Geometric Representation Theory (Lecture 8)
Posted by John Baez
This week in the Geometric Representation Theory seminar we take another of my favorite things — Pascal’s triangle — and wave two magic wands over it: the wand of categorification, and the wand of $q$deformation! When we do this, the humble binomial coefficient
$\binom{n}{k}$
magically transforms into the Grassmannian
$\binom{n}{k}_F$
namely the set of all $k$dimensional subspaces of an $n$dimensional vector space over the field $F$. And, the famous recursive formula for the binomial coefficients:
$\binom{n}{k} \; = \; \binom{n1}{k} \;+ \; \binom{n1}{k1}$
mutates into an interesting fact about Grassmannians… but one we’ll only understand fully when we bring Hecke operators into the game.
I wrote about some of this back in week188 of This Week’s Finds. But, it’s much nicer to see it as part of the grand program we’re engaged in now: systematically categorifying and $q$deforming huge tracts of mathematics with the help of geometric representation theory.
By the way: has anybody ever plotted the ‘$q$deformed Gaussian’ you’d get by graphing the $q$binomial coefficients in the $n$th row of the $q$deformed Pascal’s triangle and then taking a suitably rescaled limit as $n \to \infty$? I’d like to see it, but I’m too busy right now to fire up Mathematica and plot it myself. And surely someone has already done this.

Lecture 8 (Oct. 23)  John Baez on the qdeformed Pascal’s triangle.
Categorifying and qdeforming the recursion relation for binomial
coefficients. If
$\binom{n}{k}_F$
stands for the set of $k$dimensional
subspaces of the vector space $F^n$, we have:
$\binom{n}{k}_F \; \cong \;
\binom{n1}{k}_F \; + \; F^{nk} \times \binom{n1}{k1}_F$
so in particular, taking $F$ to be the field with $q$ elements, we
obtain this relation for $q$binomial coefficients:
$\binom{n}{k}_q \; = \;
\binom{n1}{k}_q \;+ \; q^{nk} \binom{n1}{k1}_q$
Using this to compute the $q$deformed Pascal’s triangle.
Symmetries of the $q$deformed Pascal’s triangle. Why the
binomial coefficient
$\binom{n}{k}$
is the number of combed
Young diagrams with ≤ $k$ columns and ≤ $nk$ rows.
Why the $q$binomial coefficient
$\binom{n}{k}_q$
is the
sum over such Young diagrams $D$ of
$q^{# \; of \; boxes \; of \; D}$
Why each term in this sum corresponds to a specific Bruhat
class in the Grassmannian of $k$dimensional subspaces of $F^n$.
The relation between Young diagrams and matrices in
reduced rowechelon form.
 Supplementary reading: John Baez, The Quantum Pascal’s Triangle

Streaming
video in QuickTime format; the URL is
http://mainstream.ucr.edu/baez_10_23_stream.mov  Downloadable video
 Lecture notes by Alex Hoffnung
Re: Geometric Representation Theory (Lecture 8)
I always wanted to understand what those free probability people were on about when they say something about the Wigner semicircular distribution being a $q = 0$ deformation of Gaussian distributions. E.g., section 2.6 of this.