Log24

Saturday, November 17, 2018

Representation

Filed under: General,Geometry — Tags: — m759 @ 7:00 AM

See as well . . .

. . . and posts tagged Alperin.

On its current homepage, the American Mathematical Society  
links to a Nov. 15 blog post illustrating the Stan Lee approach
to mathematics:

Stories: "Math needs more stories. All kinds of stories…" 

See too Mathematics and Narrative in this  journal.

Saturday, August 25, 2018

Point at Infinity

Filed under: General,Geometry — Tags: , — m759 @ 5:20 AM

In literature —

http://www.log24.com/log/pix18/180825-Point_Omega-cover.jpg

In film —

http://www.log24.com/log/pix18/180825-Borrego-script-Instagram-foto-only-500w.jpg

In mathematics —

http://www.log24.com/log/pix18/180824-Alperin-Groups_and_Representations-1995-p61-Further_Exercises.jpg

Thursday, August 16, 2018

Mathematics and Narrative

Filed under: General,Geometry — Tags: — m759 @ 11:00 PM

(Continued)

Mathematics:

Alperin, Groups and Representations, p. 61

Narrative:

The Tale of the Flux Capacitor

Wednesday, August 15, 2018

An Illusion of Brilliance

Filed under: General,Geometry — Tags: , — m759 @ 5:25 PM

" . . . the 3 by 3, the six-sided, three-layer configuration of
the original Rubik’s Cube, which bestows an illusion of brilliance
on those who can solve it."

— John Branch in the online New York Times  today,
     "Children of the Cube":

https://www.nytimes.com/2018/08/15/sports/
cubing-usa-nationals-max-park.html

Cube-solving, like other sports, allows for displays of
impressive and admirable skill, if not "brilliance."

The mathematics — group theory — that is sometimes associated
with Rubik's Cube is, however, not  a sport.  See Rubik + Group
in this journal.

http://www.log24.com/log/pix18/180815-Alperin-Bell-preface-1995.gif

An Exceptional Isomorphism

Filed under: General,Geometry — Tags: , , — m759 @ 5:40 AM

Why PSL(2,7) is isomorphic to GL(3.2)

From previous posts on this topic —

http://www.log24.com/log/pix18/180813-Structure_and_Sense-post-160606.gif

http://www.log24.com/log/pix18/180813-Knight_Moves-080116-page-top.gif

Tuesday, August 14, 2018

D8ing

Filed under: General,Geometry — Tags: , , , — m759 @ 12:06 AM

(Continued)

http://www.log24.com/log/pix18/180813-Ezra_Brown-PSL(2,7)-GL(2,3)-Oct-2009.jpg

http://www.log24.com/log/pix18/180814-Knight_Moves-archived-May_9_2008-500w.jpg

Thursday, August 9, 2018

True Grids

Filed under: General,Geometry — Tags: , — m759 @ 7:59 PM

From a search in this journal for "True Grid,"
a fanciful description of  the 3×3 grid —

"This is the garden of Apollo,
the field of Reason…."
John Outram, architect    

A fanciful instance of the 4×2 grid in
a scene from the film "The Master" —

IMAGE- Joaquin Phoenix, corridor scene in 'The Master'

A fanciful novel referring to the number 8,
and a not -so-fanciful reference:

http://www.log24.com/log/pix18/180809-The_EIght-and-coordinates-for-PSL(2,7)-actions-500w.jpg

Illustrated above are Katherine Neville's novel The Eight  and the
"knight" coordinatization of the 4×2 grid from a page on the exceptional
isomorphism between PSL(3,2) (alias GL(3,2)) and PSL(2,7) — groups
of, respectively, degree 7 and degree 8.

Literature related to the above remarks on grids:

Ross Douthat's New York Times  column yesterday purported, following
a 1946 poem by Auden, to contrast students of the humanities with
technocrats by saying that the former follow Hermes, the latter Apollo.

I doubt that Apollo would agree.

Wednesday, August 8, 2018

8/8

Filed under: General,Geometry — Tags: , , , — m759 @ 6:00 AM

From mathoverflow.net on Dec. 7, 2016 —

The exceptional isomorphism between
PGL(3,2) and PSL(2,7): geometric origin?

Essentially the same question was asked earlier at

math.stackexchange.com on Aug. 2, 2010.

See also this  journal in November 2017 —

"Read something that means something."
                — New Yorker  ad

'Knight' octad labeling by the 8 points of the projective line over GF(7) .

Background — Relativity Problem in Log24.

Tuesday, September 20, 2011

Relativity Problem Revisited

Filed under: General,Geometry — Tags: , , , — m759 @ 4:00 AM

A footnote was added to Finite Relativity

Background:

Weyl on what he calls the relativity problem

IMAGE- Weyl in 1949 on the relativity problem

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

– Hermann Weyl, 1949, "Relativity Theory as a Stimulus in Mathematical Research"

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

– Hermann Weyl, 1946, The Classical Groups , Princeton University Press, p. 16

…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on  coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M 24 (containing the original group), acts on the larger array.  There is no obvious solution to Weyl's relativity problem for M 24.  That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or symbol-strings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M 24. ….

Footnote of Sept. 20, 2011:

* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols.  His abstract for a 1990 paper says that in his construction "The generators of M 24 are defined… as permutations of twenty-four 7-cycles in the action of PSL2(7) on seven letters…."

