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

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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

AST134J_Diagram

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?

Updates: Some additional things.

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. 44106783ZA_ma2 

Why? 

Links

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.

Amplituhedron May Shape the Future of Physics – Discover

The Reference Frame: Amplituhedron:

Blogs

Blogposts guest post by Trnka on Carrol’ blog

The Amplituhedron and Twistors | Not Even Wrong

Shtetl-Optimized » Blog Archive » The Unitarihedron:

Update on the Amplituhedron | 4 gravitons

Q/A sites

What is the actual significance of the amplituhedron?

co.combinatorics – What is the amplituhedron?

The amplituhedron minus the physics – MathOverflow

nLab

amplituhedron in nLab

Facebook

Amplituhedron | Facebook

Videotaped lectures

Alexander Postnikov: The Combinatorics of the Grassmanian

alex-mid2

Video1, Video2, Video3, Video4

The Amplituhedron – YouTube

The Amplituhedron | Nima Arkani-Hamed – YouTube

Slides of lectures

The Amplituhedron

  Thomas Lam: Totally nonnegative Grassmannian and the amplituhedron

Tricia Hersh, Regular cell complexes in total positivity

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16 Responses to The Simplex, the Cyclic polytope, the Positroidron, the Amplituhedron, and Beyond

  1. Allen Knutson says:

    I’m not sure if this is what you’re looking for in your Bruhat order questions, but Lam, Speyer, and I show that the entire poset of positroid strata is an order ideal in affine Bruhat order (i.e. is a union of intervals [kkkkkkkk, w] where kkkkkkkkk is the kth minimal element in affine Bruhat order). This is in our paper with juggling in the title.

    • Gil Kalai says:

      Dear Allen, That’s very interesting. Actually I did not thing about ideals in the Bruhat order but it is certainly very natural to think about them. One of my motivation comes from the fantasy to define analogs of Kazhdan-Lusztig polynomials (and R-polynomials) for more general regular CW-spheres. For the case that intersection of cells is a cell (namely we have the lattice property) the standard (toric) g-polynomial should work. But more generally the situation is very mysterious and is it not clear that this fantasy is not ill-founded.

      Does your result have bearing on the question if the poset is a regular CW-sphere?

      Actually I also did not understand what kkkkkkkkk is and what the running index of the union (is it k?)

      • Extended affine Bruhat order is on the set of affine permutations {f : ZZ -> ZZ bijective, with f(i+n) = f(i) + n for all i}. It is not a Coxeter group; rather it is the semidirect product of the subgroup {f(i) = i+k for all i, k in ZZ} (a copy of ZZ) with the Coxeter subgroup {average(f(i)-i) = 0}. This Bruhat order has one minimum for each k, and these minima are the elements of that first subgroup. The way that jugglers indicate an element f of this group is to give the list f(1)-1, f(2)-2, …, f(n)-n.

        The running index of the union is w; there is one for each k-subset S of {1…n}, defined by w(i) = i (for i not in S) or w(i) = i+n (for i in S).

    • Gil Kalai says:

      Many thanks, Allen, I see. Something strange happened here and four years disappeared on this comment thread. Meanwhile I forgot some of the things. I do hope to return to the amplituhedron and related objects soon. I wish to understand the positroid in as simple terms as possible, and also I wonder if the (tree, I suppose) amplituhedron which is a map of the positroid has the same skeleton as the positroid up to some dimension. (I.e. it is “neighborly”.)

    • Gil Kalai says:

      Another natural question is: Is there an analog for the positroidron and amplituhedron when you replace the Grassmanian with the MacPhersonian (namely consider all oriented matroids and not onlt representable ones).

