Symmetric products, linear representations and trace identities

Abstract We give the equations of n -th symmetric product $$X^n/S_n$$ X n / S a flat affine scheme $$X=\mathrm {Spec}\,A$$ = Spec A over commutative ring F . As consequence, we find closed immersion into coarse moduli space parameterizing -dimensional linear representations A This is done by exhibiting an isomorphism between tensors and generated coefficients characteristic polynomial polynomials in commuting generic matrices giving Using this derive associated reduced schemes infinite field. When zero show that express it term traces.