M Theory Lesson 132
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.
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.
posted by Kea | 9:56 AM
9 Comments:
Kea, your graphics keep getting nicer. For me to understand these things, I think your discussion needs to be about 5x longer. And no words with more than 3 syllables.
Carl, watch the lectures. They go at a very slow pace.
Hi Kea,
This diagram is consistent with a group of bisected [n=4-]Polykites.
From Polyform.
Kites are associated with Penrose tiles.
In turn there is a link to Tessellation.
There may be some way of relating these to The Dedekind tessellation?
Doug, as discussed many times, the modular group (of the Dedekind picture) is associated to braids on three strands, which is a q analogue of the group S3.
Hi Kea, RE: your references:
1 - In Witten, “Quantum Gravity Partition Functions in Three Dimensions” uses the Dedekind tessellation in figure 4, page 67 with “phase transitions” and figure 3, page 49 with a discussion of the Hawking-Page phase transition, both associated with SL(2,Z).
2 - In Kontsevich, “Notes on motives in finite characteristic”, reference 7 appears related to such as Max-plus algebra
Sigh. Yes, Doug, we are also interested in A-infinity structures, braided tensor categories and the like.
Hi Kea,
Unifying twistor strings with spinor loops?
I have found a dynamic graphic simulation on Wiki which is consistent with David Hestenes concept of Zitterbewegung.
Nonsymmetric velocity time dilation by Cleonis [28 January 2006, cc-by-sa-2.5] appears to be consistent with an electron [red] traveling about a proton [blue].
This “gif” might also be interpreted as a single planet revolving about a moving star.
The star could be a galactic core with only one star shown revolving and with a little thought experiment, one may be able to imagine at least one planet revolving about this star.
If the “red” were a system of 8 planets about a star or a galaxy of stars about the galactic core then one might even imagine a virtual torus.
If one looks at
a - figure 11, page 120 Kapustin, Witten, ‘EM Duality And The Geometric Langlands‘ there is a resemblance to the gif;
b - then at figure 2, page 13 of Maloney, Witten ‘Partition Functions’ the gif forms a virtual cylinder resembling AdS3.
Tokamak basics from Pitts, Buttery and Pinches, ’Fusion: the way ahead’ demonstrates the relation of the helix and torus to electromagnetism in plasma physics.
There are helical equations of Fourier Transforms from Hamish Meikle.
The 3D helix can be decomposed into a 2D sine, 2D cosine and 1D loop.
The 1D loop resembles the nearly circular, elliptical planetary equations of Newton.
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