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Hilbert symbols, class groups and quaternion algebras

Ted Chinburg, Eduardo Friedman (2000)

Journal de théorie des nombres de Bordeaux

Let B be a quaternion algebra over a number field k . To a pair of Hilbert symbols { a , b } and { c , d } for B we associate an invariant ρ = ρ R [ 𝒟 ( a , b ) ] , [ 𝒟 ( c , d ) ] in a quotient of the narrow ideal class group of k . This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders 𝒟 ( a , b ) and 𝒟 ( c , d ) in B associated to { a , b } and { c , d } . If a = c , we compute ρ R ( [ 𝒟 ( a , b ) ] , [ 𝒟 ( c , d ) ] ) by means of arithmetic in the field k ( a ) . The problem of extending this algorithm to the general case leads to studying a finite graph associated...

Historical notes on loop theory

Hala Orlik Pflugfelder (2000)

Commentationes Mathematicae Universitatis Carolinae

This paper deals with the origins and early history of loop theory, summarizing the period from the 1920s through the 1960s.

Holonomy groups of complete flat manifolds

Michał Sadowski (2007)

Banach Center Publications

We present short direct proofs of two known properties of complete flat manifolds. They say that the diffeomorphism classes of m-dimensional complete flat manifolds form a finite set S C F ( m ) and that each element of S C F ( m ) is represented by a manifold with finite holonomy group.

Holonomy groups of flat manifolds with the R ∞ property

Rafał Lutowski, Andrzej Szczepański (2013)

Fundamenta Mathematicae

Let M be a flat manifold. We say that M has the R ∞ property if the Reidemeister number R(f) is infinite for every homeomorphism f: M → M. We investigate relations between the holonomy representation ρ of M and the R ∞ property. When the holonomy group of M is solvable we show that if ρ has a unique ℝ-irreducible subrepresentation of odd degree then M has the R ∞ property. This result is related to Conjecture 4.8 in [K. Dekimpe et al., Topol. Methods Nonlinear Anal. 34 (2009)].

Homogeneous representations of Khovanov–Lauda Algebras

Alexander Kleshchev, Arun Ram (2010)

Journal of the European Mathematical Society

We construct irreducible graded representations of simply laced Khovanov–Lauda algebras which are concentrated in one degree. The underlying combinatorics of skew shapes and standard tableaux corresponding to arbitrary simply laced types has been developed previously by Peterson, Proctor and Stembridge. In particular, the Peterson–Proctor hook formula gives the dimensions of the homogeneous irreducible modules corresponding to straight shapes.

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