The n-Category Café

Geometric Representation Theory (Lecture 16)

Posted by John Baez

Sick of Christmas shopping? Tired of the crowded malls, the rush, the commercialism? Take a break and learn some math! It’s free, it’s fun, and it’s good for you.

This time in the Geometric Representation Theory seminar, I start by quickly fixing the mistake I made when attempting to state the ‘Fundamental Theorem of Hecke Operators’ in lecture 14.

But then I begin the harder and more interesting job of trying to explain what’s really going on! Namely, ‘groupoidification’.

This is what our seminar is really about. We were just taking our time getting there, building up some of the key examples we’ll be using.

The first step is to see that groups acting on sets give groupoids. This nicely fits the idea of a groupoid as a ‘set with built-in symmetries’.

• Lecture 16 (Nov. 20) - John Baez on Hecke operators and groupoidification. Correcting the mistake from last time: a quick fix is easy, but the real solution requires ‘groupoidification’. For starters, this means replacing a group $G$ acting on a set $S$ by a groupoid $S//G$, the ‘weak quotient’ or ‘action groupoid’. Object of $S//G$ are just elements of $S$, while morphisms are of the form $(g,s): s \to g s$.

  Examples: suppose $D$ is an $n$-box uncombed Young diagram. Then the group $G = n!$ acts on the set $S$ of $D$-flags on the $n$-element set, and $S//G$ is equivalent to the groupoid of ‘D-flagged sets’. Similarly, for any field $F$, the group $G = GL(n,F)$ acts on the set $S$ of $D$-flags on the vector space $F^n$, and $S//G$ is equivalent to the groupoid of ‘$D$-flagged vector spaces’.

  • Streaming video in QuickTime format; the URL is
    http://mainstream.ucr.edu/baez_11_20_stream.mov
  • Downloadable video
  • Lecture notes by Alex Hoffnung
  • Lecture notes by Apoorva Khare