I’ve written a series of posts about representation theory on my other site:
Representation Theory – Basic Definitions
Direct Sums and Tensor Products
Irreducible and Indecomposable Representations
The Group Ring and the Regular Representation
The emphasis is on illustrating all the above things in Sage. It’s pretty elementary, and uses linear algebra more than anything else. I haven’t touched character theory yet; that will be covered in the next series of posts.
I’ve written an in-browser Sage interact script that generates the subgroup lattice for all groups of size up to 32. Click the image to access it!
(Click picture for an online harmonograph generator)
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 is light),
What should you do (what should you say)
To make it all right (to make it okay)?
Just pick up a pen (a pencil will do),
Set up a swing (or
three, or two),
And while the world spins (or comes to a still),
In your own little room (or on top of a hill),
Let your pendulum sway (in its time, in its way),
And watch as the pen draws your worries away!
Lattice of subgroups of the Dicyclic Group of order 12
Here’s my second subgroup lattice post:
Subgroup Lattices with Sage – Coloring Vertices. It shows how to color a poset according to properties that you’d like to highlight.
At the bottom of that post, you’ll also find a nice interactive section where you can play around with various groups and their subgroups. It looks like this:
These posts are pretty time-consuming to write, so the next post might be quite a while later. Have fun playing with the interactive subgroup demo!
Lattice of subgroups of the Dihedral Group of order 8.
I’ve started a new blog for demonstrating Sage code that can be run directly in the browser. I might write about that process later.
For now, here’s the first of a series of posts about creating and experimenting with the lattice of subgroups in Sage:
Lattice of Subgroups in Sage.