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I'll begin by explaining what a planar algebra is, and then show you
how the representation theory of SU(2) and SU(3) can be given a
"finite presentation by generators and relations" as a planar algebra.
This may be familiar to some, as the Temperley-Lieb algebra for SU(2)
or Kuperberg's spider for SU(3). Next, I'll explain my work on
generalising this to all SU(n). We'll start with a category of
diagrams, generated by some trivalent vertices, and a surjective map
to the representation theory; the difficulty will be understanding the
relations amongst these diagrams. The main trick is to remember that
SU(n) sits inside SU(n+1), and conversely representations of SU(n+1)
break up (or "branch") as representations of SU(n). I'll explain how
to understand the combinatorics of branching in terms of my diagrams,
and how to use this to "lift" relations for diagrams from one level to
the next.
After the seminar we will take the speaker to lunch. See the Algebra Seminar web page for information about other seminars in the series. Anthony Henderson anthonyh@maths.usyd.edu.au. |