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High-Dimensional Expanders from Chevalley Groups

Authors Ryan O'Donnell, Kevin Pratt

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ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • High-dimensional expanders
  • simplicial complexes
  • group theory

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Abstract

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.

Cite As Get BibTex

Ryan O'Donnell and Kevin Pratt. High-Dimensional Expanders from Chevalley Groups. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 18:1-18:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CCC.2022.18

Author Details

Ryan O'Donnell
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Kevin Pratt
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA

References

  1. Herbert Abels and Stephan Holz. Higher generation by subgroups. Journal of Algebra, 160(2):310-341, 1993. URL: https://doi.org/10.1006/jabr.1993.1190.
  2. Vedat Alev, Fernando Jeronimo, Dylan Quintana, Shashank Srivastava, and Madhur Tulsiani. List decoding of direct sum codes. In Proceedings of the 14th annual Symposium on Discrete Algorithms (SODA), pages 1412-1425, 2020. Google Scholar
  3. Vedat Levi Alev, Fernando Granha Jeronimo, and Madhur Tulsiani. Approximating constraint satisfaction problems on high-dimensional expanders. In Proceedings of the 60th annual Symposium on Foundations of Computer Science (FOCS), pages 180-201, 2019. URL: https://doi.org/10.1109/FOCS.2019.00021.
  4. Vedat Levi Alev and Lap Chi Lau. Improved analysis of higher order random walks and applications. In Proceedings of the 52nd annual Symposium on Theory of Computing (STOC), pages 1198-1211, 2020. Google Scholar
  5. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid. In Proceedings of the 51st annual Symposium on Theory of Computing (STOC), pages 1-12, 2019. Google Scholar
  6. Michael Aschbacher. The status of the classification of the finite simple groups. Notices of the American Mathematical Society, 51(7):736-740, 2004. Google Scholar
  7. László Babai, William Kantor, and Alexander Lubotsky. Small-diameter Cayley graphs for finite simple groups. European Journal of Combinatorics, 10(6):507-522, 1989. URL: https://doi.org/10.1016/S0195-6698(89)80067-8.
  8. Cristina Ballantine. Ramanujan type buildings. Canadian Journal of Mathematics, 52(6):1121-1148, 2000. URL: https://doi.org/10.4153/CJM-2000-047-4.
  9. Oren Becker. Symmetric unique neighbor expanders and good LDPC codes. Discrete Applied Mathematics. The Journal of Combinatorial Algorithms, Informatics and Computational Sciences, 211:211-216, 2016. URL: https://doi.org/10.1016/j.dam.2016.04.022.
  10. Anders Björner. Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings. Advances in Mathematics, 52(3):173-212, 1984. URL: https://doi.org/10.1016/0001-8708(84)90021-5.
  11. Emmanuel Breuillard, Ben Green, and Terence Tao. Suzuki groups as expanders. Groups, Geometry, and Dynamics, 5(2):281-299, 2011. URL: https://doi.org/10.4171/GGD/128.
  12. Élie Cartan. Sur la structure des groupes de transformations finis et continus. PhD thesis, Université de Paris, 1894. Google Scholar
  13. Roger Carter. Simple groups of Lie type. John Wiley & Sons, 1989. Google Scholar
  14. Donald I. Cartwright, Patrick Solé, and Andrzej Żuk. Ramanujan geometries of type Ã_n. Discrete Mathematics, 269(1-3):35-43, 2003. URL: https://doi.org/10.1016/S0012-365X(02)00748-3.
  15. Arjeh Cohen, Scott Murray, and Don Taylor. Computing in groups of Lie type. Mathematics of Computation, 73(247):1477-1498, 2004. URL: https://doi.org/10.1090/S0025-5718-03-01582-5.
  16. Yotam Dikstein and Irit Dinur. Agreement testing theorems on layered set systems. In Proceedings of the 60th annual Symposium on Foundations of Computer Science (FOCS), pages 1495-1524, 2019. URL: https://doi.org/10.1109/FOCS.2019.00088.
