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Character Theory Basics

This post illustrates some of SageMath’s character theory functionality, as well as some basic results about characters of finite groups.

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Animated GIFs

I really should be posting about character theory, but I got distracted making some aesthetic changes to this blog (new icon and favicon!) and creating animations like this:


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The Group Ring and the Regular Representation

In the previous post, we saw how to decompose a given group representation into irreducibles. But we still don’t know much about the irreducible representations of a (finite) group. What do they look like? How many are there? Infinitely many?

In this post, we’ll construct the group ring of a group. Treating this as a vector space, we get the regular representation, which turns out to contain all the irreducible representations of $G$!

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