Book contents
- Frontmatter
- Contents
- Preface
- Part I Finite Abelian Groups
- Part II Finite Nonabelian Groups
- 15 Fourier Transform and Representations of Finite Groups
- 16 Induced Representations
- 17 The Finite ax + b Group
- 18 The Heisenberg Group
- 19 Finite Symmetric Spaces–Finite Upper Half Plane Hq
- 20 Special Functions on Hq – K-Bessel and Spherical
- 21 The General Linear Group (Expression not displayed)
- 22 Selberg's Trace Formula and Isospectral Non-isomorphic Graphs
- 23 The Trace Formula on Finite Upper Half Planes
- 24 Trace Formula For a Tree and Ihara's Zeta Function
- References
- Index
21 - The General Linear Group (Expression not displayed)
from Part II - Finite Nonabelian Groups
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- Part I Finite Abelian Groups
- Part II Finite Nonabelian Groups
- 15 Fourier Transform and Representations of Finite Groups
- 16 Induced Representations
- 17 The Finite ax + b Group
- 18 The Heisenberg Group
- 19 Finite Symmetric Spaces–Finite Upper Half Plane Hq
- 20 Special Functions on Hq – K-Bessel and Spherical
- 21 The General Linear Group (Expression not displayed)
- 22 Selberg's Trace Formula and Isospectral Non-isomorphic Graphs
- 23 The Trace Formula on Finite Upper Half Planes
- 24 Trace Formula For a Tree and Ihara's Zeta Function
- References
- Index
Summary
How could they have been so blind? Now that it had been pointed out to them, it was all too perfectly obvious. The structure of C60 was not only the most wonderfully symmetrical molecular structure they had ever contemplated, it was also absurdly commonplace. A modern soccer ball is 20 white leather hexagons and 12 black leather pentagons stitched together, with each pentagon surrounded by five hexagons. It has 60 vertices; 60 points where the corners of the pentagons and the hexagons meet along the seams. How many times had each of them looked at a soccer ball without really registering these simple facts?
Baggott [1996, p. 70]We have found many reasons to study the representations of GL(2, q). In Chapter 13, for example, we discussed the molecule buckminsterfullerene or C60 and noted that an understanding of the spectral lines of this molecule requires a knowledge of the representations of A5 ≅ PSL(2, 5). In the last chapter we found that an understanding of the representations of GL(2, q) seems necessary to bound the spherical functions on the finite upper half plane. Also the Ramanujan graphs of Lubotzky, Phillips, and Sarnak [1988] are Cayley graphs for either PGL(2, q) or PSL(2, q), with prime q, using generating sets with p + 1 elements (p denoting a different prime). See also Lubotzky [1994, pp. 96 ff] and Sarnak [1990, pp. 73 ff]. Finally, the representations of SL(2, q) are needed in the paper of Lafferty and Rockmore [1992] where spectra of degree four Cayley graphs of SL(2, q) are studied.
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- Information
- Fourier Analysis on Finite Groups and Applications , pp. 362 - 384Publisher: Cambridge University PressPrint publication year: 1999