Total positivity of transformation matrices for uniform subdivisions
이 뉴스, 어떠셨어요?
한 번의 탭으로 반응을 남겨요 · 로그인 불필요
Abstract
The transformation of the $h$-vector of a finite simplicial complex under an $\mathcal{F}$-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 a 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$_2$.