3.1 Motivation and further remarks

We discuss the relationship of Theorem 3.5 with some results present in the literature on amplituhedra.

The following conjecture states that the notions of $Z$-admissibility, $Z$-compatibility, and $Z$-triangulation do not depend on the choice of $Z\in \operatorname {Gr}_{>0}(k+m,n)$. It is widely believed to be true in the physics literature.

Therefore, assuming Conjecture 3.8, the notion of, say, $Z$-compatibility does not depend on $Z$, and thus gives rise to an interesting binary relation on the set $\tilde {S_n}(-k,n-k)$.

Let us now explain some motivation that led to Theorem 3.5. For integers $a,b,c\geq 1$, define

\[ M(a,b,c):=\prod_{p=1}^a\prod_{q=1}^b\prod_{r=1}^c \frac{p+q+r-1}{p+q+r-2} \]

to be the number of plane partitions that fit into an $a\times b\times c$ box [Reference MacMahonMac04]. This expression is symmetric in $a,b,c$. In [Reference Karp, Williams and ZhangKWZ17, Conjecture 8.1], the authors noted that in all cases where a (conjectural) triangulation of $\mathcal {A}_{n,k,m}(Z)$ is known, it has precisely $M(k,\ell ,m/2)$ cells, and they conjectured that for all $n,k,m$, there exists a triangulation of the amplituhedron with $M(k,\ell ,m/2)$ cells. For example, for $m=4$, the number of terms in the BCFW recurrence, which correspond to the cells in the conjectural BCFW triangulation of $\mathcal {A}_{n,k,m}(Z)$, equals

\[ M(k,\ell,2)=\frac1{n-3}{n-3\choose k+1}{n-3\choose k} \]

(a Narayana number), see [Reference Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov and TrnkaABC16] [Equation (17.7)]. We generalize [Reference Karp, Williams and ZhangKWZ17, Conjecture 8.1] as follows.

This conjecture holds for $k=1$ (see e.g. [Reference RambauRam97]), and therefore by Theorem 3.5 it holds for $\ell =1$ as well. Additionally, we have experimentally verified it in the case $k=\ell =m=2$, $n=6$, where there are $120$ triangulations of the amplituhedron, and each of them has size $M(2,2,1)=6$. We discuss this example in more detail in § 13.4. See also [Reference Karp, Williams and ZhangKWZ17, § 8] for a list of other cases where Conjecture 3.9 holds in a weaker form.

5.1 The canonical form for the top cell

The Grassmannian $\operatorname {Gr}(k,n)$ is a $k(n-k)$-dimensional manifold. It can be covered by various coordinate charts. For example, one can represent any generic element $X\in \operatorname {Gr}(k,n)$ as the row span of a $k\times n$ block matrix $[\operatorname {Id}_k\mid A]$ for a unique $k\times (n-k)$ matrix $A$. Here $\operatorname {Id}_k$ denotes the $k\times k$ identity matrix. Alternatively, any generic $X$ is determined by its Plücker coordinates lying in a cluster $\{\Delta _J\mid J\in \mathcal {C}\}$ (see [Reference Fomin and ZelevinskyFZ02, Reference ScottSco06] and Definition 11.3), where $\mathcal {C}$ is a collection of $k(n-k)+1$ subsets of $[n]$, each of size $k$, satisfying certain properties. Here we view $\{\Delta _J\mid J\in \mathcal {C}\}$ as an element of the $k(n-k)$-dimensional projective space. There is a (rational) differential $k(n-k)$-form $\omega _{k,n}$ on $\operatorname {Gr}(k,n)$ with remarkable properties called the canonical form. Before we give its definition, we note that $\omega _{k,n}$ is a rational top form (i.e. an $r$-form on an $r$-dimensional manifold), and on the chart $[\operatorname {Id}_k\mid A]$ it can therefore be written as $f(A)\,d^{k\times (n-k)}(A)$, where $f(A)$ is some rational function of the entries of $A$, and

\[ d^{k\times(n-k)}(A):=da_{1,1}\wedge da_{1,2}\wedge\dots\wedge da_{k,n-k}. \]

Thus defining $\omega _{k,n}$ amounts to giving a formula for $f(A)$. For $X\in \operatorname {Gr}(k,n)$, define

\[ \Delta_{k}^{\operatorname{circ}}(X):=\prod_{i=1}^n \Delta_{\{i,i+1,\dots,i+k-1\}}(X) \]

to be the product of all circular (that is, cyclically consecutive) $k\times k$ minors of $X$, where the indices are taken modulo $n$.

