arXiv subjects

Yongzhi Luan

Publications and source records attributed to Yongzhi Luan.

3 recordsLinked to original sources

arXiv2026

A classification of finite simply reducible groups of order at most 2000

We give a computer-assisted classification, up to isomorphism, of all nontrivial finite simply reducible groups of order at most 2000. There are 7889 such groups, all of even order. We provide representatives and a table of their numbers at each order. Order 1024, which is not available as a catalogue of group objects in the SmallGroups library of the computer algebra system GAP, requires a separate construction. Combining descendant generation with a classification of quadratic maps over $\mathbb F_2$, we obtain exactly 803 groups of this order, including 221 of nilpotency class two. We also derive exact counting formulas on several infinite families of orders and explicit bounds from structural subclasses.

math.GR↗

arXiv2026

A new proof for the partition algorithm of the annihilator varieties of highest weight modules

Let $L(λ)$ be a simple highest weight module of a classical Lie algebra $\mathfrak{g}$ with highest weight $λ-ρ$, where $ρ$ is half the sum of positive roots. Joseph proved that the associated variety of the annihilator ideal of $L(λ)$ (also called the annihilator variety) is the Zariski closure of a nilpotent orbit in $\mathfrak{g}^*$. Recently, Bai--Ma--Wang introduced a partition algorithm to describe this corresponding nilpotent orbit for a given highest weight module $L(λ)$. In this paper, we present a new direct proof of Bai--Ma--Wang's partition algorithm using Sommers duality.

math.RT↗

arXiv2024

Dirac cohomology, branching laws and Wallach modules

The idea of using Dirac cohomology to study branching laws was initiated by Huang, Pandzić and Zhu in 2013 [HPZ]. One of their results says that the Dirac cohomology of $π$ completely determines $π|_{K}$, where $π$ is any irreducible unitarizable highest weight $(\mathfrak{g}, K)$ module. This paper aims to develop this idea for the exceptional Lie groups $E_{6(-14)}$ and $E_{7(-25)}$: we recover the $K$-spectrum of the Wallach modules from their Dirac cohomology.

math.RT↗