In this work, we study matroids over a domain and several classical combinatorial and algebraic invariants related. We define their Grothendieck-Tutte polynomial $T_{\mathcal{M}}(x,y)$, extending the definition given by Fink and Moci in 2016, and we show that such polynomial has the classical deletion-contraction property. Moreover, we study the set of independencies for a realizable matroid over a domain, generalizing the definition of \emph{poset of torsions} $Gr(\mathcal{M})$ given by the second author in 2017. This is a union of identical simplicial posets as for (quasi-)arithmetic matroids. The new notions harmonize naturally through the face module $N_\mathcal{M}$ of the matroid over a domain. Whenever $Gr(\mathcal{M})$ is a finite poset, the Hilbert series $N_\mathcal{M}(t)$ of its face module is a specialization of the Tutte polynomial $T_{\mathcal{M}}(x,y)$. Further, for arrangements of codimension-one abelian subvarities of an elliptic curve admitting complex multiplication, we extend certain results of Bibby and we provide an algebraic interpretation of the elliptic Tutte polynomial.