See "Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups," by R.T. Curtis,  Mathematical Proceedings of the Cambridge Philosophical Society  (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.

Some related articles by Curtis:

R.T. Curtis, "Natural Constructions of the Mathieu groups," Math. Proc. Cambridge Philos. Soc.  (1989), Vol. 106, pp. 423-429

R.T. Curtis. "Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups M 12  and M 24" In Proceedings of 1990 LMS Durham Conference 'Groups, Combinatorics and Geometry'  (eds. M. W. Liebeck and J. Saxl),  London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396

R.T. Curtis, "A Survey of Symmetric Generation of Sporadic Simple Groups," in The Atlas of Finite Groups: Ten Years On , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57

Wednesday, August 19, 2009

Wednesday August 19, 2009

Filed under: General,Geometry — Tags: , — m759 @ 10:30 AM

Group Actions, 1984-2009

From a 1984 book review:

"After three decades of intensive research by hundreds of group theorists, the century old problem of the classification of the finite simple groups has been solved and the whole field has been drastically changed. A few years ago the one focus of attention was the program for the classification; now there are many active areas including the study of the connections between groups and geometries, sporadic groups and, especially, the representation theory. A spate of books on finite groups, of different breadths and on a variety of topics, has appeared, and it is a good time for this to happen. Moreover, the classification means that the view of the subject is quite different; even the most elementary treatment of groups should be modified, as we now know that all finite groups are made up of groups which, for the most part, are imitations of Lie groups using finite fields instead of the reals and complexes. 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."

— Jonathan L. Alperin,
   review of books on group theory,
   Bulletin (New Series) of the American
   Mathematical Society
10 (1984) 121, doi:
   10.1090/S0273-0979-1984-15210-8
 

A more specific example:


Actions of GL(2,3) on a 3x3 coordinate-array

The same example
at Wolfram.com:

Ed Pegg Jr.'s program at Wolfram.com to display a large number of actions of small linear groups over finite fields

Caption from Wolfram.com:
 
"The two-dimensional space Z3×Z3 contains nine points: (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), and (2,2). The 48 invertible 2×2 matrices over Z3 form the general linear group known as GL(2, 3). They act on Z3×Z3 by matrix multiplication modulo 3, permuting the nine points. More generally, GL(n, p) is the set of invertible n×n matrices over the field Zp, where p is prime. With (0, 0) shifted to the center, the matrix actions on the nine points make symmetrical patterns."

Citation data from Wolfram.com:

"GL(2,p) and GL(3,3) Acting on Points"
 from The Wolfram Demonstrations Project,
 http://demonstrations.wolfram.com/GL2PAndGL33ActingOnPoints/,
 Contributed by: Ed Pegg Jr"

As well as displaying Cullinane's 48 pictures of group actions from 1985, the Pegg program displays many, many more actions of small finite general linear groups over finite fields. It illustrates Cullinane's 1985 statement:

"Actions of GL(2,p) on a p×p coordinate-array have the same sorts of symmetries, where p is any odd prime."

Pegg's program also illustrates actions on a cubical array– a 3×3×3 array acted on by GL(3,3). For some other actions on cubical arrays, see Cullinane's Finite Geometry of the Square and Cube.
 

Saturday, August 6, 2005

Saturday August 6, 2005

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM
For André Weil on
the seventh anniversary
of his death:

 A Miniature
Rosetta Stone

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

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

Thursday, August 4, 2005

Thursday August 4, 2005

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM
Visible Mathematics, continued

Today’s mathematical birthdays:
Saunders Mac Lane, John Venn,
and Sir William Rowan Hamilton.

It is well known that the quaternion group is a subgroup of GL(2,3), the general linear group on the 2-space over GF(3), the 3-element Galois field.

The figures below illustrate this fact.

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

Related material: Visualizing GL(2,p)

“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

Wednesday, May 4, 2005

Wednesday May 4, 2005

Filed under: General,Geometry — Tags: , — m759 @ 1:00 PM
The Fano Plane
Revisualized:

 

 The Eightfold Cube

or, The Eightfold Cube

Here is the usual model of the seven points and seven lines (including the circle) of the smallest finite projective plane (the Fano plane):
 
The image “http://www.log24.com/theory/images/Fano.gif” cannot be displayed, because it contains errors.
 

Every permutation of the plane's points that preserves collinearity is a symmetry of the  plane.  The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group  PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)

The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle.  It does not, however, indicate where the other 162 symmetries come from.  

Shown below is a new model of this same projective plane, using partitions of cubes to represent points:

 

Fano plane with cubes as points
 
The cubes' partitioning planes are added in binary (1+1=0) fashion.  Three partitioned cubes are collinear if and only if their partitioning planes' binary sum equals zero.

 

The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.

 

Fano plane group - generating permutations

For a proof that such permutations generate the 168 symmetries, see Binary Coordinate Systems.

 

(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations.  But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results.  This illustrates the difference between affine and projective spaces over the binary field GF(2).  In a related 2x2x2 cubic model of the affine 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices.  This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices.  Such translations in the affine 3-space have no effect on the projective plane, since they leave each of the plane model's seven partitions– the "points" of the plane– invariant.)

To view the cubes model in a wider context, see Galois Geometry, Block Designs, and Finite-Geometry Models.

 

For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.

For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way.  For more on the mathematics of cubic models, see Solomon's Cube.

 

For a large downloadable folder with many other related web pages, see Notes on Finite Geometry.

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