[seqfan] Re: Quantum Rotation Correlation Sequences

[Neil Sloane]

There is a lot about Theodor Molien in Charlie Curtis's
book "Pioneers of Representation Theory".

Molien's theorem gives a generating function for the
dimensions of the space of invariants of degree n, for most of the finite
groups one encounters in daily life.

For an introduction, see my paper  Error-Correcting Codes and Invariant
Theory: New Applications of a Nineteenth-Century Technique, Amer. Math.
Monthly, 84 (1977),
pp. 82-107, on my home page (see below), publications, item #46.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Fri, Nov 28, 2014 at 8:13 PM, Brad Klee <bradklee at gmail.com> wrote:

> Hi Seqfans,
>
> In order to discuss new interpretations of old sequences, and possibly a
> few new noteworthy sequences, I would like to draw your attention to
> Chapter 5 of Bill Harter's awesome book, which is available for free
> on-line.
>
> "Principles of Symmetry, Dynamics, and Spectroscopy"
> http://www.uark.edu/ua/modphys/markup/PSDS_Info.html
>
> In chapter 5, Harter shows how to construct correlation tables relating
> continuous symmetry groups and discrete symmetry groups. The continuous
> rotational symmetry groups are infinite-dimensional, and they also have an
> infinite number of representations. The discrete groups are finite
> dimensional, and have a finite number of representations. The correlation
> tables then relate a finite number of representations to an infinite number
> of representations. This sets up a finite number of integer sequences as
> the columns of the correlation table.
>
> Construction of the correlation table for O3 ( 3-dimensional orthogonal
> group ) and octahedral group is described on page 382 - 386 ( 69 - 73 pdf
> ). You might also read around to find out more about the physical
> interpretation of these sequences.
>
> In the attached notebook, I have computer 500 terms for both octahedral and
> icosahedral, and compared these sequences to OEIS to find a few interesting
> results. The notebook and CSV files are attached.
>
> The following findings are correct up to 500 entries:
>
> 1. The "A" series for both icosahedral and octahedral are somewhat related
> as the Molien series of the binary group.
>
> Octahedral: https://oeis.org/A008647
> Icosahedral: https://oeis.org/A008651
>
> 1a. But who in the history of mathematics is Molien!?!?!
> Emmy Noether writes:
> "*The most general theorems about algebras go back to Molien.*"
> http://www-history.mcs.st-andrews.ac.uk/Biographies/Molin.html
>
> 2. One sequence occurs for both icosahedral group ( "H" Representation )
> and octahedral group ( "E" Representation ).
>
> https://oeis.org/A008615
>
> 3. Other series that do not occur in the OEIS appear to have periodic first
> differences, which is not all that surprising. As the notebook shows, the
> periods seem to be:
>
> 12 for octahedral A2
> 10 for icosahedral T1 & T2
> 15 for icosahedral G
>
> Does this mean there is automatically a generating polynomial? Who knows...
>
> If there is any interest to get rotational correlations into OEIS, this
> code could be extended to other finite symmetry groups with relative ease.
> Maybe it would be worthwhile to look at tetrahedral ( I think that is AKA
> alternating group 4 ) ?
>
> Please send questions comments.
>
> Thanks,
>
> Brad
>
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