<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -vector inequalities under weak maps

We study the behavior of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -vectors associated to matroid complexes under weak maps, or inclusions of matroid polytopes. Specifically, we show that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -vector of the order complex of the lattice of flats of a matroid is component-wise non-increasing under a weak map. This result extends to the flag <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> -vector. We note that the analogous result also holds for independence complexes and rank-preserving weak maps.