  17. Yotam Dikstein, Irit Dinur, Yuval Filmus, and Prahladh Harsha. Boolean function analysis on high-dimensional expanders. In Proceedings of the 22nd annual International Conference on Randomization and Computation (RANDOM), volume 116, pages Art. No. 38, 20, 2018. Google Scholar
  18. Irit Dinur, Shai Evra, Ron Livne, Alexander Lubotzky, and Shahar Mozes. Locally testable codes with constant rate, distance, and locality, 2021. Announced at URL: https://www.youtube.com/watch?v=pjc6GCRFnpg.
  19. Irit Dinur, Yuval Filmus, Prahladh Harsha, and Madhur Tulsiani. Explicit SoS lower bounds from high-dimensional expanders. In Proceedings of the 12th Innovations in Theoretical Computer Science Conference (ITCS), volume 185, pages 38:1-38:16, 2021. Google Scholar
  20. Irit Dinur, Prahladh Harsha, Tali Kaufman, Inbal Livni Navon, and Amnon Ta-Shma. List-decoding with double samplers. SIAM Journal on Computing, 50(2):301-349, 2021. URL: https://doi.org/10.1137/19M1276650.
  21. Irit Dinur and Tali Kaufman. High dimensional expanders imply agreement expanders. In Proceedings of the 58th annual Symposium on Foundations of Computer Science (FOCS), pages 974-985, 2017. Google Scholar
  22. Mikhail Ershov and Andrei Jaikin-Zapirain. Property (T) for noncommutative universal lattices. Inventiones Mathematicae, 179(2):303-347, 2010. Google Scholar
  23. Mikhail Ershov, Andrei Jaikin-Zapirain, and Martin Kassabov. Property (T) for groups graded by root systems. Memoirs of the American Mathematical Society, 249(1186):v+135, 2017. URL: https://doi.org/10.1090/memo/1186.
  24. Shai Evra, Tali Kaufman, and Gilles Zémor. Decodable quantum LDPC codes beyond the square root distance barrier using high dimensional expanders. In Proceedings of the 61st annual Symposium on Foundations of Computer Science (FOCS), pages 218-227, 2020. Google Scholar
  25. Jacob Fox, Mikhail Gromov, Vincent Lafforgue, Assaf Naor, and János Pach. Overlap properties of geometric expanders. Journal für die Reine und Angewandte Mathematik. (Crelle’s Journal), 671:49-83, 2012. URL: https://doi.org/10.1515/crelle.2011.157.
  26. The GAP Group. GAP - Groups, Algorithms, and Programming, Version 4.11.1, 2021. URL: https://www.gap-system.org.
  27. Howard Garland. p-adic curvature and the cohomology of discrete subgroups of p-adic groups. Annals of Mathematics. Second Series, 97:375-423, 1973. URL: https://doi.org/10.2307/1970829.
  28. Peter Garst. Cohen-Macaulay complexes and group actions. PhD thesis, Unviersity of Wisconsin-Madison, 1979. Google Scholar
  29. Brian Hall. Lie groups, Lie algebras, and representations: an elementary introduction. Springer, 2003. Google Scholar
  30. Sergei Haller and Max Horn. Unipot - a system for computing with elements of unipotent subgroups of Chevalley groups, 2018. URL: https://www.gap-system.org/Packages/unipot.html.
  31. Prahladh Harsha and Ramprasad Saptharishi. A note on the elementary HDX construction of Kaufman-Oppenheim. Technical Report 1912.11225, arXiv, 2019. Google Scholar
  32. David Hill. Prove that the sum of all simple roots is a root. Mathematics Stack Exchange, 2016. (version: 2016-04-26). URL: https://math.stackexchange.com/q/1760142.
  33. Robert Howlett, Leanne Rylands, and Donald Taylor. Matrix generators for exceptional groups of Lie type. Journal of Symbolic Computation, 31(4):429-445, 2001. URL: https://doi.org/10.1006/jsco.2000.0431.
  34. James Humphreys. Introduction to Lie algebras and representation theory. Springer-Verlag, 1972. Google Scholar
  35. James Humphreys. Linear algebraic groups. Springer-Verlag, 1995. Google Scholar
  36. Martin Kassabov. Symmetric groups and expander graphs. Inventiones Mathematicae, 170(2):327-354, 2007. URL: https://doi.org/10.1007/s00222-007-0065-y.
