Let Φ be an irreducible root system (other than G₂) of rank at least 2, let 𝔽 be a finite field with p = char 𝔽 > 3, and let G(Φ,𝔽) be the corresponding Chevalley group. We describe a strongly explicit high-dimensional expander (HDX) family of dimension rank(Φ), where G(Φ,𝔽) acts simply transitively on the top-dimensional faces; these are λ-spectral HDXs with λ → 0 as p → ∞. 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 ≥ 2, and four exceptional constructions of dimension 4, 6, 7, and 8.
@InProceedings{odonnell_et_al:LIPIcs.CCC.2022.18, author = {O'Donnell, Ryan and Pratt, Kevin}, title = {{High-Dimensional Expanders from Chevalley Groups}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {18:1--18:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.18}, URN = {urn:nbn:de:0030-drops-165802}, doi = {10.4230/LIPIcs.CCC.2022.18}, annote = {Keywords: High-dimensional expanders, simplicial complexes, group theory} }
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