scholarly journals Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups

2015 ◽  
Vol 137 (4) ◽  
pp. 1013-1044 ◽  
Author(s):  
Serge Cantat ◽  
Keiji Oguiso
Keyword(s):  
Coxeter Groups ◽  
Automorphism Groups ◽  
Birational Automorphism ◽  
Calabi Yau Manifolds

Automorphism groups of simply reducible groups

2020 ◽  
pp. 2150088
Author(s):  
Yongzhi Luan
Keyword(s):  
Computer Algebra System ◽  
Coxeter Groups ◽  
Eigenvalue Problems ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Molecular Symmetry ◽  
Dihedral Groups ◽  
Inner Automorphism ◽  
The Moment

Simply reducible groups are closely related to the eigenvalue problems in quantum theory and molecular symmetry in chemistry. Classification of simply reducible groups is still an open problem which is interesting to physicists. Since there are not many examples of simply reducible groups in literature at the moment, we try to find some examples of simply reducible groups as candidates for the classification. By studying the automorphism and inner automorphism groups of symmetric groups, dihedral groups, Clifford groups and Coxeter groups, we find some new examples of candidates. We use the computer algebra system GAP to get most of these automorphism and inner automorphism groups.

scholarly journals Coxeter groups as automorphism groups of solid transitive 3-simplex tilings

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 557-577 ◽  
Author(s):  
Milica Stojanovic
Keyword(s):  
Symmetry Group ◽  
Hyperbolic Space ◽  
Coxeter Groups ◽  
Automorphism Groups ◽  
Order Parameters ◽  
Reflection Groups ◽  
Isometry Groups ◽  
Polar Plane ◽  
Side Face ◽  
The Absolute

In the papers of I.K. Zhuk, then more completely of E. Moln?r, I. Prok, J. Szirmai all simplicial 3-tilings have been classified, where a symmetry group acts transitively on the simplex tiles. The involved spaces depends on some rotational order parameters. When a vertex of a such simplex lies out of the absolute, e.g. in hyperbolic space H3, then truncation with its polar plane gives a truncated simplex or simply, trunc-simplex. Looking for symmetries of these tilings by simplex or trunc-simplex domains, with their side face pairings, it is possible to find all their group extensions, especially Coxeter?s reflection groups, if they exist. So here, connections between isometry groups and their supergroups is given by expressing the generators and the corresponding parameters. There are investigated simplices in families F3, F4, F6 and appropriate series of trunc-simplices. In all cases the Coxeter groups are the maximal ones.

scholarly journals Birational automorphism groups and differential equations

1990 ◽  
Vol 119 ◽  
pp. 1-80 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Critical Point ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Rigorous Proof ◽  
Differential Galois Theory ◽  
General Attitude ◽  
Transcendental Functions ◽  
Birational Automorphism

Painlevé studied the differential equations y″ = R(y′ y, x) without moving critical point, where R is a rational function of y′ y, x. Most of them are integrated by the so far known functions. There are 6 equations called Painlevé’s equations which seem to be irreducible or seem to define new transcendental functions. The simplest one among them is y″ = 6y2 + x. Painlevé declared on Comptes Rendus in 1902-03 that y″ = 6y2 + x is irreducible. It seems that R. Liouville pointed out an error in his argument. In fact there are discussions on this subject between Painlevé and Liouville on Comptes Rendus in 1902-03. In 1915 J. Drach published a new proof of the irreducibility of the differential equation y″ = 6y2 + x. The both proofs depend on the differential Galois theory developed by Drach. But the differential Galois theory of Drach contains errors and gaps and it is not easy to understand their proofs. One of our contemporaries writes in his book: the differential equation y″ = 6y2 + x seems to be irreducible dans un sens que on ne peut pas songer à préciser. This opinion illustrates well the general attitude of the nowadays mathematicians toward the irreducibility of the differential equation y″ = 6y2 + x. Therefore the irreducibility of the differential equation y″ = 6y2 + x remains to be proved. We consider that to give a rigorous proof of the irreducibility of the differential equation y″ = 6y2 + x is one of the most important problem in the theory of differential equations.

On outer automorphism groups of coxeter groups

1997 ◽  
Vol 93 (1) ◽  
pp. 499-513 ◽  
Author(s):  
R. B. Howlett ◽  
P. J. Rowley ◽  
D. E. Taylor
Keyword(s):  
Coxeter Groups ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Outer Automorphism Groups

scholarly journals Automorphism Groups of Compact Complex Surfaces

2019 ◽  
Author(s):  
Yuri Prokhorov ◽  
Constantin Shramov
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Rational Curve ◽  
Complex Surfaces ◽  
Compact Complex Surfaces ◽  
Birational Automorphism

Abstract We study automorphism groups and birational automorphism groups of compact complex surfaces. We show that the automorphism group of such a surface $X$ is always Jordan, and the birational automorphism group is Jordan unless $X$ is birational to a product of an elliptic and a rational curve.

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