We give the equations of the n-th symmetric product Xn/Sn of a flat affine scheme X=SpecA over a commutative ring F. As a consequence, we find a closed immersion into the coarse moduli space parameterizing n-dimensional linear representations of A. This is done by exhibiting an isomorphism between the ring of symmetric tensors over A and the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices giving representations of A. Using this we derive an isomorphism of the associated reduced schemes over an infinite field. When the characteristic is zero we show that this isomorphism is an isomorphism of schemes and we express it in term of traces.
Symmetric products, linear representations and trace identities / Vaccarino, Francesco. - In: BEITRAGE ZUR ALGEBRA UND GEOMETRIE. - ISSN 0138-4821. - ELETTRONICO. - (2021). [10.1007/s13366-021-00577-0]
Titolo: | Symmetric products, linear representations and trace identities | |
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Data di pubblicazione: | 2021 | |
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Digital Object Identifier (DOI): | http://dx.doi.org/10.1007/s13366-021-00577-0 | |
Appare nelle tipologie: | 1.1 Articolo in rivista |
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http://hdl.handle.net/11583/2888233