# What is the amplituhedron?

The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces a remarkable new way for computations in quantum field theory based on volume computations of certain polyhedra. This and related works may also have profound implications for the foundation of particle physics. It is related to various beautiful mathematics, and, in particular, to the combinatorics of certain stratifications of the Grassmanians.(See also the Quanta Magazine article, A Jewel at the Heart of Quantum Physics, by Natalie Wolchover, and Nima Arkani-Hamed’s on-line SUSY 2013 video lecture The Amplituhedron.)

The amplituhedron is remarkable new subsequent geometric object which comes in this study extending the notion of "positive grassmanian." (I did not see it explicitly defined in the above paper, at least not by this name.) It is very briefly described in Arkani-Hamed's lecture.

My question is quite simple:

## what is the mathematical definition of the amplituhedron?

An older-sister MO question: The amplituhedron minus the physics

Update: The paper The amplituhedron by Nima Arkani-Hamed and Jaroslav Trnka is now on the arxive.

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Thanks Gil for asking this question; I tried to skim the talk, wikipedia site etc. after mathoverflow.net/questions/142841/… --- but am waiitng to see a simplified definition. –  Suvrit Sep 27 '13 at 16:08
I understand that this is an exciting problem connecting physics with recent advances in algebraic combinatorics (and it is close to my personal interests.) But I don't see how this question is not a duplicate of Joseph O'Rourke's earlier question. If you really want to know what the amplituhedron is (from a math standpoint), it might be worth contacting Alex Postnikov directly. –  Sam Hopkins Sep 27 '13 at 17:18
Dear Sam, yes this is almost a duplicate of Joe's question that I simply did not see. (A few different links, images, and tags). Regarding your second point: In a few cases I asked questions on MO where I can also get an answer by directly asking a colleague. The advantage of asking on MO is that this way more people are exposed to the information and also it is nice to test the MO platform which is a relatively new thing. Of course, in a few cases I both posted on MO and asked colleagues personally. –  Gil Kalai Sep 27 '13 at 17:37
I welcome Gil's more focused and knowledgeable question, as my question ("The amplituhedron minus the physics") has faded in visibility. –  Joseph O'Rourke Sep 27 '13 at 19:50
It is somewhat unsettling that we have already had two questions decorated with several links to papers on an object and we are still asking for its precise definition! :-) –  Mariano Suárez-Alvarez Sep 28 '13 at 1:17

I would think that this presentation by Jaroslav Trnka, given here in Utrecht, goes at least some way towards a mathematical definition of the amplituhedron.

To skip the physical motivation, start at page 13; jargon abbreviations such as NMHV = "next-to-maximally helicity-violating" can be ignored (they relate only to the physical significance of the construction). The construction of the amplituhedron $P_{n,k,m}$ is summarized on page 23. What follows on later pages is the description how to associate a form $\Omega_{n,k,m}$ to the space $P_{n,k,m}$ and use this to calculate the required physical quantity (a scattering amplitude).

My attempt to parse a definition of the amplituhedron from Trnka's presentation:

For given integers $k,n,m$ (with $n\geq k+m$) take a $k\times n$ real matrix $C\in G_{+}(k,n)$ and a $(k+m)\times n$ real matrix $Z\in G_{+}(k+m,n)$. Here $G_+(k,n)$ is the positive Grassmannian space of $k\times n$ matrices with all $k\times k$ minors $>0$. (The $k\times n$ matrices are identified modulo the simultaneous action of a $k\times k$ matrix on each of the column vectors.)

Associated to these two positive Grassmannians is the $k\times (k+m)$ real matrix $Y$ having matrix elements

$$Y_{\alpha}^{\beta}=\sum_{p=1}^{n}C_{p}^{\alpha}Z_{p}^{\beta}.$$

By varying $C\in G_{+}(k,n)$ at fixed $Z\in G_{+}(k+m,n)$, the matrix $Y$ varies over a space $P_{n,k,m}$. This space is called the amplituhedron.

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Many thanks, Carlo. I still dont see what precisely is $P_{n,k,m}$. –  Gil Kalai Sep 27 '13 at 22:46
Noob question probably, but when you write "all minors >0", are you imposing a canonical ordering on the columns? Because otherwise they can't all be >0, since they change sign when swapping two columns. –  Federico Poloni Sep 28 '13 at 13:49
@FedericoPoloni --- indeed, the minors are ordered in ascending order of the columns: the $(k\times n)$ matrix $C$ is in the positive Grasmannian $G_{+}(k,n)$ if for each set of $k$ integers $1\leq a_1<a_2<\cdots a_k\leq n$ the $k\times k$ submatrix of $C$ consisting of the ordered columns $(a_1,a_2,\ldots a_k)$ has a positive determinant. –  Carlo Beenakker Sep 28 '13 at 14:30
@suvrit --- as far as my very_limited understanding goes, strict positivity of the minors ensures that the matrix $Y$ lies inside the amplituhedron $P_{n,k,m}$; a vanishing minor produces a matrix $Y$ on the boundary. For the physics application one is interested in the volume bounded by the amplituhedron, so it should not make a difference if the positive Grassmannian is replaced by a nonnegative Grasmannian. –  Carlo Beenakker Sep 28 '13 at 17:40
Sorry for making this thread long, but I wanted to alert the readers that as per Logan's answer here: mathoverflow.net/a/143421/8430, the above $Y=CZ^T$ is not the whole amplitudhedron, just the tree-case ... so I'll wait for more updates before I shoot off more comments! –  Suvrit Sep 28 '13 at 18:04