Burnside ring in sage?

Martin R

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Jul 19, 2021, 12:36:30 PM7/19/21

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I need a (possibly quick and dirty) implementation of the Burnside ring of group actions (see https://en.wikipedia.org/wiki/Burnside_ring, mostly for the symmetric group).

Has anyone done this already? I guess one could wrap gap, but I'd be very grateful if I wouldn't have to do it myself :-)

Best,

Martin

Vinay Wagh

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Jul 24, 2021, 9:10:01 AM7/24/21

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Dear Martin,

I dont have the sage implementation, but I have a "working" implementation of Burnside Rings in GAP using homalg. The algorithms implemented are based on the discussion from:

M. Kreuzer, D. P. Patil. Computational aspects of Burnside rings, part I: the ring structure , Beitr Algebra Geom (2017) 58, 427 – 452.

Do let me know if this would be of any use to you.

-- VInay

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Dima Pasechnik

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Jul 24, 2021, 10:29:51 AM7/24/21

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Hi Vinay,
is your package publicly available?
If not, please consider making it public, or even better, one of GAP packages on
https://gap-packages.github.io/

Dima

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Vinay Wagh

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Jul 25, 2021, 7:06:26 AM7/25/21

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Dear Dima,

The code is still in the "preliminary" stage! The status of "package" is too far away :-D.

As the code was written mainly for some demonstration to the students, it is very simple and not at all optimised. As I have said earlier, this code uses the algorithms from the article quoted above. Attaching the code.

-- VInay

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burnside.g

Martin R

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Jul 28, 2021, 7:55:03 AM7/28/21

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Dear Vinay!

Thank you for sharing your code! I managed to install it, but I don't understand how to use it yet. Could you show an easy example for how I can create an element of the ring? E.g., how can I create a group action of the symmetric group and transform it into an element of the ring?