The family of high rank arithmetic groups is a class of groups playing an important role in various areas of mathematics.
It includes SL(n,Z), for n>2 , SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more.
A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasi-isometric rigidity.
The basis of elements of the highest weight representations of affine Lie algebra of type A can be labeled in three different ways, my multipartitions, by piecewise linear paths in the weight space, and by canonical basis elements. The entire infinite basis is recursively generated from the highest weight vector of operators f_i from the Chevalley basis of the affine Lie algebra, and organized into a crystal called a Kashiwara crystal. We describe cases where one can move between the different labelings in a non-recursive fashion, particularly when the crystal has some symmetry.
The Einstein Institute of Mathematics was founded in 1925, and has since become one of the world's leading research institutes. The Institute offers B.Sc., M.Sc., and Ph.D. studies, hosts postdocs and visitors from around the world and promotes mathematical education through various seminars, conferences, workshops and other programs for students, teachers and the general public. Our research areas currently include the foundations of mathematics, logic, algebra, group theory, representation theory, differential geometry, algebraic topology, analysis, ergodic theory and dynamical systems, number theory, probability, combinatorics and game theory.