See the title in this journal.
Such generation occurs both in Euclidean space …
… and in some Galois spaces —
In Galois spaces, some care must be taken in defining "reflection."
See the title in this journal.
Such generation occurs both in Euclidean space …
… and in some Galois spaces —
In Galois spaces, some care must be taken in defining "reflection."
Material related to the previous post, "Symmetry" —
This is the group of "8 rigid motions
generated by reflections in midplanes"
of "Solomon's Cube."
Material from this journal on May 1, the date of Golomb's death —
"Weitere Informationen zu diesem Themenkreis
finden sich unter http://www.encyclopediaofmath.org/
index.php/Cullinane_diamond_theorem und
http://finitegeometry.org/sc/gen/coord.html ."
The nonCoxeter simple reflection group of order 168
is a counterexample to the statement that
"Every finite reflection group is a Coxeter group."
The counterexample is based on a definition of "reflection group"
that includes reflections defined over finite fields.
Today I came across a 1911 paper that discusses the counterexample.
Of course, Coxeter groups were undefined in 1911, but the paper, by
Howard H. Mitchell, discusses the simple order168 group as a reflection group .
(Naturally, Mitchell's definition of "reflection" and his statement that
"The discussion of the binary groups
applies also to the case p = 2."
should be approached with care.)
A review of this topic might be appropriate for Jessica Fintzen's 2012 fall tutorial at Harvard
on reflection groups and Coxeter groups. The syllabus for the tutorial states that
"finite Coxeter groups correspond precisely to finite reflection groups." This statement
is based on Fintzen's definition of "reflection group"—
"Reflection groups are— as their name indicates—
groups generated by reflections across
hyperplanes of R^{n} which contain the origin."
For some background, see William Kantor's 1981 paper "Generation of Linear Groups"
(quoted at the finitegeometry.org page on the simple order168 counterexample).
Kantor discusses Mitchell's work in some detail, but does not mention the
simple order168 group explicitly.
Today's New York Lottery numbers:
Continuing the Serious Hardy Apology sequence,
here is a reference to volume number 231 in the
Springer Graduate Texts in Mathematics series—
For some less serious work, see posts on 4403 (4/4/03)
as well as posts numbered 550 and 764.
CHAPTER V
"This is an account of the discrete groups generated by reflections…."
— Regular Polytopes , by H.S.M. Coxeter (unabridged and corrected 1973 Dover reprint of the 1963 Macmillan second edition)
"In this article, we begin a theory linking hyperplane arrangements and invariant forms for reflection groups over arbitrary fields…. Let V be an ndimensional vector space over a field F, and let G ≤ Gl_{n }(F) be a finite group…. An element of finite order in Gl(V ) is a reflection if its fixed point space in V is a hyperplane, called the reflecting hyperplane. There are two types of reflections: the diagonalizable reflections in Gl(V ) have a single nonidentity eigenvalue which is a root of unity; the nondiagonalizable reflections in Gl(V ) are called transvections and have determinant 1 (note that they can only occur if the characteristic of F is positive)…. A reflection group is a finite group G generated by reflections."
— Julia Hartmann and Anne V. Shepler, "Reflection Groups and Differential Forms," Mathematical Research Letters , Vol. 14, No. 6 (Nov. 2007), pp. 955971
"… the class of reﬂections is larger in some sense over an arbitrary ﬁeld than over a characteristic zero ﬁeld. The reﬂections in Gl(V ) not only include diagonalizable reﬂections (with a single nonidentity eigenvalue), but also transvections, reﬂections with determinant 1 which can not be diagonalized. The transvections in Gl(V ) prevent one from developing a theory of reﬂection groups mirroring that for Coxeter groups or complex reﬂection groups."
— Julia Hartmann and Anne V. Shepler, "Jacobians of Reflection Groups," Transactions of the American Mathematical Society , Vol. 360, No. 1 (2008), pp. 123133 (Pdf available at CiteSeer.)
See also A Simple Reflection Group of Order 168 and this morning's Savage Logic.
Rigor
“317 is a prime, not because we think so,
or because our minds are shaped in one way
rather than another, but because it is so,
because mathematical reality is built that way.”
– G. H. Hardy,
A Mathematician’s Apology
The above photo is taken from
a post in this journal dated
March 10, 2010.
This was, as the Pope might say,
the dies natalis of a master gameplayer–
New York Times, March 16, 2010–
Tim Holland, Backgammon Master, Tim Holland, who was widely considered the world’s greatest backgammon player during that ancient board game’s modern heyday, in the 1960s and ’70s, died on March 10 at his home in West Palm Beach, Fla. He was 79. <<more>> 
In Holland's honor, a post
from Columbus Day, 2004—
Tuesday October 12, 200411:11 PM Time and Chance
Today’s winning lottery numbers

A quote from Holland on backgammon–
"It’s the luck factor that seduces everyone
into believing that they are good,
that they can actually win,
but that’s just wishful thinking."
For those who are, like G.H. Hardy,
suspicious of wishful thinking,
here is a quote and a picture from
Holland's ordinary birthday, March 3—
"The die is cast." — Caesar
Jeremy Gray, Plato's Ghost: The Modernist Transformation of Mathematics, Princeton, 2008–
"Here, modernism is defined as an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated— indeed, anxious— rather than a naïve relationship with the daytoday world, which is the de facto view of a coherent group of people, such as a professional or disciplinebased group that has a high sense of the seriousness and value of what it is trying to achieve. This brisk definition…."
Brisk? Consider Caesar's "The die is cast," Gray in "Solomon's Cube," and yesterday's post—
This is the group of "8 rigid motions
generated by reflections in midplanes"
of Solomon's Cube.
Related material:
"… the action of G_{168} in its alternative guise as SL(3; Z/2Z) is also now apparent. This version of G_{168} was presented by Weber in [1896, p. 539],* where he attributed it to Kronecker."
— Jeremy Gray, "From the History of a Simple Group," in The Eightfold Way, MSRI Publications, 1998
Here MSRI, an acronym for Mathematical Sciences Research Institute, is pronounced "Misery." See Stephen King, K.C. Cole, and Heinrich Weber.
*H. Weber, Lehrbuch der Algebra, Vieweg, Braunschweig, 1896. Reprinted by Chelsea, New York, 1961.
From the conclusion of Weyl's Symmetry —
One example of Weyl's "structureendowed entity" is a partition of a sixelement set into three disjoint twoelement sets– for instance, the partition of the six faces of a cube into three pairs of opposite faces.
The automorphism group of this facespartition contains an order8 subgroup that is isomorphic to the abstract group C_{2}×C_{2}×C_{2} of order eight–
The action of Klein's simple group of order 168 on the Cayley diagram of C_{2}×C_{2}×C_{2} in yesterday's post furnishes an example of Weyl's statement that
"… one may ask with respect to a given abstract group: What is the group of its automorphisms…?"
The Sept. 8 entry on nonEuclidean* blocks ended with the phrase “Go figure.” This suggested a MAGMA calculation that demonstrates how Klein’s simple group of order 168 (cf. Jeremy Gray in The Eightfold Way) can be visualized as generated by reflections in a finite geometry.
* i.e., other than Euclidean. The phrase “nonEuclidean” is usually applied to only some of the geometries that are not Euclidean. The geometry illustrated by the blocks in question is not Euclidean, but is also, in the jargon used by most mathematicians, not “nonEuclidean.”
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