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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 2nd 2019
   • (edited Sep 2nd 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexRichard Williamson asks me to say again what the next smallest step would be in the little coding project on some basic representation theory that we have been talking about earlier. And to say it here on the nForum in case anyone else feels like joining in. Recall, this is about the the experimental maths exploration that we started in _[[schreiber:The image of the Burnside ring in the Representation ring|The image of the Burnside ring in the Representation ring]]_. We have Python code [there](https://arxiv.org/src/1812.09679v1/anc) which reads in a finite group, and spits out the "image of beta" in terms of a list of vectors $\{V_1, V_2, \cdots\}$ that serve as its basis in the vector space of representation characters. The next little coding step, still missing, is to automate the detection of how much this image fails to be all of the integral characters: Make the program 1. read in (from GAP) the list of irreducible integral characters $\{\rho^{int}_1, \rho^{int}_2, \cdots\}$ over the reals, 1. check whether there is a linear iso $Span_{\mathbb{Z}}(\{V_1, V_2, \cdots\}) \simeq Span_{\mathbb{Z}}(\{\rho^{int}_1, \rho^{int}_2, ...\})$ between their span and the image of beta. Yes or no. Once this comparison is fully automated, the goal is to apply it to tons of examples. But the first little step is just to automate this comparison. This is the step which in our examples so far appears after the lines "Hence the cokernel of $\beta$ is...". For instance bottom of [p. 21](https://ncatlab.org/schreiber/files/ImageofBeta190902.pdf#page=21). For starters, we just need "Is it zero or not?"

   Richard Williamson asks me to say again what the next smallest step would be in the little coding project on some basic representation theory that we have been talking about earlier. And to say it here on the nForum in case anyone else feels like joining in.

   Recall, this is about the the experimental maths exploration that we started in The image of the Burnside ring in the Representation ring.

   We have Python code there which reads in a finite group, and spits out the “image of beta” in terms of a list of vectors $\{V_1, V_2, \cdots\}$ that serve as its basis in the vector space of representation characters.

   The next little coding step, still missing, is to automate the detection of how much this image fails to be all of the integral characters:

   Make the program

   1. read in (from GAP) the list of irreducible integral characters $\{\rho^{int}_1, \rho^{int}_2, \cdots\}$ over the reals,

   2. check whether there is a linear iso $Span_{\mathbb{Z}}(\{V_1, V_2, \cdots\}) \simeq Span_{\mathbb{Z}}(\{\rho^{int}_1, \rho^{int}_2, ...\})$ between their span and the image of beta. Yes or no.

   Once this comparison is fully automated, the goal is to apply it to tons of examples. But the first little step is just to automate this comparison.

   This is the step which in our examples so far appears after the lines “Hence the cokernel of $\beta$ is…”. For instance bottom of p. 21. For starters, we just need “Is it zero or not?”