Symmetric products, linear representations and the commuting scheme I: isomorphisms and embeddings
نویسنده
چکیده
We show that the symmetric product of a flat affine scheme over a commutative ring can be embedded into the quotient by the general linear group of the scheme of commuting matrices. We also prove that the symmetric product of the affine space is isomorphic to the above quotient when the base ring is a characteristic zero field. Over an infinite field of arbitrary characteristic the quotient of the reduced scheme associated to the scheme of commuting matrices will be showed to be again isomorphic to the symmetric product of the affine space showing that the invariant part of the radical of the commuting scheme is zero over a characteristic zero field. We show also that the symmetric product of an affine flat scheme can be embedded into the quotient by the general linear group of the tuples of matrices exhibiting a surjective morphism from the invariant of matrices to the multisymmetric functions that hold for any a base ring. ∗The author is supported by the Research Grant nr.199/2004. MSC:14L30,13A50,14A15
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Linear representations, symmetric products and the commuting scheme
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