Pub Date : 2023-12-02DOI: 10.1007/s00031-023-09829-w
Haremy Zuñiga
Let (Gamma ) be a discrete subgroup of (text {PU}(1,n)). In this work, we look at the induced action of (Gamma ) on the projective space (mathbb {P}(wedge ^{k+1}mathbb {C}^{n+1})) by the Plücker embedding, where (wedge ^{k+1}) denotes the exterior power. We define a limit set for this action called the k-Chen-Greenberg limit set, which extends the classical definition of the Chen-Greenberg limit set (L(Gamma )), and we show several of its properties. We prove that its Kulkarni limit set is the union taken over all (pin L(Gamma )) of the projective subspace generated by all k-planes that contain p or are contained in (p^{perp }) via the Plücker embedding. We also prove a duality between both limit sets.
{"title":"The Limit Set for Representations of Discrete Subgroups of $$text {PU}(1,n)$$ by the Plücker Embedding","authors":"Haremy Zuñiga","doi":"10.1007/s00031-023-09829-w","DOIUrl":"https://doi.org/10.1007/s00031-023-09829-w","url":null,"abstract":"<p>Let <span>(Gamma )</span> be a discrete subgroup of <span>(text {PU}(1,n))</span>. In this work, we look at the induced action of <span>(Gamma )</span> on the projective space <span>(mathbb {P}(wedge ^{k+1}mathbb {C}^{n+1}))</span> by the Plücker embedding, where <span>(wedge ^{k+1})</span> denotes the exterior power. We define a limit set for this action called the <i>k</i>-Chen-Greenberg limit set, which extends the classical definition of the Chen-Greenberg limit set <span>(L(Gamma ))</span>, and we show several of its properties. We prove that its Kulkarni limit set is the union taken over all <span>(pin L(Gamma ))</span> of the projective subspace generated by all <i>k</i>-planes that contain <i>p</i> or are contained in <span>(p^{perp })</span> via the Plücker embedding. We also prove a duality between both limit sets.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"33 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508437","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}
Pub Date : 2023-11-18DOI: 10.1007/s00031-023-09825-0
Laura Escobar, Martha Precup, John Shareshian
After proving that every Schubert variety in the full flag variety of a complex reductive group G is a general Hessenberg variety, we show that not all such Schubert varieties are adjoint Hessenberg varieties. In fact, in types A and C, we provide pattern avoidance criteria implying that the proportion of Schubert varieties that are adjoint Hessenberg varieties approaches zero as the rank of G increases. We show also that in type A, some Schubert varieties are not isomorphic to any adjoint Hessenberg variety.
{"title":"Which Schubert Varieties are Hessenberg Varieties?","authors":"Laura Escobar, Martha Precup, John Shareshian","doi":"10.1007/s00031-023-09825-0","DOIUrl":"https://doi.org/10.1007/s00031-023-09825-0","url":null,"abstract":"<p>After proving that every Schubert variety in the full flag variety of a complex reductive group <i>G</i> is a general Hessenberg variety, we show that not all such Schubert varieties are adjoint Hessenberg varieties. In fact, in types A and C, we provide pattern avoidance criteria implying that the proportion of Schubert varieties that are adjoint Hessenberg varieties approaches zero as the rank of <i>G</i> increases. We show also that in type A, some Schubert varieties are not isomorphic to any adjoint Hessenberg variety.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"61 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508439","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}
Pub Date : 2023-11-17DOI: 10.1007/s00031-023-09828-x
Mehdi Lejmi, Nicoletta Tardini
A hypercomplex structure (I, J, K) on a manifold M is said to be (C^infty )-pure-and-full if the Dolbeault cohomology (H^{2,0}_{partial }(M,I)) is the direct sum of two natural subgroups called the (overline{J})-invariant and the (overline{J})-anti-invariant subgroups. We prove that a compact hypercomplex manifold that satisfies the quaternionic version of the (dd^c)-Lemma is (C^infty )-pure-and-full. Moreover, we study the dimensions of the (overline{J})-invariant and the (overline{J})-anti-invariant subgroups, together with their analogue in the Bott-Chern cohomology. For instance, in real dimension 8, we characterize the existence of hyperkähler with torsion metrics in terms of the dimension of the (overline{J})-invariant subgroup. We also study the existence of special hypercomplex structures on almost abelian solvmanifolds.
