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Let be a quaternion algebra over a number field . To a pair of Hilbert symbols and for we associate an invariant in a quotient of the narrow ideal class group of . This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders and in associated to and If , we compute by means of arithmetic in the field The problem of extending this algorithm to the general case leads to studying a finite graph associated...
This paper deals with the origins and early history of loop theory, summarizing the period from the 1920s through the 1960s.
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 and that each element of is represented by a manifold with finite holonomy group.
Let M be a flat manifold. We say that M has the 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 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 property. This result is related to Conjecture 4.8 in [K. Dekimpe et al., Topol. Methods Nonlinear Anal. 34 (2009)].
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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