论文标题
雪佛兰团体的高维扩张器
High-Dimensional Expanders from Chevalley Groups
论文作者
论文摘要
令$φ$是至少$ 2 $的不可约的根系（除$ g_2 $以外的$ g_2 $），让$ \ m \ m arthbb {f} $是一个有限的字段，具有$ p = \ operatorName {char} \ mathbb {f}> 3 $，然后让$ \ mathrm {g mathrm {g}（c c $ be）我们描述了一个尺寸$ \ mathrm {rank}（φ）$的强烈明确的高维膨胀剂（HDX）家族，其中$ \ mathrm {g}（φ，\ Mathbb {F}）这些是$λ$ -Spectral HDXS，带有$λ\至0 $ AS $ p \ to \ infty $。这概括了Kaufman和Oppenheim的结构（Stoc 2018），该结构与$φ= a_d $相对应。我们的工作为任何尺寸$ \ ge 2 $的光谱HDXS提供了三个新的家族，以及四个尺寸的特殊结构$ 4 $，$ 6 $，$ 7 $和$ 8 $。
Let $Φ$ be an irreducible root system (other than $G_2$) of rank at least $2$, let $\mathbb{F}$ be a finite field with $p = \operatorname{char} \mathbb{F} > 3$, and let $\mathrm{G}(Φ,\mathbb{F})$ be the corresponding Chevalley group. We describe a strongly explicit high-dimensional expander (HDX) family of dimension $\mathrm{rank}(Φ)$, where $\mathrm{G}(Φ,\mathbb{F})$ acts simply transitively on the top-dimensional faces; these are $λ$-spectral HDXs with $λ\to 0$ as $p \to \infty$. This generalizes a construction of Kaufman and Oppenheim (STOC 2018), which corresponds to the case $Φ= A_d$. Our work gives three new families of spectral HDXs of any dimension $\ge 2$, and four exceptional constructions of dimension $4$, $6$, $7$, and $8$.
