k-Adjoint of hyperplane arrangements

In this paper, we introduce the k-adjoint of a given hyperplane arrangement $A$ associated with rank-k elements in the intersection lattice $L\left(A\right)$, which generalizes the classical adjoint proposed by Bixby and Coullard. The k-adjoint of $A$ induces a decomposition of the Grassmannian, which we call the $A$-adjoint decomposition. Inspired by the work of Gelfand, Goresky, MacPherson, and Serganova, we generalize the matroid decomposition and refined Schubert decomposition of the Grassmannian from the perspective of $A$. Furthermore, we prove that these three decompositions are exactly the same decomposition. A notable application involves providing a combinatorial classification of all the k-dimensional restrictions of $A$. Consequently, we establish the antitonicity of some combinatorial invariants, such as Whitney numbers of the first kind and the independence numbers.