M Theory Lesson 136

Having blogged a bit already about Geometric Representation Theory, I was going to leave it to people to watch this lecture series for themselves. But lecture 8 is way too cool to ignore!

Here we see how paths in a $q$ deformed Pascal triangle can be counted. First note that in M Theory we usually draw the triangle as a quadrant in a plane, on which we consider paths for the noncommutative Fourier transform. A step to the right picks up multiples of a power of $q$, whereas a step up simply multiplies the entry below by 1. In this way one obtains polynomials in $q$ with integer coefficients. Even though $q$ starts out labelling the number of elements in a finite field, we recall counting trees in a similar fashion, but ending up with complex roots of unity. Physicists may smell a sneaky Wick rotation in the shrubbery.