MAT-Freeness is not combinatorial
Torsten Hoge, Gerhard Roehrle
TL;DR
The paper shows that MAT-freeness for hyperplane arrangements is not determined by combinatorial data alone, exhibiting field-dependence through a concrete example. It introduces MAT*-freeness as a lattice-based, combinatorial variant and proves its combinatorial status, connecting it to additive freeness and extending the discussion to MAT2-freeness. The work situates MAT-freeness among other freeness notions, clarifying when field extension or infinitude changes outcomes, and highlights practical criteria via MAT-partitions. Overall, it demonstrates that certain freeness notions resist purely combinatorial characterization, while offering robust, lattice-driven alternatives.
Abstract
We show that the notion of MAT-freeness for hyperplane arrangements depends on the underlying field. In particular, MAT-freeness is not combinatorial.