      • Let’s understand some things first on the geometry, and how it informs the combinatorics.
        Let M be a space with two finite decompositions into locally closed subsets, \calY and \calZ.
        Assume that the strata Y in \calY are irreducible, and that \calZ refines \calY, i.e. each Y is a union of various Z from \calZ.
        Then there’s a map \calZ -> \calY called “what stratum am I in?”, and a map from \calY -> \calZ called “which of those strata Z is open dense in me?”. Obviously the composite \calY -> \calZ -> \calY is the identity, so one can **mistake** \calY for a subset of \calZ.

        This mistake is the source of the terminology “positroid”. The positroid stratification is much coarser, thankfully, than the abominable matroid decomposition. So one could think of a positroid as a special kind of (representable) matroid and that’s what Postnikov did. But the matroid stratum associated to that positroid is smaller than the “positroid stratum” (the first is only open dense in the second).

        There’s something very curious, which is that the map {representable matroids} -> {positroids} given by the geometry does have a natural extension to all matroids, essentially taking C |-> {(the lex-first base in the ith rotation of C), for i =1..n}, which isn’t coming from geometry when the matroid stratum is empty.

        Anyway, the matroid decomposition is so horrible (strata not equidimensional, closure of a stratum not a union of others, even beyond the question of which strata are nonempty) that I have trouble imagining what one could want the analogue of the amplituhedron to be.

    • Gil Kalai says:

      Dear Allen, I see. I thought that the matroid stratification (or the join of all Schubert decompositions for all orderings of the standard basis), is extremely terrible but becomes nice when we restrict to the positive part of the Grassmanian. So on the Macphersonian side I want to start with points in cyclic position (where all d tuples has + sign) and consider all order types so that signs correspond to every d-subsets is either + or 0. (representable and not representable). But maybe this is a bad idea.

      • It becomes nice exactly because almost all the strata disappear entirely. I’m confused about your order types idea — won’t having + (not 0) in some order mean you’ll have – in some adjacent order, off by a simple reflection?

    • Gil Kalai says:

      Alan, let me try: we start with a set of n points in R^d in cyclic position, in other words, d by n matrices with all square minors are positive. The set of these matrices (thought of as vectors in the appropriate Grassmanian) is an open set and its closure contains some boundary strata, described by the sign patterns of the minors (only + and 0) and these strata are nice. (unlike the case for arbitrary +/-/0 signs which gives horrible stratification).

      Now lets think about the same objects when we consider oriented matroids. The above description refer to representable case. So I somehow want to add to the description above also configurations which are (in some sense) in the “boundary” of the cyclic-position-body but which are not representable. I though that maybe this could make sense for the Macphersonian that record all oriented matroids (representale or not). Also for the Macphersonian there should be a “large” cell representing the cyclic position.

      So the vague idea is that in the Macphersonian there is a subobject analogous to the positive Grassmanian, (and the even vaguer sentiment is that the Macphersonian is fun 🙂 .)

      • I think I see the point. Call an oriented matroid an oriented positroid if it has the right positivity properties. I think the issue will be that, any oriented matroid with these combinatorial properties is actually representable, so, we don’t get a surprising new object like you want to define.

        I have a very slow project with Puck Rombach about showing that positroids are not just representable, but graphical, indeed planar-graphical.

    • Gil Kalai says:

      I see ! I realize you plan to prove much more but is the fact that every oriented positroid representable, easy? (What is the proof/ a link?)

  2. Vít Tuček says:

    Thanks for interesting read.

    What do you mean by “project using an n by k totally positive matrix”?

  3. Gil Kalai says:

    The paper Decompositions of amplituhedra by Steven N. Karp, Lauren K. Williams, and Yan X Zhang, is a great resource. http://front.math.ucdavis.edu/1708.09525 Looking at it I understand a few things better (and do not understand some things I wrote above). I will try to come back to the amplituhedron later. Meanwhile let me remark that there are two objects of interest: The amplituhedron, and triangulations of the amplituhedron. for the case where the abplituhedron reduces to the cyclic polytope, the triangulations are those considered in 1997 by Rambau’s work on triangulations of cyclic polytopes and higher Bruhat orders.

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