Is there an existing issue for this?

• I have searched the existing issues for a bug report that matches the one I want to file, without success.

Problem Description

The Burnside ring of a group encodes the permutation representations of the group. This could be implemented as a parent in Sage.

Proposed Solution

Implement a parent class for the Burnside ring of a finite group using GAP under the hood. This should also include a method which attempts to express a given representation in the Burnside ring by first checking that all the character values are positive (a necessary but not sufficient condition for a representation to be a permutation representation) and then attempting to express it as an element of the Burnside ring.

Alternatives Considered

One could work with symmetric functions directly or by hand/in an ad hoc manner, but implementing a parent class for the Burnside ring of a group is the way that makes sense in Sage.

Additional Information

No response