Explicit construction of a Ramanujan $(n_1,n_2,\ldots,n_{d-1})$-regular hypergraph
Alireza Sarveniazi
Source: Duke Math. J. Volume 139, Number 1
(2007), 141-171.
Abstract
Using the main properties of the skew polynomial rings $\mathbb{F}_{q^d}\{\tau\}$ and some related rings, we describe the explicit construction of Ramanujan hypergraphs, which are certain simplicial complexes introduced in the author's thesis [29] (see also [30]) as generalizations of Ramanujan graphs. Such hypergraphs are described in terms of Cayley graphs of various groups. We give an explicit description of our hypergraph as the Cayley graph of the groups $\mathrm{PSL}_d(\mathbb{F}_r)$ and $\mathrm{PGL}_d(\mathbb{F}_r)$ with respect to a certain set of generators, over a finite field $\mathbb{F}_r$ with $r$ elements
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1184341240
Digital Object Identifier: doi:10.1215/S0012-7094-07-13913-9
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References
K. Brown, Buildings, Springer, New York, 1989.
Mathematical Reviews (MathSciNet): MR0969123
L. Carlitz, On polynomials in a Galois field, Bull. Amer. Math. Soc. 38 (1932), 736--744.
Mathematical Reviews (MathSciNet): MR1562499
Digital Object Identifier: doi:10.1090/S0002-9904-1932-05519-7
Project Euclid: euclid.bams/1183496202
—, On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), 137--168.
Mathematical Reviews (MathSciNet): MR1545872
Zentralblatt MATH: 0012.04904
Digital Object Identifier: doi:10.1215/S0012-7094-35-00114-4
Project Euclid: euclid.dmj/1077488973
—, A set of polynomials, Duke Math. J. 6 (1940), 486--504.
Mathematical Reviews (MathSciNet): MR0001946
Digital Object Identifier: doi:10.1215/S0012-7094-40-00639-1
Project Euclid: euclid.dmj/1077491918
—, Some special functions over GF$(q,x)$, Duke Math. J. 27 (1960), 139--158.
Mathematical Reviews (MathSciNet): MR0206348
Zentralblatt MATH: 0097.25904
Digital Object Identifier: doi:10.1215/S0012-7094-60-02714-9
Project Euclid: euclid.dmj/1077469034
D. I. Cartwright, Harmonic Functions on Buildings of, Type $\tilde{A}_n$, Sympos. Math. 39, Cambridge Univ. Press, Cambridge, 1999.
Mathematical Reviews (MathSciNet): MR1802428
D. I. Cartwright and T. Steger, A family of $\tilde{A}_n$-groups, Israel J. Math. 103 (1998), 125--140.
Mathematical Reviews (MathSciNet): MR1613560
Zentralblatt MATH: 0923.51010
Digital Object Identifier: doi:10.1007/BF02762271
G. Davidoff, P. Sarnak, and A. Vallete, Elementary Number Theory, Group Theory, and Ramanujan Graphs, London Math. Soc. Stud. Texts 55, Cambridge Univ. Press, Cambridge, 2003.
Mathematical Reviews (MathSciNet): MR1989434
Zentralblatt MATH: 1032.11001
J. D. Dixon, M. P. F. Du Sautoy, A. Mann, and D. Segal, Analytic Pro-p-Groups, London Math. Soc. Lecture Note Ser. 157, Cambridge Univ. Press, Cambridge, 1991.
Mathematical Reviews (MathSciNet): MR1152800
V. G. Drinfel'D [drinfeld], Proof of the Petersson conjecture for $\rm{GL}(2)$ over a global field of characteristic p (in Russian), Funktsional. Anal. i Prilozhen. 22, no. 1 (1988), 34--54.; English translation in Funct. Anal. Appl. 22 (1988), 28--43.
Mathematical Reviews (MathSciNet): MR0936697
B. Farb and R. K. Dennis, Noncommutative Algebra, Grad. Texts in Math. 144, Springer, New York, 1993.
Mathematical Reviews (MathSciNet): MR1233388
Zentralblatt MATH: 0797.16001
N. Jacobson, Finite-Dimensional Division Algebras over Fields, Springer, Berlin, 1996.
Mathematical Reviews (MathSciNet): MR1439248
Zentralblatt MATH: 0874.16002
T. Y. Lam, A First Course in Noncommutative Rings, Grad. Texts in Math. 131, Springer, New York, 1986.
Mathematical Reviews (MathSciNet): MR1125071
S. Lang, Algebra, Addison Wesley, Reading, Mass., 1965.
Mathematical Reviews (MathSciNet): MR0197234
—, $\rm{SL}_2(R)$, reprint of the 1975 ed., Grad. Texts in Math. 105, Springer, New York, 1985.
Mathematical Reviews (MathSciNet): MR0803508
W.-C. W. Li, Ramanujan hypergraphs, Geom. Funct. Anal. 14 (2004), 380--399.
