#10305: Add rings for the center of the symmetric group algebras
-------------------------------------+-------------------------------------
Reporter: mguaypaq | Owner: mguaypaq
Type: enhancement | Status: needs_work
Priority: minor | Milestone: sage-5.12
Component: combinatorics | Resolution:
Keywords: combinatorics, | Merged in:
rings, symmetric functions | Reviewers: Travis Scrimshaw,
Authors: Mathieu Guay- | Mathieu Guay-Paquet
Paquet, Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | Stopgaps:
Dependencies: |
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Comment (by mguaypaq):
Hi Travis,
Replying to [comment:11 tscrim]:
> Hey Matthieu,
>
> Here's everything that I could do with the patch. I've reworked it to
use the same framework as the symmetric functions, allowed the SGA center
for a fixed n, and added in coercions to the SGA when appropriate.
Great! I'll have a look as you suggest.
> - Could you put in some references for the Farahat-Higman/partial class
algebras?
Sure. In the meantime, here's the bibtex from mathscinet for the original
paper by Farahat and Higman:
{{{
@article {MR0103935,
AUTHOR = {Farahat, H. K. and Higman, G.},
TITLE = {The centres of symmetric group rings},
JOURNAL = {Proc. Roy. Soc. London Ser. A},
FJOURNAL = {Proceedings of the Royal Society. London. Series A.
Mathematical, Physical and Engineering Sciences},
VOLUME = {250},
YEAR = {1959},
PAGES = {212--221},
ISSN = {0962-8444},
MRCLASS = {20.00},
MRNUMBER = {0103935 (21 \#2697)},
MRREVIEWER = {T. Nakayama},
}
}}}
The partial class algebra is used and described in a recent paper by
Féray, but originally defined by Ivanov and Kerov. Both articles are
available in English on arXiv, and the mathscinet bibtex is:
{{{
@article {MR2854640,
AUTHOR = {F{\'e}ray, Valentin},
TITLE = {Partial {J}ucys-{M}urphy elements and star factorizations},
JOURNAL = {European J. Combin.},
FJOURNAL = {European Journal of Combinatorics},
VOLUME = {33},
YEAR = {2012},
NUMBER = {2},
PAGES = {189--198},
ISSN = {0195-6698},
MRCLASS = {05A05 (05A15)},
MRNUMBER = {2854640 (2012k:05013)},
DOI = {10.1016/j.ejc.2011.09.035},
URL = {http://dx.doi.org/10.1016/j.ejc.2011.09.035},
}
@article {MR1708561,
AUTHOR = {Ivanov, V. and Kerov, S.},
TITLE = {The algebra of conjugacy classes in symmetric groups, and
partial permutations},
JOURNAL = {Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov.
(POMI)},
FJOURNAL = {Rossi\u\i skaya Akademiya Nauk. Sankt-Peterburgskoe
Otdelenie.
Matematicheski\u\i\ Institut im. V. A. Steklova. Zapiski
Nauchnykh Seminarov (POMI)},
VOLUME = {256},
YEAR = {1999},
NUMBER = {Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 3},
PAGES = {95--120, 265},
ISSN = {0373-2703},
MRCLASS = {20B30 (05A05)},
MRNUMBER = {1708561 (2000g:20010)},
DOI = {10.1023/A:1012473607966},
URL = {http://dx.doi.org/10.1023/A:1012473607966},
}
}}}
> - From what I could find, it seems like the Farahat-Higman algebra is
actually just the parts for a fixed `n`, not allowing `n` to vary. Thus we
would need to take the terms just corresponding to partitions `n` in the
larger product. Is this correct?
In the current version, the ground field `R` given to the constructor
`FarahatHigmanAlgebra` is first transformed into a polynomial ring of the
form `R['t']`, and then the `n` is the variable `t` from this new
polynomial ring. A better design would be to pass the ring `R['t']`
directly to the constructor, and let the parameter be chosen arbitrarily
from this ring (not just the generator `t`). This is one of the points
that made me think a better design is in order, because it creates some
problems down the line.
Regarding the bound on partitions based on `n`: strictly speaking, in the
FH algebra, the basis includes *all* partitions, not just those where
`size + length <= n`. If you want to get the center of the symmetric group
algebra for a fixed integer `n`, say `n = 4`, then you need to do two
things:
1. Start from the Farahat-Higman algebra over `R['t']` with parameter `t`,
and specialize `t` to `4`.
2. Map every basis element where the partition has `size + length > 4` to
zero.
The second step is where the bound on partitions comes up, but at that
point it's no longer the Farahat-Higman algebra itself, but rather a
homomorphic image. Does that make sense?
> - On line 426 of `farahat_higman.py`, you had originally left that
sentence as incomplete and I couldn't figure out how to finish it. Could
you finish writing that paragraph?
Hah, what I just wrote above is what I meant to write in the
documentation. I think it's a very muddled and unclear explanation,
though, so I should improve it. Let me know if it makes no/some/complete
sense.
> (Due to the amount of changes by just reorganizing code, I felt it was
better to post a new version of the patch than a review patch.)
Good call.
> For reviewing this patch, we'll do a cross-review where I review your
changes and you review mine. Sound good?
That works.
Cheers,
Mathieu
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Ticket URL: <http://trac.sagemath.org/ticket/10305#comment:12>
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