In this talk we will explain how we used methods coming from non-associative algebras and from computational algebra to obtain the full description of the ring of invariants of pairs of \( 4\times 4 \) matrices, which could not be solved using the standard representation theory methods. Moreover, we will talk about its connection with the Calogero-Moser spaces and the Hilbert scheme of points. (Joint work with Farkhod Eshmatov and Rustam Turdibaev.)
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