Mathematical Reviews (MathSciNet): MR2060199
Zentralblatt MATH: 1084.05047
Digital Object Identifier: doi:10.1007/s00039-004-0461-z
A. Lubotzky, R. Phillips, and P. Sarnak, ``Explicit expanders and the Ramanujan Conjectures'' in Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing (Berkeley, Calif., 1986), Assoc. for Computing Machinery, New York, 1986, 240--246.
—, Ramanujan graphs, Combinatorica 8 (1988), 261--277.
Mathematical Reviews (MathSciNet): MR0963118
Digital Object Identifier: doi:10.1007/BF02126799
A. Lubotzky, B. Samuels, and U. Vishne, Explicit constructions of Ramanujan complexes of type $\tilde{A}_d$, European J. Combin. 26 (2005), 965--993.
Mathematical Reviews (MathSciNet): MR2143204
Zentralblatt MATH: 1135.05038
Digital Object Identifier: doi:10.1016/j.ejc.2004.06.007
—, Ramanujan complexes of type $\tilde{A}_d$, Israel J. Math. 149 (2005), 267--299.
Mathematical Reviews (MathSciNet): MR2191217
Zentralblatt MATH: 1087.05036
Digital Object Identifier: doi:10.1007/BF02772543
C. C. Macduffee, An Introduction to Abstract Algebra, Wiley, New York, 1940.
Mathematical Reviews (MathSciNet): MR0003591
M. Morgenstern, Existence and explicit construction of $q+1$ regular Ramanujan graphs for every prime power q, J. Combin. Theory Ser. B 62 (1994), 44--62.
Mathematical Reviews (MathSciNet): MR1290630
Zentralblatt MATH: 0814.68098
Digital Object Identifier: doi:10.1006/jctb.1994.1054
O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc. 35 (1933), 559--584.; Errata, Trans. Amer. Math. Soc. 36 (1934), 275. ${\!}$;
Mathematical Reviews (MathSciNet): MR1501703
Mathematical Reviews (MathSciNet): MR1501741
JSTOR: links.jstor.org
—, Theory of non-commutative polynomials, Ann. of Math. (2) 34 (1933), 480--508.
Mathematical Reviews (MathSciNet): MR1503119
Digital Object Identifier: doi:10.2307/1968173
JSTOR: links.jstor.org
M. Ronan, Lectures on Buildings, Perspect. Math. 7, Academic Press, Boston, 1989.
Mathematical Reviews (MathSciNet): MR1005533
—, Buildings: Main ideas and applications, I: Main ideas, Bull. London Math. Soc. 24 (1992), 1--51.; II: Arithmetic groups, buildings and symmetric spaces, Bull. London Math. Soc. 24 (1992), 97--126. ${\!}$;
Mathematical Reviews (MathSciNet): MR1139056
Mathematical Reviews (MathSciNet): MR1148671
Zentralblatt MATH: 0753.51009
Digital Object Identifier: doi:10.1112/blms/24.1.1
M. Rosen, Number Theory in Function Fields, Grad. Texts in Math. 210, Springer, New York, 2002.
Mathematical Reviews (MathSciNet): MR1876657
P. Sarnak, Some Applications of Modular Forms, Cambridge Tracts in Math. 99, Cambridge Univ. Press, Cambridge, 1990.
Mathematical Reviews (MathSciNet): MR1102679
A. Sarveniazi, Ramanujan $(n_1,n_2,\ldots,n_{d-1})$-regular hypergraphs based on Bruhat-Tits buildings of type $\tilde{A}_{d-1}$, Ph.D. dissertation, University of Göttingen, Göttingen, Germany, 2003.
—, Ramanujan $(n_1,n_2,\ldots,n_{d-1})$-regular hypergraphs, I: Definition and abstract construction, preprint, 2004.
W. Scharlau, Quadratic and Hermitian Forms, Grundlehren Math. Wiss. 270, Springer, Berlin, 1985.
Mathematical Reviews (MathSciNet): MR0770063
J.-P. Serre, A Course in Arithmetic, Grad. Texts in Math. 7, Springer, New York, 1973.
Mathematical Reviews (MathSciNet): MR0344216
—, Linear Representations of Finite Groups, Grad. Texts in Math. 42, Springer, New York, 1977.
Mathematical Reviews (MathSciNet): MR0450380
—, Trees, Springer, New York, 1980.
Mathematical Reviews (MathSciNet): MR0607504
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Kanô Memorial Lectures 1, Publ. Math Soc. Japan 11, Princeton Univ. Press, Princeton, 1971.
Mathematical Reviews (MathSciNet): MR0314766
A. Terras, Fourier Analysis on Finite Groups and Applications, London Math. Soc. Stud. Texts 43, Cambridge Univ. Press, Cambridge, 1999.
Mathematical Reviews (MathSciNet): MR1695775
Zentralblatt MATH: 0928.43001
M.-F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Math. 800, Springer, Berlin, 1980.
Mathematical Reviews (MathSciNet): MR0580949
A. Weil, Basic Number Theory, Grundlehren Math. Wiss. 144, Springer, New York, 1967.
Mathematical Reviews (MathSciNet): MR0234930
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