August 6, 2005
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For André Weil on
the seventh anniversary
of his death:A Miniature
Rosetta Stone
In a 1940 letter to his sister Simone, André Weil discussed a sort of “Rosetta stone,” or trilingual text of three analogous parts: classical analysis on the complex field, algebraic geometry over finite fields, and the theory of number fields.John Baez discussed (Sept. 6, 2003) the analogies of Weil, and he himself furnished another such Rosetta stone on a much smaller scale:
“… a 24-element group called the ‘binary
tetrahedral group,’ a 24-element group called ‘SL(2,Z/3),’ and
the vertices of a regular polytope in 4 dimensions called the
’24-cell.’ The most important fact is that these are all the
same thing!”For further details, see Wikipedia on the 24-cell, on special linear groups, and on Hurwitz quaternions,
The group SL(2,Z/3), also known as “SL(2,3),” is of course derived from
the general linear group GL(2,3). For the relationship of this
group to the quaternions, see the Log24 entry for August 4 (the
birthdate of the discoverer of quaternions, Sir William Rowan Hamilton).The 3×3 square shown above may, as my August 4 entry indicates, be
used to picture the quaternions and, more generally, the 48-element
group GL(2,3). It may therefore be regarded as the structure
underlying the miniature Rosetta stone described by Baez.“The typical example of a finite group is GL(n,q), the
general linear
group of n dimensions over the field with q elements. The student who
is introduced to the subject with other examples is being completely
misled.”– J. L. Alperin, book review,
Bulletin (New Series) of the American
Mathematical Society 10
(1984), 121
