scholarly journals Rigidity of maximal holomorphic representations of Kähler groups

2015 ◽  
Vol 26 (14) ◽  
pp. 1550113 ◽  
Author(s):  
Marco Spinaci
Keyword(s):  
Riemann Surface ◽  
Lie Group ◽  
Complete Classification ◽  
Schwarz Lemma ◽  
Connected Components ◽  
Equivariant Map ◽  
Kähler Groups ◽  
Diagonal Embedding

We investigate representations of Kähler groups [Formula: see text] to a semisimple non-compact Hermitian Lie group [Formula: see text] that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor–Wood inequality similar to those found by Burger–Iozzi and Koziarz–Maubon. Thanks to the study of the case of equality in Royden’s version of the Ahlfors–Schwarz lemma, we can completely describe the case of maximal holomorphic representations. If [Formula: see text], these appear if and only if [Formula: see text] is a ball quotient, and essentially reduce to the diagonal embedding [Formula: see text]. If [Formula: see text] is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, which thus appear as preferred elements of the respective maximal connected components.

scholarly journals Unitary spherical representations of Drinfeld doubles

2018 ◽  
Vol 2018 (742) ◽  
pp. 157-186 ◽  
Author(s):  
Yuki Arano
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
Complete Classification ◽  
Unitary Representations ◽  
Compact Lie Group ◽  
Drinfeld Double ◽  
Quantum Analogue ◽  
Property T ◽  
Simply Connected

Abstract We study irreducible spherical unitary representations of the Drinfeld double of the q-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of \mathrm{SU}_{q}(3) , we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group \mathrm{SU}_{q}(2n+1) has property (T), which also implies central property (T) of the dual of \mathrm{SU}_{q}(2n+1) .

scholarly journals Complete Classification of Degree 7 for Genus 1

2021 ◽  
pp. 594-603
Author(s):  
Peshawa M. Khudhur
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Symmetric Group ◽  
Compact Riemann Surface ◽  
Complete Classification ◽  
Conjugacy Classes ◽  
Branched Cover ◽  
Transitive Subgroup ◽  
Genus One

Assume that  is a meromorphic fuction of degree n where X is compact Riemann surface of genus g. The meromorphic function gives a branched cover of the compact Riemann surface X. Classes of such covers are in one to one correspondence with conjugacy classes of r-tuples (  of permutations in the symmetric group , in which  and s generate a transitive subgroup G of  This work is a contribution to the classification of all primitive groups of degree 7, where X is of genus one.

scholarly journals On the Balmer spectrum for compact Lie groups

2019 ◽  
Vol 156 (1) ◽  
pp. 39-76
Author(s):  
Tobias Barthel ◽  
J. P. C. Greenlees ◽  
Markus Hausmann
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Complete Classification ◽  
Prime Ideals ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Finite Set

We study the Balmer spectrum of the category of finite $G$-spectra for a compact Lie group $G$, extending the work for finite $G$ by Strickland, Balmer–Sanders, Barthel–Hausmann–Naumann–Nikolaus–Noel–Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of $G$. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes $p$.

scholarly journals Connected Components of the Hurwitz Space for the Symmetric Group of Degree 7

2020 ◽  
Vol 24 (2) ◽  
pp. 129
Author(s):  
Haval M. Mohammed Salih
Keyword(s):  
Symmetric Group ◽  
Monodromy Group ◽  
Riemann Sphere ◽  
Complete Classification ◽  
Connected Components ◽  
Branch Points ◽  
Genus Zero ◽  
Computational Tools ◽  
Hurwitz Space

The Hurwitz space  is the space of genus  covers of the Riemann sphere  with branch points and the monodromy group . Let be the symmetric group . In this paper, we enumerate the connected components of . Our approach uses computational tools, relying on the computer algebra system GAP and the MAPCLASS package, to find the connected components of . This work gives us the complete classification of  primitive genus zero symmetric group of degree seven. 

scholarly journals A Boothby–Wang Theorem for Besse Contact Manifolds

2020 ◽  
Author(s):  
Marc Kegel ◽  
Christian Lange
Keyword(s):  
Lie Group ◽  
Group Actions ◽  
Contact Manifold ◽  
Complete Classification ◽  
Contact Manifolds ◽  
Lie Group Actions ◽  
Oxford University ◽  
Reeb Flows ◽  
Sasakian Geometry

