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Discrete Applied Mathematicsยท 2026Q2

On the total positivity of transformation matrices

Yanxin Liu, Jianxi Mao

Short summary

The transformation matrix for the barycentric subdivision of a simplicial complex is proven to be totally positive, confirming a conjecture by Mu and Welker.

AI-generated from the title and abstract; the full text is not read.

Key points

  • A new combinatorial proof confirms the total positivity of the transformation matrix for the barycentric subdivision of a simplicial complex.
  • The total positivity of the transformation matrix for interval subdivisions is proven.
  • A sufficient condition for the transformation matrix of an โ„ฑ-uniform subdivision to be totally positive of order 2 (TP 2 ) is established.
  • The transformation matrix of the ๐‘Ÿ-colored barycentric subdivision is shown to be TP 2.

AI-generated from the title and abstract; the full text is not read.

Abstract

The transformation of the โ„Ž -vector of a finite simplicial complex under an โ„ฑ -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 โ„ฑ -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 ๐‘Ÿ -colored barycentric subdivision is TP 2 .

The authors' abstract, as published at the source. Discrete Applied Mathematics, 2026 ยท DOI โ†—

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Field: Discrete Mathematics and Combinatorics

Discrete Mathematics and CombinatoricsMathematics