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Holiday Harmonograph
(Guest post from the Annals of Harmonography)When it’s snowing outside (or maybe not),And your feet are cold (or maybe hot),When it’s dark as day (or bright as night),And your heart is heavy (and head...
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Lattice of Subgroups III - Coloring Edges
This post will cover the coloring of edges in the lattice of subgroups of a group. Coloring edges is almost as simple as coloring vertices, so we’ll start with that. Generating small groupsAs we’ve...
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Subgroup Explorer
I’ve written a subgroup lattice generator for all groups of size up to 32. It’s powered by Sage and GAP, and allows you to view the lattice of subgroups or subgroup conjugacy classes of a group from...
View ArticleRepresentation Theory in Sage - Basics
This is the first of a series of posts about working with group representations in Sage.Basic DefinitionsGiven a group $G$, a linear representation of $G$ is a group homomorphism $\rho: G \to...
View ArticleDirect Sums and Tensor Products
In this short post, we will show two ways of combining existing representations to obtain new representations.RecallIn the previous post, we saw two representations of $D_4$: the permutation...
View ArticleIrreducible and Indecomposable Representations
Following up from the questions I asked at the end of the previous post, I’ll define (ir)reducible and (in)decomposable representations, and discuss how we might detect them. Unlike previous posts,...
View ArticleDecomposing Representations
In this post, we’ll implement an algorithm for decomposing representations that Dixon published in 1970.As a motivating example, I’ll use the permutation matrix representation of $D_4$ that we saw in...
View ArticleThe 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...
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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:I’m not putting this in a...
View ArticleCharacter Theory Basics
This post illustrates some of SageMath’s character theory functionality, as well as some basic results about characters of finite groups. Basic Definitions and PropertiesGiven a representation...
View ArticleThe Weyl Algebra and $\mathfrak{sl}_2$
I’ve been away from this blog for quite a while - almost a year, in fact! My excuses are my wedding and the prelims (a.k.a. quals), as well as all the preparation that had to go into them (although, to...
View ArticleNoncommutative Algebras in Sage
In this post, I’ll demonstrate 3 ways to define non-commutative rings in Sage. They’re essentially different ways of expressing the non-commutative relations in the ring:Via g_algebra: define the...
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Distributive Laws
I’ve been participating in the Kan Extension Seminar II, and this week it’s my turn to post about Jon Beck’s “Distributive Laws” at the n-Category Cafe!The post uses lots of string diagrams for monads,...
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