-orbit decomposition of the flag variety of <span> <span>({{,textrm{SL},}}_3)</span> </span>.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156879","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}

{"title":"Local Cohomology of Modular Invariant Rings","authors":"Kriti Goel, Jack Jeffries, Anurag K. Singh","doi":"10.1007/s00031-024-09851-6","DOIUrl":"https://doi.org/10.1007/s00031-024-09851-6","url":null,"abstract":"<p>For <i>K</i> a field, consider a finite subgroup <i>G</i> of <span>({text {GL}}_n(K))</span> with its natural action on the polynomial ring <span>(R:= K[x_1,dots ,x_n])</span>. Let <span>(mathfrak {n})</span> denote the homogeneous maximal ideal of the ring of invariants <span>(R^G)</span>. We study how the local cohomology module <span>(H^n_{mathfrak {n}}(R^G))</span> compares with <span>(H^n_{mathfrak {n}}(R)^G)</span>. Various results on the <i>a</i>-invariant and on the Hilbert series of <span>(H^n_mathfrak {n}(R^G))</span> are obtained as a consequence.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124995","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}

{"title":"Translation Surfaces in Lie Groups with Constant Gaussian Curvature","authors":"Xu Han, Zhonghua Hou","doi":"10.1007/s00031-024-09852-5","DOIUrl":"https://doi.org/10.1007/s00031-024-09852-5","url":null,"abstract":"<p>Let <i>G</i> be an <i>n</i>-dimensional <span>((nge 3))</span> Lie group with a bi-invariant Riemannian metric. We prove that if a surface of constant Gaussian curvature in <i>G</i> can be expressed as the product of two curves, then it must be flat. In particular, we can essentially characterize all such surfaces locally in the three-dimensional case.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140098054","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}

{"title":"Interval Exchange Transformations Groups: Free Actions and Dynamics of Virtually Abelian Groups","authors":"Nancy Guelman, Isabelle Liousse","doi":"10.1007/s00031-024-09849-0","DOIUrl":"https://doi.org/10.1007/s00031-024-09849-0","url":null,"abstract":"<p>In this paper, we study groups acting freely by IETs. We first note that a finitely generated group admits a free IET action if and only if it is virtually abelian. Then, we classify the free actions of non-virtually cyclic groups showing that they are “conjugate” to actions in some specific subgroups <span>(G_n)</span>, namely <span>(G_n simeq (mathcal {G}_2)^{n}rtimes mathcal S_{n})</span> where <span>(mathcal {G}_2)</span> is the group of circular rotations seen as exchanges of 2 intervals and <span>(mathcal S_{n})</span> is the group of permutations of <span>({1,...,n})</span> acting by permuting the copies of <span>(mathcal {G}_2)</span>. We also study non-free actions of virtually abelian groups, and we obtain the same conclusion for any such group that contains a conjugate to a product of restricted rotations with disjoint supports and without periodic points. As a consequence, we get that the group generated by <span>(fin G_n)</span> periodic point free and <span>(gnotin G_{n})</span> is not virtually nilpotent. Moreover, we exhibit examples of finitely generated non-virtually nilpotent subgroups of IETs; some of them are metabelian, and others are not virtually solvable.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019767","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}