Subgroup Explorer

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!

High school deconvolution

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.