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Rational points and Coxeter group actions on the cohomology of toric varieties

Gustav I. Lehrer (2008)

Annales de l’institut Fourier

We derive a simple formula for the action of a finite crystallographic Coxeter group on the cohomology of its associated complex toric variety, using the method of counting rational points over finite fields, and the Hodge structure of the cohomology. Various applications are given, including the determination of the graded multiplicity of the reflection representation.

Realizations of Loops and Groups defined by short identities

Anthony Donald Keedwell (2009)

Commentationes Mathematicae Universitatis Carolinae

In a recent paper, those quasigroup identities involving at most three variables and of “length” six which force the quasigroup to be a loop or group have been enumerated by computer. We separate these identities into subsets according to what classes of loops they define and also provide humanly-comprehensible proofs for most of the computer-generated results.

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