Geometric Representation Theory (Lecture 6)
Posted by John Baez
Where would a wizard be without his magic wands?
In mathematics, a ‘magic wand’ is any systematic process that you can apply to big chunks of interesting mathematics and get new, more interesting mathematics. Or — more magical still — it’s a mysterious bunch of tricks that feel like they’d be part of a systematic process if only we understood them better.
What are some magic wands? One of the most famous was stolen from physicists: it’s called quantization. Muttering one of several cryptic spells, you can wave this wand over any mathematical concept related to classical mechanics, and hope that — POOF! — it suddenly transforms into an analogous concept related to quantum mechanics. We’ve had huge success with this over the last century, but it’s still poorly understood.
Another magic wand is categorification: replacing any number by a set with that number of elements, replacing any set by a category whose set of isomorphism classes it is, and so on. You could almost say this blog is a shrine to categorification. It too, is still poorly understood. Perhaps when a magic wand’s powers become fully understood, it ceases to count as ‘magic’!
Yet another magic wand is $q$deformation — closely related to quantization but not the same. It’s a way of modifying mathematical entities that depends on a parameter $q$. Sometimes this parameter has the physical meaning of $exp(i\hbar)$… but sometimes it’s better to think of it as a power of a prime number! In fact, $q$deformation was discovered by Gauss long before the quantum was a twinkle in Planck’s eye.
When you have two magic wands at your disposal, you can ask if they commute. First wave one, then the other. First wave the other, then the one. Does the same magic occur? Or at least isomorphic magics?
In lecture 6 of the Geometric Representation Theory seminar, I wave two magic wands — categorification and $q$deformation — at a humble mathematical entity: the binomial coefficient. It seems they commute. But, puzzles abound!

Lecture 6 (Oct. 16) 
John Baez on categorifying and $q$deforming the theory of multinomial
coefficients.
A surprising fact: $q$multinomial coefficients are actually polynomials
in $q$ with natural number coefficients. It suffices to prove this
for $q$binomial coefficients, since any $q$multinomial coefficient is a product of
$q$binomial coefficients. Since a $q$binomial coefficient is the number of points in a Grassmanian over $F_q$ (the field with $q$ elements), it’s enough to decompose this
Grassmannian into ‘Bruhat classes’, and show that each of these
is isomorphic (as a set) to $F_q^k$ for some k.
For this, we show that each Bruhat class corresponds to a set of
matrices in
reduced
row echelon form. Example: the Young diagram
Another surprising fact: any $q$multinomial is actually a ‘palindromic’ polynomial in $q$. The closure of a Bruhat class is called a Schubert cell, and this palindromic property follows from Poincaré duality, since Schubert cells are a basis for the cohomology of the Grassmannian over $\mathbb{C}$.

Streaming
video in QuickTime format; the URL is
http://mainstream.ucr.edu/baez_10_16_stream.mov  Downloadable video
 Lecture notes by Alex Hoffnung
 Lecture notes by Chris Rogers

Streaming
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Re: Geometric Representation Theory (Lecture 6)
Have you ever thought of using magic wands from electrical engineering [EE]?
Paul AM Dirac had no degree in physics, but a BS in EE and PhD in mathematics. This allowed him to do things that were considered not to be rigorous.
http://nobelprize.org/nobel_prizes/physics/laureates/1933/diracbio.html
Richard Bellman developed dynamic programming or optimization.
http://en.wikipedia.org/wiki/Dynamic_programming
When a colleague remarked that this was not rigorous, Bellman reportedly responded, “Of course not. It’s not even precise. A good principle should guide the intuition.”