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October 19, 2007

Geometric Representation Theory (Lecture 3)

Posted by John Baez

Any structure on a set has some group of symmetries. But you can also work backwards. Given the symmetries, you can figure out the structure those symmetries preserve!

Last time in the Geometric Representation Theory seminar, Jim Dolan introduced the ‘orbi-simplex’ as an easy way to do this. Start with a group GG acting on a set SS. Form a simplex with SS as vertices. Mod out by the action of GG. This, in a nutshell, is the orbi-simplex.

This time, Jim will show how to stare at an orbi-simplex and read off a logical theory — a bunch of predicates and axioms — describing a structure on SS whose symmetries form exactly the group GG.

Jim also introduces the all-important concept of ‘Hecke operator’: an intertwining operator between permutation representations coming from an invariant binary relation between their underlying GG-sets. And, it turns out Hecke operators from a permutation representation to itself come from certain edges in the orbi-simplex!

  • Lecture 3 (Oct. 4) - James Dolan on the orbi-simplex. Pictures of orbi-simplices for subgroups of 3!, the group of all permutations of the 3-element set. How the simplices in an orbi-simplex get labelled by Young diagrams DD: a DD-labelled simplex in the orbi-simplex of a subgroup GS!G \subseteq S! is a GG-orbit in the space of DD-flags on SS. Example: the DD-labelled simplices in the orbi-simplex for the 3-element cyclic subgroup of 3!. How DD-labelled simplices in the orbi-simplex correspond to atomic invariant DD-ary predicates, and how to read off the axioms these predicates satisfy, recovering an axiomatic theory whose model on SS has GG as symmetries.

    The relation to traditional representation theory. Theorem: let GG be a subgroup of S!S! for some finite set SS, and let RR be the corresponding permutation representation of GG on S\mathbb{C}^S. Then the space of intertwining operators from RR to RR has a basis given by the orbits of GG on S×SS \times S — that is, atomic GG-invariant binary relations on SS. These operators are called ‘Hecke operators’. Apart from the diagonal orbit {(s,s):sS}\{(s,s): s \in S\}, the orbits in S×SS \times S correspond to certain edges in the orbi-simplex — namely, those labelled by this Young diagram:
    Example Young Tableaux in SVG
Posted at October 19, 2007 2:36 AM UTC

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Re: Geometric Representation Theory (Lecture 3)

Given the symmetries, you can figure out the structure those symmetries preserve!

One may take this philosophy further, and argue that the most interesting structures are those preserved by the the most interesting symmetries, say simple Lie algebras. Victor Kac makes this point in math/9912235, p20:

“Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds.”

Posted by: Thomas Larsson on October 19, 2007 8:24 AM | Permalink | Reply to this
Read the post Geometric Representation Theory (Lecture 5)
Weblog: The n-Category Café
Excerpt: James Dolan on how to get intertwining operators called "Hecke operators" from relations between types of geometrical figures.
Tracked: October 22, 2007 11:37 PM

Re: Geometric Representation Theory (Lecture 3)

I watched lectures 2 and 3 today. Although probably everything gets clarified later on, I would still like to make a comment and ask a question - expressing thoughts improves my understanding :-). I hope someone answers.

And BTW: I still can’t open the streams.

Comment: the proof of a theorem about bijection between G-morphisms of representation C^S, where G acts on S, and orbits of SxS was, frankly, too much of a hand-waving. One can do it honestly like this: In C^S one has a basis consisting of elements of S. Thus for every (a,b)\in SxS one can consider a morphism which sends a to b, and all the other elements of S sends to 0. Denote this morphism also as (a,b). It is usually not a G-morphism, but the following is a G-morphism (one easily checks it on the basis of C^S):
(a,b) + (ga,gb) + … +(ha,hb)
where the summands are [all distinct elements of the orbit of (a,b)]. So one has a map
{Orbits in SxS} –> G-morphisms of C^S.
It’s obvious that this map is injective. And if one takes a G-morphism F and writes it “in the basis”, i.e. as a sum of some (i,j) morphisms then one sees also surjectivity, by induction on number of (i,j)’s: take any (a,b) from this sum, make a G-morphism f out of it, substract suitable multiplicity of f from F to get a G-morphism which is written as a shorter sum of (i,j)’s.

Sort-of-question: I’m not convinced by the motivation for the orbi-simplex (however, my whole knowledge of logic comes from this).
If the task one poses is to find a structure such that G is a group of all automorphisms preserving this structure than one can take only a small subset of predicates James proposes to take: namely, only one |S|-ary predicate, where |S| is a number of elements of S, consisting of the orbit of (s,…,t) in S^{|S|}, where s,…,t are all elements of S. As far as I understand, isomorphism is a bijection f such that if (x,y,…,z) fulfill the predicate P then also (fx,fy,…,fz) fulfill P. So, if one takes an isomorphism f of S with above structure, one sees that it must act on S just like some element of a group. OR, I messed sth up :-)

I like orbi-simplex because its edges give G-morphisms. I don’t understand why its induced structure on S is cool.

Posted by: sirix on February 4, 2008 1:23 AM | Permalink | Reply to this

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