In Geometric Representation Theory lecture 2
James Dolan draws a 2-simplex with a barycentric subdivision, looking like a hexagon with a central point, or rather a cube with a missing hidden vertex. This picture is labelled with Young diagrams associated to the group $S_3$ of permutations on three letters. Diagrams of height 1 are associated with vertices, diagrams of height 2 with edges and diagrams of height 3 with faces. Note that horizontal lists are unordered. So the six faces represent the elements of the group.
Dolan wants to think of these diagrams as axiomatic theories
in a categorical sense. Such diagrams, and their subdiagrams, are associated with sequences of subspaces of the three element set, in analogy with flag spaces for vector spaces. Sets are just a classical kind of vector space. This is clear when counting vectors in vector spaces over finite
fields, as discussed by Baez in lecture 1
. For vector spaces over $F$ the permutation groups would naturally be replaced by the example of $GL(n,F)$.
The example above generalises as expected. Before we know it, we'll probably be doing motivic cohomology and Langlands geometry using Hecke pictures. Of course, Kontsevich
has already been thinking about such things.