Consider the variety Comm_n of pairs of commuting n by n matrices, it is invariant under the action of GL_n on the space of pairs of all matrices, by simultaneous conjunction. The natural algebraic structure to associate with is equivariant Borel-Moore homology (or K-theory) of the variety. It turns out the direct sum by n of homologies of varieties Comm_n is equipped with an associative multiplication, given by convolution. The conjugation action has a fixed point (0,0), it’s embedding induces the pullback map of the algebra in hand into the direct sum of algebras of symmetric polynomials, equipped itself with a shuffle multiplication for which the map is an algebra homomorphism. In the paper (arXiv: 2108.08779), where one considers arbitrary finite quiver, Neguţ proves the image of the morphism is given by polynomials satisfying certain vanishing conditions. In this talk I’ll give the overview of the subject and announce the computation made for general reductive groups. This is a work in progress.