Representation Theory in Sage

I’ve written a series of posts about representation theory on my other site:

  1. Representation Theory – Basic Definitions
  2. Direct Sums and Tensor Products
  3. Irreducible and Indecomposable Representations
  4. Decomposing Representations
  5. 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.

High school deconvolution

aurora_smal_animated

One of my favorite formulas is the factorization of 1-x^n:

1 - x^n = (1 - x)(1 + x + x^2 + \dots + x^{n-1}).

This is a fairly ubiquitous formula that most people would have seen in high school, perhaps in various guises. A common special case is (1-x)(1+x) = 1- x^2, which is itself a special case of the “sum of squares” formula (a-b)(a+b) = a^2 - b^2.

The irreducible factors of 1 + x + x^2 + \dots + x^{n-1} are the cyclotomic polynomials which have many interesting links to number theory.

In other areas, substituting something nice for x leads to interesting formulas/results. When x is some prime power p^e (usually written q), the quantity

\frac{1-q^n}{1-q} = 1 + q + \dots + q^{n-1}

is the q-analog of n. The reason for the name is that as we let q \to 1, we get

\lim_{q \to 1} \frac{1-q^n}{1-q} = n.

This observation forms the starting point of the study of combinatorial q-analogs. Toggling between q=1 and q=p^e allows one to view sets as vector spaces over the “field” of 1 element.

However, I won’t be writing about that today. Instead, this post is about a curious connection between that simple formula above and deconvolution.

Continue reading

Subgroup Lattices II – Coloring Vertices

Lattice of subgroups of the Dicyclic Group of order 12
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:

interact_demo

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!