Positive determinacy of h-Shuhan matrices with $h<2$

TL;DR

This work defines the $h$-Shuhan matrices as a real generalization of generalized Cartan matrices via (S1)–(S3) and studies positivity notions when $h<2$, linking generalized positivity to the positive semi-definiteness of a symmetrized form. It develops explicit coefficient sequences $a_n^h,b_n^h,d_n^h,e_j^h$ and derives precise $h$-thresholds for finite/affine types (eg, $A_n^h,D_n^h,E_6^h,E_7^h,E_8^h,B_n^h,F_4^h,G_2^h$) using $2\cos$-based bounds, while introducing $\mu_n$ as the largest root parameter with $\hat{b}_n^h=0$ and showing $\mu_n$ grows with $n$ yet remains below $\epsilon\approx 2.04998$. The results yield comprehensive classifications: for $0\le h<2$, generalized positivity of $H$ is characterized in terms of $H_\sigma$ within specific Dynkin-type lists, and for $h\ge \epsilon$ the matrices $B_n^h$ become generalized positive-definite for all $n$, implying finite-type behavior under $H+(2-h)E$. The findings connect the h-Shuhan framework to finite and affine Cartan types and provide sharp, computable thresholds for positivity across many twisted and untwisted cases, offering a structured atlas of positivity regimes in this generalized setting.

Abstract

In this paper, we define h-Shuhan matrix, which is the generalization of the generalized Cartan matrix, and find the h-Shuhan matrices for all positive semi-definite ( or generalized positive semi-definite, virtual positive semi-definite) with $h<2$. Furthermore, we know that the largest eigenvalue of the matrix $\hat{B}_{n}^{h}$ increases with $n$, but it is always less than $h$ plus a constant $ε\approx2.04998$.

Theorems & Definitions (27)

• Definition 1.1
• Lemma 1.2
• Lemma 1.3
• Proposition 1.4
• Remark 1.5
• Lemma 1.6
• Proposition 1.7
• Definition 1.8
• Lemma 2.1
• Proposition 2.2