In the last post, we discussed an operation $\newcommand{\box}{\square} \omega$ on skew Littlewood-Richardson Young tableaux with one marked inner corner, defined as a commutator of rectification and shuffling. As a continuation, we’ll now discuss where this operator arises in geometry.

Recall from our post on Schubert calculus that we can use Schubert varieties to answer the question:

Given four lines $\ell_1,\ell_2,\ell_3,\ell_4\in \mathbb{C}\mathbb{P}^3$, how many lines intersect all four of these lines in a point?

In a recent and fantastic collaboration between Jake Levinson and myself, we discovered new links between several different geometric and combinatorial constructions. We’ve weaved them together into a beautiful mathematical story, a story filled with drama and intrigue.

So let’s start in the middle.

Those of you who played with little puzzle toys growing up may remember the “15 puzzle”, a $4\times 4$ grid of squares with 15 physical squares and one square missing. A move consisted of sliding a square into the empty square. The French name for this game is “jeu de taquin”, which translates to “the teasing game”.

*This is a continuation of Part 1 of this series of posts on $q$-analogs.*

Another important area in which $q$-analogs come up is in combinatorics. In this context, $q$ is a formal variable, and the $q$-analog is a generating function in $q$, but viewed in a different light than usual generating functions. We think of the $q$-analog of as ``$q$-counting’’ a set of weighted objects, where the weights are given by powers of $q$.

Hi, I’m Maria and I’m a $q$-analog addict. The theory of $q$-analogs is a little-known gem, and in this series of posts I’ll explain why they’re so awesome and addictive!

So what is a $q$-analog? It is one of those rare mathematical terms whose definition doesn’t really capture what it is about, but let’s start with the definition anyway:

**Definition:** A *$q$-analog* of a statement or expression $P$ is a statement or expression $P_q$, depending on $q$, such that setting $q=1$ in $P_q$ results in $P$.

So, for instance, $2q+3q^2$ is a $q$-analog of $5$, because if we plug in $q=1$ we get $5$.

Education is a difficult task, it really is. Teaching takes a few tries to get the hang of. Writing textbooks is even harder. And math is one of those technical fields in which human error is hard to avoid. So usually, when I see a mistake in a math text, it doesn’t bother me much.

But some things just hurt my soul.

No correspondence between the integers and rationals? Yes there is, Example 2! Yes there is!