## The Simplex, the Cyclic polytope, the Positroidron, the Amplituhedron, and Beyond

A quick schematic road-map to these new geometric objects. The  positroidron can be seen as a cellular structure on the nonnegative Grassmanian – the part of the real Grassmanian G(m,n) which corresponds to m by n matrices with all m by m minors non-negative. The cells in the cellular structure of the positroidron correspond to those matrices with the same (+,0) pattern for m by m minors. When m=1 we get a (spherical) simplex. When we project the positroidron using an n by k totally positive matrix we get for m=1 the cyclic polytope, and for general m the  amplituhedron. When we project using general matrices we obtain general polytopes for m=1, and an interesting extension of polytopes proposed by Thomas Lam for general m.

Alex Postnikov’s recent lectures series in our Midrasha was an opportunity to understand slightly better some remarkable combinatorial objects that drew much attention recently. So here is my tentative partial understanding (based on Postnikov’s talks and some very useful explanation by Alex and by Lauren Williams) of it and some natural questions we can ask.

## Cellular structures on the Grassmanian.

### Schubert decomposition

The Grassmanian G(m,n), – the space of m-dimensional subspaces on an n dimensionsl space, can be seen as equivalence classes of m by n matrices under the action of GL(m). (So far, we can assume that the underlying field is arbitrary.)

The parts of the Schubert decomposition of G(m,n) correspond to all matrices with a given m-subset S with A as the  lexicographically first non zero m by m minor.

Over the reals or the complex numbers this give an important cellular structure. The parts are indeed topologically open cells. Their closures are in general not homeomorphic to closed cells and can be quite (excitingly) complicated. The intersection of two cells can also be rather (excitingly) complicated.

### The matroid decomposition

Given an n by m matrix we can regard its corresponding (representable) matroid. The bases of the matroid are those  m-subsets S of [n] such that the corresponding m by m minor is non-zero. Two matrices belong to the same part in the matroid decomposition of G(m,n) if their corresponding matroids are the same. The matroid decomposition is the common refinement of the Schubert decomposition for all n! permutations of the n columns.

Parts in the matroid decomposition are realization spaces of matroids and they are very complicated and a whole area of universality theorems (with pioneering contributions by Nicolai Mnev) is devoted to that.

### Postnikov decomposition: The cyclic refinement of the Schubert decomositions

Now, let us consider the common refinement of all Schubert decomposition with respect to the n cyclic permutation on [n]. It turns out that this is an important decomposition of the Grassmanian and we will call it the Postnikov decomposition.

## The nonnegative Grassmanian over the reals

From now on  we will work over the reals. Given an n by m matrix we can regard its corresponding oriented matroid as a map from m-subsets S of [n] to {+.-,0} which assigns to S the sign of the corresponding m by m minor.  The possitive Grassmanian in G(m,n) consists of those matrices where all m by m minors are positive. The nonnegative Grassmanian consist of all matrices where all m by m minors are nonnegative. A positroid is a matroid which correspond to a matrix in the nonnegative Grassmanian. For the nonnegative Grassmanian, the Postnikov decomposition coincide with the matroid decomposition and they give us a remarkable cellular objects that we call the positroidron.  (The common name is “the (positroid) cell decomposition of the positive (or non-negative) Grassmannian.”)

### The triangle: m=1, n=2

When m=1 the positroidron is simply a simplex, or more precisely a certain spherical simplex.

### An example m=2, n=4.

The postroidron is a four-dimensional CW complex that can be described as follows. The 2-skeleton is the 2 skeleton of the octahedron. First you add two of the three squares in the equator. (The symmetry breaking allowing to choose these two comes from the preferred cyclic ordering.) Then  for each square you add the two pyramids on both its sides. (These are curved pyramids so they do not overlap.) Finally there is an additional four-dimensional face: the positive Grassmantan is a solid body whose boundary is the union of these four pyramids.

The combinatorial structure of the positroidron was explored by Alex Postnikov, and further combinatorial descriptions by several authors followed. It can be expressed in terms of certain planar graphs and it is closely related to certain permutation statistics.

Postnikov’s original paper is: Total positivity, Grassmannians, and networks

## The Amplituhedron and Lamtopes

### The cyclic polytope and general polytopes.

Every convex polytope is a projection of the simplex. When we project using an n by k totally positive matrix  we obtain the cyclic polytopes (A matrix is totally positive if all its minors are positive. Here it is enough to consider k by k minors.) Cyclic polytopes were discovered by Caratheodory at the early 20th century, and later rediscovered by David Gale who determined their combinatorial structure.

### The amplituhedron and Lamtopes

The amplituhedron is a simple extension of the cyclic polytope for m>1. It is simply the projection of the positroidron under a totally positive matrix. This is a generalization of the cyclic polytope.

Thomas Lam (according to Alex Postnikov) proposed to consider not just the Amplituhedron but arbitrary projections of the positroidron.

### The amplituhedron and physics

Computations of scattering amplitudes using the positive Grassmanians offers substantial simplification compared to “naive” Feynman diagrams computations. The computations are based on summing up certain volumes on cells of the positroidron, and the sums extend over cells which correspond to faces of the amplituhedron. This suggests that the cells of the amplituhedron have some conceptual significance. (It may also allow further computing improvement in the future.) Nima Arkani-Hamed and Jaroslav Trnka regard the amplituhedron as a key geometric object for understanding the emergence of locality and unitarity in quantum physics (or something beyond quantum physics).

### Representability

The questions if a matroid is representable over a field, or if an oriented matroid is representable over the reals are very basic. A 1987 conjecture by da Silva asserts that (abstract) Positroids are representable and this was proved by Federico Ardila,  Rincón, and Williams. The abstract analog of the Grassmanian (based on abstract oriented matroids rather than just representable ones) is called McPhersonian.

