On the total positivity of transformation matrices

The transformation of the ℎ -vector of a finite simplicial complex under an ℱ -uniform subdivision is encoded by a transformation matrix. Mu and Welker conjectured that the transformation matrix of the barycentric subdivision is totally positive. In this paper, we give a new combinatorial proof of this conjecture. We also prove the total positivity of the transformation matrix of the interval subdivision. In addition, we establish a sufficient condition for the transformation matrix of an ℱ -uniform subdivision to be totally positive of order 2 (TP 2 ), thereby partially answering a question of Mu and Welker. As an application, we show that the transformation matrix of the 𝑟 -colored barycentric subdivision is TP 2 .