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Another 80-dimensional extremal lattice

Mark Watkins (2012)

Journal de Théorie des Nombres de Bordeaux

We show that the unimodular lattice associated to the rank 20 quaternionic matrix group SL 2 ( F 41 ) ⊗ S ˜ 3 ⊂ GL 80 ( Z ) is a fourth example of an 80-dimensional extremal lattice. Our method is to use the positivity of the Θ -series in conjunction with an enumeration of all the norm 10 vectors. The use of Aschbacher’s theorem on subgroups of finite classical groups (reliant on the classification of finite simple groups) provides one proof that this lattice is distinct from the previous three, while computing the inner product...

Applications de Gauss et pléthysme

Laurent Manivel (1997)

Annales de l'institut Fourier

Les représentations irréductibles de Gl ( n , ℂ ) sont décrites par les foncteurs de Schur, dont la composition définit le pléthysme. Sa compréhension est un problème important en théorie des invariants, ou bien en relation avec les représentations des groupes symétriques.Nous proposons dans cet article une approche géométrique du problème. Généralisant les plongements classiques de Veronese et de Segre, nous construisons des plongements de variétés de drapeaux dans d’autres variétés de drapeaux, sur lesquels...

Automorphic realization of residual Galois representations

Robert Guralnick, Michael Harris, Nicholas M. Katz (2010)

Journal of the European Mathematical Society

We show that it is possible in rather general situations to obtain a finite-dimensional modular representation ρ of the Galois group of a number field F as a constituent of one of the modular Galois representations attached to automorphic representations of a general linear group over F , provided one works “potentially.” The proof is based on a close study of the monodromy of the Dwork family of Calabi–Yau hypersurfaces; this in turn makes use of properties of rigid local systems and the classification...

Automorphism groups of polycyclic-by-finite groups and arithmetic groups

Oliver Baues, Fritz Grunewald (2006)

Publications Mathématiques de l'IHÉS

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic...

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