[seqfan] Re: Generating the symmetric group with minimal products of involutions
Brad Klee
bradklee at gmail.com
Wed Jan 4 19:30:58 CET 2017
Hi,
Aha! All row lengths equal 3 for n g.t. 2. I thought some more about this,
and it does reduce to the most elementary fact, that a product of
reflections is a rotation. Drawing a few n-point polygons, it's easy to see
why this is so.
There are just so many facts in group theory, and who can remember them
all? But as long as we are on the subject, here's another one ( my personal
favorite ):
e + v + f = |G| + 2
for tetrahedral, icosahedral, and cubic symmetry. Furthermore, on the same
symmetries:
e = v + f - 2 = |G|/2
from which follows Euler's equation
(e + v + f) - 2 * e = v + f - e = 2
The first and second lesser-known relations are particularly interesting in
the context of representation theory. The first suggests the existence of
an "almost regular" (|G|+2)-dimensional representation, while the second
suggests that the irreducible decomposition of the (e)-rep can be obtained
by some transformation from the irreducible decomposition of the (v+f)-rep.
* * * Chalenge. For each polyhedron symmetry: Accomplish the (-1)
transformation from (e)-rep to (v+f)-rep using the Hamiltonian circuit
around the adjacency-graph. Have you ever seen that before?
I found this out recently, but it seems like A. Ceulemans was already onto
a similar idea in 1999:
https://lirias.kuleuven.be/bitstream/123456789/34201/1/article.pdf
Euler's formula generalizes a few different ways, so it's a possibility
that the idea of an "almost regular" representation is not just limited to
the polyhedron groups. Possibly grounds for discovery, including more
sequences.
Unfortunately, these finite groups are not much use for time-transforming
wavepackets . . .
Cheers,
Brad
On Wed, Jan 4, 2017 at 7:36 AM, Richard J. Mathar <mathar at mpia-hd.mpg.de>
wrote:
> In response to http://list.seqfan.eu/pipermail/seqfan/2016-
> December/017171.html
> I figured out that each element of order >=3 of the symmetric group S_n
> can be written
> as a product of 2 involutions. I added an associated comment to A066052
> showing
> that permutations of order >=3 can be written as a product of 2
> involutions. I did
> not submit the associated array mentioned in http://list.seqfan.eu/
> pipermail/seqfan/2016-December/017168.html
> because that would be trivially limited to
> T(n,0)=1
> T(n,1)=A001189(n), n>=2.
> T(n,2)=A066052(n), n>=3.
> with row sums A000142(n).
>
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> Seqfan Mailing list - http://list.seqfan.eu/
>
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