This is not necessarily the same construction as the Culler-Shalen character variety (generated by evaluations of traces), although when the they do agree since Procesi has shown that in this case the ring of invariants is in fact generated by only traces.
There are several version of those, including Hodge algebras of Corrado de Concini, David Eisenbud, and Claudio Procesi.
With Claudio Procesi: Topics in Hyperplane Arrangements, Polytopes and Box-Splines, Springer, 2010
De Mari, C. Procesi, and M. Shayman, Hessenberg varieties, Trans.
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In geometry, Hessenberg varieties, first studied by De Mari, Procesi, and Shayman, are a family of subvarieties of the full flag variety which are defined by a Hessenberg function h and a linear transformation X.
It was Adriano Garsia's idea to construct an appropriate module in order to prove positivity (as was done in his previous joint work with Procesi on Schur positivity of Kostka–Foulkes polynomials).
Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402 .
Claudio Monteverdi | Claudio Roditi | Claudio Arrau | Claudio Reyna | Cláudio Taffarel | Claudio Scimone | Claudio Procesi | Claudio Baglioni | Claudio Lolli | Claudio Cecchetto | Claudio Magris | Claudio Naranjo | Sergio Claudio dos Santos | Claudio Marrero | Claudio Marchisio | Claudio Chiappucci | Claudio Bravo | Claudio Bisio | Claudio Vargas | Claudio Treves | Claudio Taffarel | Claudio Stampi | Claudio Scarchilli | Claudio Scajola | Claudio Saracini | Claudio Sanchez | Claudio R. M. Costa | Claudio Ranieri | Claudio Prieto | Claudio Osorio |