AbstractA Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal $$S^1$$ S 1 -orbibundles over integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomological condition, this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry (Boyer and Galicki in Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008). We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result. More precisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizer groups. By introducing all relevant orbifold notions in this equivariant way, we avoid patching constructions with orbifold charts. As an application, and building on work by Cristofaro-Gardiner–Mazzucchelli, we deduce a complete classification of closed Besse contact 3-manifolds up to strict contactomorphism.

scholarly journals Equipopularity Classes of 132-Avoiding Permutations

10.37236/3675 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Lynn Chua ◽  
Krishanu Roy Sankar
Keyword(s):  
Generating Functions ◽  
Complete Classification ◽  
Connected Components ◽  
Plane Tree ◽  
Plane Trees ◽  
Set Of Permutations ◽  
Spine Structure

The popularity of a pattern $p$ in a set of permutations is the sum of the number of copies of $p$ in each permutation of the set. We study pattern popularity in the set of 132-avoiding permutations. Two patterns are equipopular if, for all $n$, they have the same popularity in the set of length-$n$ 132-avoiding permutations. There is a well-known bijection between 132-avoiding permutations and binary plane trees. The spines of a binary plane tree are defined as the connected components when all edges connecting left children to their parents are deleted, and the spine structure is the sorted sequence of lengths of the spines. Rudolph shows that patterns of the same length are equipopular if their associated binary plane trees have the same spine structure. We prove the converse of this result using the method of generating functions, which gives a complete classification of 132-avoiding permutations into equipopularity classes.

scholarly journals Thin Lehman Matrices and Their Graphs

10.37236/437 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jonathan Wang
Keyword(s):  
Positive Integer ◽  
Maximum Clique ◽  
Circulant Matrix ◽  
Complete Classification ◽  
Connected Components ◽  
Connected Component ◽  
Johnson Graph ◽  
Induced Subgraph ◽  
The Matrix

Two square $0,1$ matrices $A,B$ are a pair of Lehman matrices if $AB^T = J+dI$, where $J$ is the matrix of all $1$s and $d$ is a positive integer. It is known that there are infinitely many such matrices when $d=1$, and these matrices are called thin Lehman matrices. An induced subgraph of the Johnson graph may be defined given any Lehman matrix, where the vertices of the graph correspond to rows of the matrix. These graphs are used to study thin Lehman matrices. We show that any connected component of such a graph determines the corresponding rows of the matrix up to permutations of the columns. We also provide a sharp bound on the maximum clique size of such graphs and give a complete classification of Lehman matrices whose graphs have at most two connected components. Some constraints on when a circulant matrix can be Lehman are also provided. Many general classes of thin Lehman matrices are constructed in the paper.

scholarly journals A criterion for discrete branching laws for Klein four symmetric pairs and its application to E6(−14)

2020 ◽  
Vol 31 (06) ◽  
pp. 2050049
Author(s):  
Haian He
Keyword(s):  
Lie Group ◽  
Simple Formula ◽  
Complete Classification ◽  
Necessary Condition ◽  
Symmetric Pair ◽  
Branching Laws ◽  
Nonidentity Element

Let [Formula: see text] be a noncompact connected simple Lie group, and [Formula: see text] a Klein four-symmetric pair. In this paper, we show a necessary condition for the discrete decomposability of unitarizable simple [Formula: see text]-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for [Formula: see text], there does not exist a unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. As an application, for [Formula: see text], we obtain a complete classification of Klein four symmetric pairs [Formula: see text], with [Formula: see text] noncompact, such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module and is also discretely decomposable as a [Formula: see text]-module for some nonidentity element [Formula: see text].

Liapunov stability and adding machines

1995 ◽  
Vol 15 (2) ◽  
pp. 271-290 ◽  
Author(s):  
Jorge Buescu ◽  
Ian Stewart
Keyword(s):  
Complete Classification ◽  
Cantor Set ◽  
Connected Components ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Continuous Maps ◽  
Liapunov Stable ◽  
Transitive Set ◽  
Adding Machines

AbstractLet X be a locally connected locally compact metric space and f: X → X a continuous map. Let A be a compact transitive set under f. If A is asymptotically stable, then it has finitely many connected components, which are cyclically permuted. If it is Liapunov stable, then A may have infinitely many connected components. Our main result states that these form a Cantor set on which f is topologically conjugate to an adding machine. A number of consequences are derived, including a complete classification of compact transitive sets for continuous maps of the interval and the Liapunov instability of the invariant Cantor set of Denjoy maps of the circle.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

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