### The Bruhat order

Important examples of regular CW deomposition of spheres come from intervals in the Bruhat order of  Coxeter groups. It will be interesting to know if such intervals also appear (as are) as intervals in the geometric objects considered here. (Definitely Bruhat intervals of the form [e,w] (where w is a Grassmannian permutation) appear as intervals in the poset of cells in the positive Grassmannian.  This is more or less immediate from the description of the poset of cells of the positive Grassmannian in terms of pairs of permutations.)

Some connections between the positroidron and the Bruhat order are known and studied in Lauren Williams’ paper Shelling totally nonnegative flag varieties . The appendix to the paper describes some early combinatorial representations of the positroidron.

## Open questions

### A basic questions:

1) Is the positroidron a regular CW complex? As far as I know it was proved already by Postnikov that the open cells are homeomorphic to an open ball and Konni Rietsch and Lauren Williams  proved that the closed cells are contractible, with boundaries homotopy-equivalent to spheres —  in Discrete Morse theory for totally non-negative flag varieties. It is conjectured that closures of open cells are homehomorphic to close balls.

The results and techniques in Hersh’s paper on regular CW complexes in total positivity are closely related.

### Further question

2)  Are the amplituhedron and Lamtopes are regular CW complexes? In fact, is it the case that their open cells are homehomorphic to balls?

3) (Perhaps well understood) how does intervals in the Bruhat order of Coxeter groups fits into the picture.

4) Describe the cellular structure of the amplituhedron.

5) Are Lamtopes closed under taking intervals and duality. If not, what would be an interesting class of objects including Lamtopes which are closed to taking intervals and to duality.

A reminder: miracles of  cyclic polytopes

a) Cyclic polytopes in d dimensions are neighborly. When you regard it as a projection of the simples, every face of the simplex of dimension  [d/2]-1 is mapped to a face of the cyclic polytope.

b) Cyclic polytopes are extremal. The upper bound theorem asserts that the maximum of k-faces for a d-polytopes with n vertices is attained by the cyclic polytope.

c) The vertices of the cyclic polytope represent a remarkable oriented matroid or “order-type.”  It is universal in the following sense: For  a set of points in $R^d$ in “cyclic position” every subset is also in cyclic position. On the other hand, for every n and d there is f(n,d) such that every set of f(n,d) points in $R^d$ contains a subset of n points in cyclic position.  In the plane this is the famous Erdos Szekeres theorem. In high dimension this is now well understood by works of Jirka Matousek and a few coauthors.

### Further question motivated by the cyclic polytope and related theory

6) Is the amplituhedron extremal among Lamtopes in terms of face numbers and other combinatorial parameters? (The upper bound theorem asserts that cyclic polytopes are extremal among polytopes.) Does the amplituhedron exhibit neighborliness? tightness?

7) Do the notions of g-polynomial [and Kazhdan-Lusztig polynomial] extend to lamtopes?

8) (Proposed by Eric Katz) Do the notions of secondary polytopes/fiber polytopes extend to lamtopes.

9)  Is the amplituhedron (in some sense) “neighborly?”

10)  Are the vertices of the amplituhedron (thought of as representing a sequence of k-spaces in some Euclidean space) universal in some sense?

1) (Feb. 22) Hersh’s work:   The paper “Regular cell complexes in total positivity” shows that certain spaces of totally nonnnegative, real matrices, stratified according to which minors are positive and which are 0, are regular CW complexes homeomorphic to closed balls having the closed intervals in Bruhat order as their posets of closure relations.  These regular CW balls arise as links of cells in the double Bruhat stratification of the totally nonnegative part of the flag variety, (as can be seen from the Marsh-Rietsch parametrization of the totally nonnegative part of the flag variety).  So this also gives some further evidence for the conjecture about the nonnegative part of the flag variety being a closed ball.  A main example of the stratified spaces Hersh studied was the totally nonnegative, real, upper triangular matrices with 1’s on the diagonal and entries just above the diagonal summing to a fixed positive constant, stratified according to which minors are positive and which are 0.  (To me this shows that the actual picture is more general than described in this post.)

2) (Feb 22) Algebraic shifting: In my early works I studied stratifications of Grassmanians G(V,m) where V itself is a kth exterior power of an n-dimensional vector space. Algebraic shifting is essentially the study of the Schubert decomposition with respect to the lexicographic ordering of base elements of the exterior algebra (which are indexed by k-subsets of [n].). In general, it is interesting to study stratifications of the Grassmanian when the vector space we start with has additional structure.

### Trivia question

When you search pictures in Google for “Bruhat order” you get this picture.

Why?

### Some Papers:

Alex Postnikov,   Total positivity, Grassmannians, and networks

Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, Jaroslav Trnka, Scattering amplitudes and the positive Grassmannian, arxiv/1212.5605

Nima Arkani-Hamed and Jaroslav Trnka  The Amplituhedron

Federico Ardila, Felipe Rincón, Lauren Williams, Positively oriented matroids are realizable

Konni Rietsch and Lauren Williams,  Discrete Morse theory for totally non-negative flag varieties.

arXiv:1408.5531 Amplituhedron cells and Stanley symmetric functions. Thomas Lam

arXiv:1111.3660 Positroid Varieties: Juggling and Geometry. Allen Knutson, Thomas Lam, David Speyer.

Patricia Hersh,  Regular cell complexes in total positivity; also available at arXiv:0711.1348.

Inventiones Mathematicae, 197 (2014), no. 1, 57–114.

Popular articles

Natalie Wolchover (September 17, 2013). “A Jewel at the Heart of Quantum Physics”. Quanta Magazine.

### The Reference Frame: Amplituhedron:

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