High-Dimensional Expanders from Chevalley Groups (2203.03705v1)

Abstract: Let $\Phi$ 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 $\mathrm{G}(\Phi,\mathbb{F})$ be the corresponding Chevalley group. We describe a strongly explicit high-dimensional expander (HDX) family of dimension $\mathrm{rank}(\Phi)$, where $\mathrm{G}(\Phi,\mathbb{F})$ acts simply transitively on the top-dimensional faces; these are $\lambda$-spectral HDXs with $\lambda \to 0$ as $p \to \infty$. This generalizes a construction of Kaufman and Oppenheim (STOC 2018), which corresponds to the case $\Phi = A_d$. Our work gives three new families of spectral HDXs of any dimension $\ge 2$, and four exceptional constructions of dimension $4$, $6$, $7$, and $8$.

• The paper extends HDX construction from A_d to multiple irreducible root systems (excluding G2) using Chevalley groups over fields with characteristic p > 3.
• The paper introduces families of high-dimensional expanders with explicit spectral gaps and exceptional cases in dimensions 4, 6, 7, and 8.
• The approach leverages Chevalley group actions on simplicial complexes to enable applications in coding theory, quantum computing, and algorithm design.

High-Dimensional Expanders from Chevalley Groups

The paper "High-Dimensional Expanders from Chevalley Groups" addresses the construction of high-dimensional expanders (HDXs) using the framework of Chevalley groups, a class of groups associated with root systems from Lie theory. The emphasis is on creating a family of HDXs that are strongly explicit and exhibit desirable spectral properties.

Key Contributions

1. Generalization of Construction: The paper extends the construction previously achieved by Kaufman and Oppenheim for the root system $A_d$ to encompass other irreducible root systems $\Phi$, excluding $G_2$, and over any field $\mathbb{F}$ with characteristic $p > 3$. This represents a significant generalization of high-dimensional expander constructions using algebraic group actions.
2. Properties of New HDX Families: The work introduces new spectral HDX families for dimensions $\geq 2$, along with some specific exceptional construction cases for dimensions 4, 6, 7, and 8. These HDXs are characterized by a parameter $\lambda$, controlling the spectral gap, which approaches 0 as $p \to \infty$.
3. Mathematical Framework: Using Chevalley groups $G(\Phi, \mathbb{F})$, the authors establish these groups act transitively on the top-dimensional faces of the simplicial complex, leading to the construction of expander families. The action is such that it allows the derivation of graph-like structures with carefully controlled eigenvalue spectra.

Implications and Potential Applications

The construction of HDXs with bounded degree and specified spectral properties has profound implications in theoretical computer science. These include, but are not limited to:

• Coding Theory: The expanders can be potentially used to generate locally testable codes (LDPC codes) that are both highly symmetric and possess near-optimal error correction capabilities through their expander properties.
• Quantum Computing: HDXs serve as foundational structures in the development of quantum codes, where constructs like Ramanujan complexes have prompted research into quantum error correction.
• Combinatorial Geometry and Algorithm Design: Expander structures are instrumental in derandomizing algorithms and proving inapproximability results.

Theoretical and Practical Significance

Theoretical contributions lie in the application of deep algebraic concepts—root systems, group actions, and expansions—to solve problems in HDXs, thus enriching combinatorial and group theoretic studies. Practically, while the paper provides constructive methods, the challenge remains in translating these methods to efficient algorithms for applications outside pure theoretical constructs, especially in areas involving large-scale computations.

Future Directions

Research could inevitably explore the following paths:

• Expansion Beyond Current Root Systems:

Although this paper excludes the $G_2$ root system due to technical constraints, future studies might tackle such complex cases, further broadening the applicability of these methods.

• Linking With More Group Variants:

Another prospect is extending the construction methodologies to other types of positional root systems or even non-Chevalley groups with particular interest in symmetric or alternating groups given their combinatorial nature.

• Optimization of Construction Techniques:

Making the current constructions more computationally feasible by optimizing the algorithms involved in constructing and traversing these complex structures would have a meaningful impact in practical scenarios.

This research pushes the boundaries of how algebraic structures like Chevalley groups can be applied to network theory and combinatorics, fostering enhanced understanding in the context of mathematical expansions.