The $\mathcal{G}$-invariant, which was introduced by Derksen, is a matroid invariant that contains all of the data in the Tutte polynomial, and far more. The theory of chromatic uniqueness and chromatic equivalence for graphs, based on the chromatic polynomial, has been

developed extensively, and there are many results for the analogous notions using the Tutte polynomial, for both graphs and matroids. The corresponding questions for the $\mathcal{G}$-invariant are ripe for exploration. This talk will survey recent results that yield non-isomorphic matroids that have the same $\mathcal{G}$-invariant, all aimed at shedding light on what minimal structure determines the $\mathcal{G}$-invariant.