HIGHEST WEIGHT VECTORS AND TRANSMUTATION

Abstract

Let G = GLn be the general linear group over an algebraically closed field k, let mathfrak{g}={mathfrak{gl}}_n be its Lie algebra and let U be the subgroup of G which consists of the upper uni-triangular matrices. Let kleft[mathfrak{g}right] be the algebra of polynomial functions on mathfrak{g} and let k{left[mathfrak{g}right]}^G be the algebra of invariants under the conjugation action of G. We consider the problem of giving finite homogeneous spanning sets for the k{left[mathfrak{g}right]}^G -modules of highest weight vectors for the conjugation action on kleft[mathfrak{g}right] . We prove a general result in arbitrary characteristic which reduces the problem to giving spanning sets for the vector spaces of highest weight vectors for the action of GLr × GLs on tuples of r × s matrices. This requires the technique called “transmutation” by R. Brylinsky which is based on an instance of Howe duality. In characteristic zero, we give for all dominant weights χ ∈ ℤn finite homogeneous spanning sets for the k{left[mathfrak{g}right]}^G -modules k{left[mathfrak{g}right]}_{upchi}^U of highest weight vectors. This result was already stated by J. F. Donin, but he only gave proofs for his related results on skew representations for the symmetric group. We do the same for tuples of n × n-matrices under the diagonal conjugation action.