{"title":"On the Invariant and Anti-Invariant Cohomologies of Hypercomplex Manifolds","authors":"Mehdi Lejmi, Nicoletta Tardini","doi":"10.1007/s00031-023-09828-x","DOIUrl":"https://doi.org/10.1007/s00031-023-09828-x","url":null,"abstract":"<p>A hypercomplex structure (<i>I</i>, <i>J</i>, <i>K</i>) on a manifold <i>M</i> is said to be <span>(C^infty )</span>-pure-and-full if the Dolbeault cohomology <span>(H^{2,0}_{partial }(M,I))</span> is the direct sum of two natural subgroups called the <span>(overline{J})</span>-invariant and the <span>(overline{J})</span>-anti-invariant subgroups. We prove that a compact hypercomplex manifold that satisfies the quaternionic version of the <span>(dd^c)</span>-Lemma is <span>(C^infty )</span>-pure-and-full. Moreover, we study the dimensions of the <span>(overline{J})</span>-invariant and the <span>(overline{J})</span>-anti-invariant subgroups, together with their analogue in the Bott-Chern cohomology. For instance, in real dimension 8, we characterize the existence of hyperkähler with torsion metrics in terms of the dimension of the <span>(overline{J})</span>-invariant subgroup. We also study the existence of special hypercomplex structures on almost abelian solvmanifolds.</p>","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"62 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508438","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}
Pub Date : 2023-09-23DOI: 10.1007/s00031-023-09824-1
Michele Carmassi
{"title":"ON THE BRUHAT $$ mathcal{G} $$-ORDER BETWEEN LOCAL SYSTEMS ON THE B-ORBITS IN A HERMITIAN SYMMETRIC VARIETY","authors":"Michele Carmassi","doi":"10.1007/s00031-023-09824-1","DOIUrl":"https://doi.org/10.1007/s00031-023-09824-1","url":null,"abstract":"","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135959830","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}
Pub Date : 2023-09-20DOI: 10.1007/s00031-023-09815-2
Daniele Angella, Simone Calamai, Francesco Pediconi, Cristiano Spotti
Abstract We extend the classical Donaldson-Fujiki interpretation of the scalar curvature as moment map in Kähler geometry to the wider framework of locally conformally Kähler geometry.
{"title":"A Moment Map for Twisted-Hamiltonian Vector Fields on Locally Conformally Kähler Manifolds","authors":"Daniele Angella, Simone Calamai, Francesco Pediconi, Cristiano Spotti","doi":"10.1007/s00031-023-09815-2","DOIUrl":"https://doi.org/10.1007/s00031-023-09815-2","url":null,"abstract":"Abstract We extend the classical Donaldson-Fujiki interpretation of the scalar curvature as moment map in Kähler geometry to the wider framework of locally conformally Kähler geometry.","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136263206","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}
Pub Date : 2023-09-09DOI: 10.1007/s00031-023-09820-5
Avery Wilson
{"title":"Compactifications of Moduli of G-Bundles and Conformal Blocks","authors":"Avery Wilson","doi":"10.1007/s00031-023-09820-5","DOIUrl":"https://doi.org/10.1007/s00031-023-09820-5","url":null,"abstract":"","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136108416","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}
Pub Date : 2023-08-04DOI: 10.1007/s00031-023-09818-z
Z. Chang, Yongjie Wang
{"title":"A First Fundamental Theorem of Invariant Theory for the Quantum Queer Superalgebra","authors":"Z. Chang, Yongjie Wang","doi":"10.1007/s00031-023-09818-z","DOIUrl":"https://doi.org/10.1007/s00031-023-09818-z","url":null,"abstract":"","PeriodicalId":49423,"journal":{"name":"Transformation Groups","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44515564","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}
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