Abstract: | Associated to a split, semisimple linear algebraic group G is a group of invariant
quadratic forms, which we denote Q(G). Namely, Q(G) is the group of quadratic
forms in characters of a maximal torus which are fixed with respect to the action
of the Weyl group of G. We compute Q(G) for various examples of products of the
special linear, special orthogonal, and symplectic groups as well as for quotients of
those examples by central subgroups. Homomorphisms between these linear algebraic groups induce homomorphisms between their groups of invariant quadratic forms. Since the linear algebraic groups are semisimple, Q(G) is isomorphic to Z^n for some n, and so the induced maps can be described by a set of integers called Rost multipliers. We consider various cases of the Kronecker tensor product map between copies of the special linear, special orthogonal, and symplectic groups. We compute the Rost multipliers of the induced map in these examples, ultimately concluding that the Rost multipliers depend only on the dimensions of the underlying vector spaces. |