[seqfan] Quantum Rotation Correlation Sequences

Brad Klee bradklee at gmail.com
Sat Nov 29 02:13:48 CET 2014


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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