M Theory Lesson 128
In Geometric Representation Theory lecture 13 you can hear James Dolan talking about braid diagrams and Hecke operators.
First, we think of Hecke operators as magic matrices with sets as elements. Composition of operators is sort of like a renormalised matrix multiplication. These can be redrawn as braid diagrams where we don't worry too much about the crossings. Matrix multiplication will be replaced by braid compositions. Dolan gave an example like this one:
The dots on the top line come from the cardinalities $R$ and $B$, whereas the dots on the bottom line come from $R'$ and $B'$. The left hand strands represent the set in the top left box. Note that the total number of dots on the top and bottom is always the same. Braids with real crossings supposedly come in handy when considering not sets but vector spaces, or rather representations of groups like $GL(n, F_{q})$ over a finite field with $q$ elements (Langlands, anyone?). This links a simple set cardinality with a knot parameter $q$. But the knotty $q$ can take many complex values, most notably a complex root of unity. Fortunately, we already know that cardinalities can also take such values.
Another example, this time for a $3 \times 3$ matrix, shows how to associate an element of $B_3$ with a matrix whose entries sum to $3$. Sticking to the set interpretation, a zero is given by an empty square.
M theorists will find such diagrams familiar by now. If you enjoyed lecture 13, in lecture 14 you can see John Baez write up the three matrices $\mathbf{1}$, $(231)$ and $(312)$ which underlie the mass Fourier transform.
First, we think of Hecke operators as magic matrices with sets as elements. Composition of operators is sort of like a renormalised matrix multiplication. These can be redrawn as braid diagrams where we don't worry too much about the crossings. Matrix multiplication will be replaced by braid compositions. Dolan gave an example like this one:
The dots on the top line come from the cardinalities $R$ and $B$, whereas the dots on the bottom line come from $R'$ and $B'$. The left hand strands represent the set in the top left box. Note that the total number of dots on the top and bottom is always the same. Braids with real crossings supposedly come in handy when considering not sets but vector spaces, or rather representations of groups like $GL(n, F_{q})$ over a finite field with $q$ elements (Langlands, anyone?). This links a simple set cardinality with a knot parameter $q$. But the knotty $q$ can take many complex values, most notably a complex root of unity. Fortunately, we already know that cardinalities can also take such values.Another example, this time for a $3 \times 3$ matrix, shows how to associate an element of $B_3$ with a matrix whose entries sum to $3$. Sticking to the set interpretation, a zero is given by an empty square.
M theorists will find such diagrams familiar by now. If you enjoyed lecture 13, in lecture 14 you can see John Baez write up the three matrices $\mathbf{1}$, $(231)$ and $(312)$ which underlie the mass Fourier transform.
posted by Kea | 10:54 AM
2 Comments:
Hi Kea,
When I click on Hecke in the first paragraph, I find an interesting paper by Kapustin and Witten, 'Electric-Magnetic Duality And The Geometric Langlands Program'.
I had expected a wiki reference?
I had not seen the 'dominoes-like' representation of braids before.
The more usual braid diagrams in 2D hint at 3D with braid crossings.
Should HEP physics begin to look at helical Fourier transforms of EE? The helix is associated with EM in solenoids.
Hi Kea, I have been reading the paper referred to in my previous comment.
On p89 figure 4 is a schematic of four-manifold with Wilson’s lines: horizontal [C], vertical [SIGMA]. I am surprised that Witten and Kapustin did not include an example of a helix as a geodesic around the cylinder. If the cylinder is virtual, then this would seem to be consistent with David Hestenes Zitterwebegung concepts
This may also be consistent with a twister theory relation to the helix as a string representation of a trajectory?
Similar, but not identical, cylindrical representations are on p120 figure [f]11; p124 f12; p129 f13; p130 f14; p149 f15; p160 f16; p164 f17; p166 f18; p170 f19 and p176 f20.
If the spherical object S is a sun / star or a proton, then the virtual cylinder could be the trajectory of a planet or electron, respectively.
Electrical engineering has appeared to provide sufficient evidence that electromagnetism is associated with a helix and geophysics has associated the magnetosphere as a torus, distorted by interaction with the heliomagnetosphere.
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