Some graph theoretical characterizations of positive definite symmetric quasi-Cartan matrices

TL;DR

This paper provides graph-based characterizations of positive definite symmetric quasi-Cartan matrices of specific Dynkin types, using constructive, purely graph-theoretical proofs based on classical inflations.

Contribution

It introduces new graphical characterizations for these matrices, relying solely on classical inflations and graph theory, enhancing understanding of their structure.

Findings

Graphical characterizations for Dynkin types A_n and D_n

Constructive proofs using classical inflations

Purely graph-theoretical approach

Abstract

We present some graphical characterizations of positive definite symmetric quasi-Cartan matrices of Dynkin type $\mathbb{A}_{n}$ and $\mathbb{D}_{n}$. Our proofs are constructive, purely graph theoretical, and almost self-contained in the sense that they rely on the classical inflations method only.