We consider problems in invariant theory related to the classification of four vector subspaces of an -dimensional complex vector space. We use castling techniques to quickly recover results of Howe and Huang on invariants. We further obtain information about principal isotropy groups, equidimensionality and the modules of covariants.
The half-liberated orthogonal group appears as intermediate quantum group between the orthogonal group , and its free version . We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between and , a non abelian discrete group playing the role of weight lattice, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that...
Let be a split semisimple linear algebraic group over a field and let be a split maximal torus of . Let be an oriented cohomology (algebraic cobordism, connective -theory, Chow groups, Grothendieck’s , etc.) with formal group law . We construct a ring from and the characters of , that we call a formal group ring, and we define a characteristic ring morphism from this formal group ring to where is the variety of Borel subgroups of . Our main result says that when the torsion index...
Let be a semisimple linear algebraic group of inner type over a field , and let be a projective homogeneous -variety such that splits over the function field of . We introduce the -invariant of which characterizes the motivic behavior of , and generalizes the -invariant defined by A. Vishik in the context of quadratic forms. We use this -invariant to provide motivic decompositions of all generically split projective homogeneous -varieties, e.g. Severi-Brauer varieties, Pfister quadrics,...
We give the definition of a kind of building for a symmetrizable Kac-Moody group over a field endowed with a discrete valuation and with a residue field containing . Due to the lack of some important property of buildings, we call it a hovel. Nevertheless, some good ones remain, for example, the existence of retractions with center a sector-germ. This enables us to generalize many results proved in the semisimple case by S. Gaussent and P. Littelmann. In particular, if , the geodesic segments...