  37. Martin Kassabov, Alexander Lubotzky, and Nikolay Nikolov. Finite simple groups as expanders. Proceedings of the National Academy of Sciences of the United States of America, 103(16):6116-6119, 2006. URL: https://doi.org/10.1073/pnas.0510337103.
  38. Tali Kaufman and Alexander Lubotzky. Edge transitive Ramanujan graphs and highly symmetric LDPC good codes. In Proceedings of the 44th annual Symposium on Theory of Computing (STOC), pages 359-366, 2012. Google Scholar
  39. Tali Kaufman and David Mass. Local-to-global agreement expansion via the variance method. In Proceedings of the 11th annual Innovations in Theoretical Computer Science Conference (ITCS), pages 74:1-74:14, 2020. Google Scholar
  40. Tali Kaufman and Izhar Oppenheim. Construction of new local spectral high dimensional expanders. In Proceedings of the 50th Annual Symposium on Theory of Computing (STOC), pages 773-786, 2018. Google Scholar
  41. Tali Kaufman and Izhar Oppenheim. High order random walks: beyond spectral gap. Combinatorica, 40(2):245-281, 2020. URL: https://doi.org/10.1007/s00493-019-3847-0.
  42. Tali Kaufman and Madhu Sudan. Algebraic property testing: the role of invariance. In Proceedings of the 40th annual Symposium on Theory of Computation (STOC), pages 403-412. ACM, New York, 2008. URL: https://doi.org/10.1145/1374376.1374434.
  43. Tali Kaufman and Ran Tessler. New cosystolic expanders from tensors imply explicit quantum LDPC codes with ω(√n log^k n) distance. In Proceedings of the 53rd annual Symposium on Theory of Computing (STOC), pages 1317-1329, 2021. Google Scholar
  44. Tali Kaufman and Avi Wigderson. Symmetric LDPC codes and local testing. Combinatorica, 36(1):91-120, 2016. URL: https://doi.org/10.1007/s00493-014-2715-1.
  45. Laurent Lafforgue. Chtoucas de Drinfeld et correspondance de Langlands. Inventiones Mathematicae, 147(1):1-241, 2002. URL: https://doi.org/10.1007/s002220100174.
  46. Folke Lannér. On complexes with transitive groups of automorphisms. Communications du Séminaire Mathématique de l'Université de Lund, 11:71, 1950. Google Scholar
  47. Wen-Ching Winnie Li. Ramanujan hypergraphs. Geometric and Functional Analysis, 14(2):380-399, 2004. URL: https://doi.org/10.1007/s00039-004-0461-z.
  48. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Explicit constructions of Ramanujan complexes of type Ã_d. European Journal of Combinatorics, 26(6):965-993, 2005. URL: https://doi.org/10.1016/j.ejc.2004.06.007.
  49. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Ramanujan complexes of type Ã_d. Israel Journal of Mathematics, 149:267-299, 2005. Probability in mathematics. URL: https://doi.org/10.1007/BF02772543.
  50. Izhar Oppenheim. Local spectral expansion approach to high dimensional expanders part I: Descent of spectral gaps. Discrete & Computational Geometry, 59(2):293-330, 2018. Google Scholar
  51. Pavel Panteleev and Gleb Kalachev. Asymptotically good quantum and locally testable classical LDPC codes. Technical Report 2111.03654, arXiv, 2021. Google Scholar
  52. Alireza Sarveniazi. Ramanujan Hypergraph Based on Bruhat-Tits Building. PhD thesis, University of Göttingen, 2004. Google Scholar
  53. Victor Shoup. New algorithms for finding irreducible polynomials over finite fields. Mathematics of Computation, 54(189):435-447, 1990. URL: https://doi.org/10.2307/2008704.
  54. Robert Steinberg. Lectures on Chevalley groups, volume 66 of University Lecture Series. American Mathematical Society, 2016. Notes prepared by John Faulkner and Robert Wilson, revised and corrected edition of the 1968 original. URL: https://doi.org/10.1090/ulect/066.
  55. Andrzej Żuk. Property (T) and Kazhdan constants for discrete groups. Geometric and Functional Analysis, 13(3):643-670, 2003. URL: https://doi.org/10.1007/s00039-003-0425-8.
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