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Bilinear forms for SL(2,q), An and similar groups.

Alexandre Turull (1992)

Publicacions Matemàtiques

The set of invariant symmetric bilinear forms on irreducible modules over fields of characteristic zero for certain groups is studied. Results are obtained under the presence in a finite group of elements of order four whose square is central. In particular, we find that the relevant modules for the groups mentioned in the title always accept an invariant symmetric bilinear form under which the module admits an orthonormal basis.

Billiard complexity in the hypercube

Nicolas Bedaride, Pascal Hubert (2007)

Annales de l’institut Fourier

We consider the billiard map in the hypercube of d . We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that n 3 d - 3 is the order of magnitude of the complexity.

Binary quadratic forms and Eichler orders

Montserrat Alsina (2005)

Journal de Théorie des Nombres de Bordeaux

For any Eichler order 𝒪 ( D , N ) of level N in an indefinite quaternion algebra of discriminant D there is a Fuchsian group Γ ( D , N ) SL ( 2 , ) and a Shimura curve X ( D , N ) . We associate to 𝒪 ( D , N ) a set ( 𝒪 ( D , N ) ) of binary quadratic forms which have semi-integer quadratic coefficients, and we develop a classification theory, with respect to Γ ( D , N ) , for primitive forms contained in ( 𝒪 ( D , N ) ) . In particular, the classification theory of primitive integral binary quadratic forms by SL ( 2 , ) is recovered. Explicit fundamental domains for Γ ( D , N ) allow the characterization...

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