TY - JOUR

T1 - Vines and MAT-labeled graphs

AU - Tran, Hung Manh

AU - Tran, Tan Nhat

AU - Tsujie, Shuhei

PY - 2024

Y1 - 2024

N2 - The present paper explores a connection between two concepts arising from different fields of mathematics. The first concept, called vine, is a graphical model for dependent random variables. This concept first appeared in a work of Joe (1994), and the formal definition was given later by Cooke (1997). Vines have nowadays become an active research area whose applications can be found in probability theory and uncertainty analysis. The second concept, called MAT-freeness, is a combinatorial property in the theory of freeness of logarithmic derivation module of hyperplane arrangements. This concept was first studied by Abe-Barakat-Cuntz-Hoge-Terao (2016), and soon afterwards investigated further by Cuntz-M\"ucksch (2020). In the particular case of graphic arrangements, the last two authors (2023) recently proved that the MAT-freeness is completely characterized by the existence of certain edge-labeled graphs, called MAT-labeled graphs. In this paper, we first introduce a poset characterization of a vine, the so-called vineposet. Then we show that, interestingly, there exists an explicit equivalence between the categories of locally regular vineposets and MAT-labeled graphs. In particular, we obtain an equivalence between the categories of regular vineposets and MAT-labeled complete graphs. Several applications will be mentioned to illustrate the interaction between the two concepts. Notably, we give an affirmative answer to a question of Cuntz-M\"ucksch that MAT-freeness can be characterized by a generalization of the root poset in the case of graphic arrangements.

AB - The present paper explores a connection between two concepts arising from different fields of mathematics. The first concept, called vine, is a graphical model for dependent random variables. This concept first appeared in a work of Joe (1994), and the formal definition was given later by Cooke (1997). Vines have nowadays become an active research area whose applications can be found in probability theory and uncertainty analysis. The second concept, called MAT-freeness, is a combinatorial property in the theory of freeness of logarithmic derivation module of hyperplane arrangements. This concept was first studied by Abe-Barakat-Cuntz-Hoge-Terao (2016), and soon afterwards investigated further by Cuntz-M\"ucksch (2020). In the particular case of graphic arrangements, the last two authors (2023) recently proved that the MAT-freeness is completely characterized by the existence of certain edge-labeled graphs, called MAT-labeled graphs. In this paper, we first introduce a poset characterization of a vine, the so-called vineposet. Then we show that, interestingly, there exists an explicit equivalence between the categories of locally regular vineposets and MAT-labeled graphs. In particular, we obtain an equivalence between the categories of regular vineposets and MAT-labeled complete graphs. Several applications will be mentioned to illustrate the interaction between the two concepts. Notably, we give an affirmative answer to a question of Cuntz-M\"ucksch that MAT-freeness can be characterized by a generalization of the root poset in the case of graphic arrangements.

KW - math.CO

KW - Primary 06A07, Secondary 52C35

U2 - 10.1017/fms.2024.124

DO - 10.1017/fms.2024.124

M3 - Article

VL - 12

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

M1 - e128

ER -