Let us consider the action of the general linear group $\mathrm{GL}_n(\mathbb{C})$ on the direct product $\mathcal{M}_n^d$
of $d$ copies of $\mathcal{M}_n$ by simultaneous conjugation sending $(X_1,\ldots, X_d)$ to $(gX_1g^{-1},\ldots,gX_dg^{-1})$
for any $g\in \mathrm{GL}_n(\mathbb{C})$ . This induces an action of $\mathrm{GL}_n(\mathbb{C})$ on the algebra $\mathbb{C}[\mathcal{M}_n^d]$ of polynomial
functions on $\mathcal{M}_n^d$. The algebra of invariants under this action, $\mathbb{C}[\mathcal{M}_n^d]^{\mathrm{GL}_n}$, is an important
object in several areas of mathematics. 
In this talk we will explain how we used methods coming from non-associative algebras
to obtain the full description of the case $\mathbb{C}[\mathcal{M}_4^2]^{\mathrm{GL}_4}$, 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.

References

[1] F. Eshmatov, X. García-Martínez, T. Normatov and R. Turdibaev. On the coordinate rings of
Calogero-Moser spaces and the invariant commuting variety of a pair of matrices. Preprint available
in arXiv:2307.06098.

[2] X. García-Martínez, T. Normatov and R. Turdibaev. The ring of invariants of pairs of $3\times 3$ matrices.
Journal of Algebra  603 (2022), 201--212.