The n-Category Café

Geometric Representation Theory (Lecture 9)

Posted by John Baez

Pick some type of structure you can put on an $n$-element set. Then the group $n!$ — the group of all permutations of this $n$-element set — acts on the set $X$ of all structures of this type. So, we get an action of $n!$ on $X$, and thus a representation of $n!$ on the vector space $\mathbb{C}^X$.

My favorite type of structure on an $n$-element set is called a ‘$D$-flag’. Remember this? $D$ can be any $n$-box Young diagram — that is, any list of natural numbers $n_1 \ge \cdots \ge n_k \gt 0$ that add up to $n$. A $D$-flag on an $n$-element set is a way of partitioning it into subsets of sizes $n_1, \dots , n_k$.

If $X$ is the set of $D$-flags on our favorite $n$-element set, then by what I’ve said, we get a representation of $n!$ on $\mathbb{C}^X$.

So, any $n$-box Young diagram gives a representation of $n!$. These are called flag representations. They’re typically reducible — big and fat. But, just like they say every fat person has a skinny person inside, trying to get out, each of these fat representations has an irreducible representation inside, trying to get out.

This time in the Geometric Representation Theory seminar, James Dolan shows how to get these skinny representations out. The trick is to chop away at the fat ones using Hecke operators. This amounts to a categorified version of the Gram–Schmidt process, where we keep chopping off the components of a vector that point along the previous vectors in our list, until we’re left with orthogonal vectors.

By this method, we get all the irreducible representations of $n!$ — one for each $n$-box Young diagram!

Near the end of this class, Jim starts discussing Hecke operators between flag representations in more detail. He describes them using certain matrices which will eventually become known as ‘crackpot matrices’… for reasons that will be clear someday.

• Lecture 9 (Oct. 25) - James Dolan on Hecke operators for the groups $n!$. Each $n$-box Young diagram $D$ gives an action of $n!$ on the set of $D$-flags on the $n$-element set. These actions give permutation representations of $n!$ called ‘flag representations’. Flag representations are usually reducible, but we can extract a complete set of irreducible representations using Hecke operators, via a categorified version of Gram–Schmidt orthonormalization. So, we obtain one irreducible representation of $n!$ for each $n$-box Young diagram. The example of 4!, continued.
  
  Given an $n$-box Young diagram $D$ with $d$ rows and an $n$-box Young diagram $E$ with $e$ rows, we can use “crackpot matrices” — $d \times e$ matrices of natural numbers with specified row and column sums — to give explicit descriptions of all the Hecke operators from one flag representation to another.
  
  
  • Streaming video in QuickTime format; the URL is
    http://mainstream.ucr.edu/baez_10_25_stream.mov
  • Downloadable video
  • Lecture notes by Alex Hoffnung
  • Lecture notes by Apoorva Khare