Posts In Category groups
Seating the first few thousand Knights
on February 3, 2010 by lieven in games, groups, Comments (0)
The odd Knights of the round table-problem asks for a specific one-to-one correspondence between two realizations of ‘the’ algebraic closure of the field of two elements.
The first identifies the multiplicative group of its non-zero elements with the group of all odd complex roots of unity, under complex multiplication. The addition on is then [...]
The odd knights of the round table
on January 28, 2010 by lieven in games, geometry, groups, numbers, Comments (0)
Here’s a tiny problem illustrating our limited knowledge of finite fields : “Imagine an infinite queue of Knights , waiting to be seated at the unit-circular table. The master of ceremony (that is, you) must give Knights and a place at an odd root of unity, say and , such that the [...]
Olivier Messiaen & Mathieu 12
on December 31, 2009 by lieven in Bourbaki, general, groups, Comments (5)
To mark the end of 2009 and 6 years of blogging, two musical compositions with a mathematical touch to them. I wish you all a better 2010!
Remember from last time that we identified Olivier Messiaen as the ‘Monsieur Modulo’ playing the musical organ at the Bourbaki wedding. This was based on the fact that his [...]
E(8) from moonshine groups
on May 15, 2009 by lieven in groups, Comments (1)
Time to wrap up this series on John Duncan’s paper Arithmetic groups and the affine E8 Dynkin diagram in which he gives a realization of the extended E(8)-Dynkin diagram (together with its isotropic root vector) from the moonshine groups, compatible with McKay’s E(8)-observation.
In the previous post we have described all 171 moonshine groups using Conway’s [...]
looking for the moonshine picture
on May 11, 2009 by lieven in groups, Comments (4)
We have seen that Conway’s big picture helps us to determine all arithmetic subgroups of commensurable with the modular group , including all groups of monstrous moonshine.
As there are exactly 171 such moonshine groups, they are determined by a finite subgraph of Conway’s picture and we call the minimal such subgraph the moonshine [...]
Conway’s big picture
on May 2, 2009 by lieven in groups, Comments (0)
Expanding (and partially explaining) the original moonshine observation of McKay and Thompson, John Conway and Simon Norton formulated monstrous moonshine :
To every cyclic subgroup of the Monster is associated a function
with and all coefficients are characters at of a representation of . These representations are the homogeneous components of [...]
the monster graph and McKay’s observation
on April 22, 2009 by lieven in groups, Comments (7)
While the verdict on a neolithic Scottish icosahedron is still open, let us recall Kostant’s group-theoretic construction of the icosahedron from its rotation-symmetry group .
The alternating group has two conjugacy classes of order 5 elements, both consisting of exactly 12 elements. Fix one of these conjugacy classes, say and construct a graph with [...]
the scottish solids hoax
on March 25, 2009 by lieven in groups, Comments (13)
A truly good math-story gets spread rather than scrutinized. And a good story it was : more than a millenium before Plato, the Neolithic Scottish Math Society classified the five regular solids : tetrahedron, cube, octahedron, dodecahedron and icosahedron. And, we had solid evidence to support this claim : the NSMS mass-produced stone replicas of [...]
Geometry of the Okubo algebra
on March 14, 2009 by lieven in geometry, groups, Comments (3)
Last week, Melanie Raczek gave a talk entitled ‘Cubic forms and Okubo product’ in our Artseminar, based on her paper On ternary cubic forms that determine central simple algebras of degree 3.
I had never heard of this strange non-associative product on 8-dimensional space, but I guess it is an instance of synchronicity that [...]
can -oids save group-theory 101?
on February 15, 2009 by lieven in groups, rants, Comments (2)
Can one base a group-theory 101 course on the notion of groupoids?
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