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 -deformation! When we do this, the humble binomial coefficient
magically transforms into the Grassmannian
namely the set of all -dimensional subspaces of an -dimensional vector space over the field . And, the famous recursive formula for the binomial coefficients:
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 -deforming huge tracts of mathematics with the help of geometric representation theory.
By the way: has anybody ever plotted the ‘-deformed Gaussian’ you’d get by graphing the -binomial coefficients in the th row of the -deformed Pascal’s triangle and then taking a suitably rescaled limit as ? 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 q-deformed Pascal’s triangle.
Categorifying and q-deforming the recursion relation for binomial
coefficients. If
stands for the set of -dimensional
subspaces of the vector space , we have:
so in particular, taking to be the field with elements, we
obtain this relation for -binomial coefficients:
Using this to compute the -deformed Pascal’s triangle.
Symmetries of the -deformed Pascal’s triangle. Why the
binomial coefficient
is the number of combed
Young diagrams with ≤ columns and ≤ rows.
Why the -binomial coefficient
is the
sum over such Young diagrams of
Why each term in this sum corresponds to a specific Bruhat
class in the Grassmannian of -dimensional subspaces of .
The relation between Young diagrams and matrices in
reduced row-echelon 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 deformation of Gaussian distributions. E.g., section 2.6 of this.