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

Holiday Harmonograph

harmonograph_loop

(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!

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!

Cabbages and Kings

We recently tried out The Walrus and the Carpenter.  The food was great, and I found out there’s such a thing as lamb salt, which P describes as “like bacon bits, but made of lamb”. You can read more about our experience here.

However, the time has come to speak of other things.

The Walrus and the Carpenter is one of three restaurants making up the title of the book A Boat, a Whale & a Walrus. The book title reminds me of the palindrome

A man, a plan, a canal – Panama!

Many people have successfully extended this palindrome with the addition of other objects such as a cat, a yak, a yam… My favourite is this one by Jon Agee:

A man, a plan, a cat, a bar, a cap, a mall, a ball, a map, a car, a bat, a canal: Panama.

Unfortunately, until the word “urlawaelahwataoba” enters the English language, we’re not going to have any palindromes starting with “a boat, a whale, a walrus”. It also doesn’t help that the fourth in this suite of restaurants is called “The Barnacle”. Have they no sympathy for palindrome makers or people with dyslexia?

To end off this random (or is it?) ramble, here’s one of mine and Mr. Richland’s Favorite Song[s]: