Refurio Anachro @RefurioAnachro@mastodon.cloud

Huh? We're baaack!

What the heck is an #Amplituhedron?

Here's the first chunk of my series about them. Expect more within a week:

https://mastodon.cloud/@RefurioAnachro/99490746199683654

The image is from this pop post:

https://www.quantamagazine.org/physicists-discover-geometry-underlying-particle-physics-20130917/

It shows a plabic #graph representation for a #permutation. At black points turn left, and turn right at white ones! This one has no zeros.

#quantum #process, #computing #scattering #amplitudes

You can find lots of fun stuff about positive Grassmannians in *Alexander Postnikov*'s paper _"Total positivity, Grassmannians, and networks"_:
https://arxiv.org/abs/math/0609764

On p53 ff he talks a bit about _plabic graphs_, a representation for permutations.

Or try this fun 4-part lecture series by *Alexander Postnikov*, _"Combinatorics of the Grassmanian"_:
https://m.youtube.com/watch?v=5m6j_yiepFM

Img src:

https://mathoverflow.net/questions/142841/the-amplituhedron-minus-the-physics

Images for Amplituhedron for my in-progress post about it.

I like this one because it's simple. This one a higher (4?) dimensional configuration projected down to 3d. That causes some edges to become internal.

https://en.wikipedia.org/wiki/File:Amplituhedron-0c.png

Regarding the description there, "Positroid" is derived from the words positive and matroid. It's a decomposition of a positive Grassmannian. More fine grained than Schubert cells, but less than matroid stratification.

#physics

Hm. Putting these in my bio isn't quite practical.
#mathematics #maths #math #fractal #geometry #topology #homology #algebra #complex #calculus #logic #set #arxiv

And there are no parts 10 and 15. I just forgot to use these numbers. Cheers!

... more later

Amplituhedron 16/

There are 6 ways to pick 2 out of 4, and we can arrange these as the vertices of an octahedron!

1010
1100 0110 0011 1001
0101

Δ13
Δ12 Δ23 Δ34 Δ14
Δ24

4 of these vertices are labeled in a cycle, and the upper and lower vertices have edges that exchange a pair of digits (look at the binary). It looks like an octahedron, but it is not! There are 2 extra square faces suspended between the 2 special top and bottom vertices!

Amplituhedron /14

...for more on relations like these take a look at _cluster algebras_:
https://en.wikipedia.org/wiki/Cluster_algebra

Amplituhedron /13

Going further, in Gr(2,4) we need to strike out two columns to obtain a minor, and therefore the coordinates need two indices. We find that Δij = -Δji so we don't need both, and it means that if there were any Δjj they'd have to be zero.

Playing with those you might find more relations:

Δ13Δ24 = Δ12Δ34 + Δ14Δ23

You can write this one graphically as a sum of pairs of chords in a circle:

(X) = (||) + (=)

Amplituhedron? 12/

Since Gr(2,3) is isomorpic to Gr(1,3), how do we get homogeneous coordinates from a 2x3 matrix?

(a b c)
(d e f)

We simply compute the determinants of the minors of size k!

Δ1 =
|b c|
|e f |

Δ2 =
|a c|
|d f |

Δ3 =
|a b|
|d e|

[Δ1:Δ2:Δ3]

Amplituhedron 11/

Okay, homogeneous coordinates for P³ or Gr(1,3) have three components, and they are written like this:

[x:y:z]

Remember, there is no point deserving the label [0:0:0] and we're allowed to scale the numbers by a factor without changing the point it refers to:

[x:y:z] = [ax:ay:az]

This is what a point in Gr(1,3) really looks like1

Amplituhedron /9

Okay, let's use three numbers and work in homogeneous coordinates, which can also be called called Plücker coordinates...

https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates

Historically, *Julius Plücker* only considered lines in space, and *Günter Grassmann* later extended the idea to also entail planes and higher dimensional subspaces. Hence the general version is sometimes called Grassmann coordinates.

https://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding

Amplituhedron 8/

...an element of (or a point in) Gr(k,n) can be written as a k × n matrix!

https://en.wikipedia.org/wiki/Grassmannian

So Gr(2,3) consists of all planes in 3-space which contain the origin, but since every such plane can also be given by a 3-vector of unit length (the surface normal), we obtain the same space as Gr(1,3): a sphere! And then, a sphere requires only two numbers to identify one of its points, so what's going on?

Amplituhedron? 7/

Let's get into the fun:

A Grassmannian Gr(k,k+m) is the space of k-dimensional subspaces in R^(k+m) (for now, but other manifolds would also work). For example, a plane containing the origin in 3 dimensions can be specified by two 3-dimensional vectors like this:

(x1 y1 z1)
(x2 y2 z2)

Amolituhedron? 6/

Btw: The amplituhedron is a recurring topic in *Nima Arkani-Hamed*'s recent work, and you can find more relevant papers by searching for him on the arxiv. Or his recent talks on youtube, in which he entertains with history of physics and deep motivational considerations, while sketching lookouts into the formal side.

Amplituhedron? 5/

These notes are mostly based on the 2017 paper by Arkani-Hamed, and Jaroslav Trnka: _"Unwinding the Amplituhedron in binary"_
https://arxiv.org/abs/math/1704.05069

And my investigations were spurred by his 2017 talk
_"Physics and Mathematics for the End of Spacetime"_:
https://m.youtube.com/watch?v=z1-QDXReDTU