Séminaire Lotharingien de Combinatoire, 91B.12 (2024), 12 pp.
Hung Manh Tran, Tan Nhat Tran and Shuhei Tsujie
Vines and MAT-Labeled Graphs
Abstract.
The present note 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 and Terao
(2016), and soon afterwards investigated further by
Cuntz and Mücksch (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. Then we show that, interestingly, there
exists an explicit equivalence between the categories of locally
regular vines and MAT-labeled graphs. In particular, we obtain an
equivalence between the categories of regular vines 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 and Mücksch that MAT-freeness can be characterized
by a generalization of the root poset in the case of graphic
arrangements.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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