Definition 5.1 The canonical form $\omega _{k,n}$ on $\operatorname {Gr}(k,n)$ is defined as follows: if $X\in \operatorname {Gr}(k,n)$ is the row span of $[\operatorname {Id}_k\mid A]$ for a $k\times (n-k)$ matrix $A$ then

\[ \omega_{k,n}(X):={\pm}\frac{d^{k\times(n-k)}(A)}{\Delta_{k}^{\operatorname{circ}}(X)}. \]

Until § 8, we view $\omega _{k,n}$ as only being defined up to a sign.

Example 5.2 Consider an element $X\in \operatorname {Gr}(2,4)$ given by the row span of

(5.1)\begin{equation} \begin{pmatrix} 1 & 0 & p & q\\ 0 & 1 & r & s \end{pmatrix}. \end{equation}

Then we have

(5.2)\begin{equation} \omega_{k,n}(X)={\pm}\frac{dp\wedge dq\wedge dr\wedge ds}{p(ps-rq)s}. \end{equation}

What makes $\omega _{k,n}$ special is that it can be alternatively computed using other coordinate charts on $\operatorname {Gr}(k,n)$ in a simple way. For the proof of the following result, see [Reference LamLam16b, Proposition 13.3].

Proposition 5.3 Suppose that the Plücker coordinates $\{\Delta _J\mid J\in \mathcal {C}\}$ form a cluster for some $\mathcal {C}=\{J_0,J_1,\dots ,J_{k(n-k)}\}$ (see Definition 11.3), and suppose that they are normalized so that $\Delta _{J_0}(X')=1$ for all $X'$ in the neighborhood of $X\in \operatorname {Gr}(k,n)$. Then

(5.3)\begin{equation} \omega_{k,n}(X)={\pm} \frac{d(\Delta_{J_1}(X))\wedge d(\Delta_{J_2}(X))\wedge\dots\wedge d(\Delta_{J_{k(n-k)}}(X))}{\Delta_{J_1}(X) \cdot\Delta_{J_2}(X)\cdots\Delta_{J_{k(n-k)}}(X)}.\end{equation}

The normalization $\Delta _{J_0}(X')=1$ means that the homogeneous coordinates $\Delta _{J_i}(X)$ become rational functions on $\operatorname {Gr}(k,n)$. Instead of normalization, one could also replace $\Delta _{J_i}(X)$ by the rational function $\Delta _{J_i}(X)/\Delta _{J_0}(X)$ in (5.3).

The form $\omega _{k,n}(X)$ depends neither on the choice of the cluster $\mathcal {C}$ nor on the choice of $J_0\in \mathcal {C}$. This result will serve as a definition of the canonical form for any positroid cell in the next section.

Example 5.4 There are two clusters in $\operatorname {Gr}(2,4)$: $\mathcal {C}:=\{\{1,2\},\{2,3\},\{3,4\},\{1,4\},\{1,3\}\}$ and $\mathcal {C}':=\{\{1,2\},\{2,3\},\{3,4\},\{1,4\},\{2,4\}\}$. For $X$ being the row span of the matrix given in (5.1), we have

\begin{gather*} \Delta_{12}(X)=1,\quad \Delta_{23}(X)={-}p,\quad \Delta_{34}(X)=ps-rq,\quad \Delta_{14}(X)=s,\\ \Delta_{13}(X)=r,\quad \Delta_{24}(X)={-}q. \end{gather*}

Letting $J_0$ be $\{1,2\}$ and applying Proposition 5.3 to $\mathcal {C}$ and $\mathcal {C}'$ yields the following two expressions for $\omega _{k,n}(X)$:

\[ \omega_{k,n}(X)={\pm}\frac{dp\wedge d(ps-rq)\wedge ds\wedge dr}{p(ps-rq)sr}={\pm}\frac{dp\wedge d(ps-rq)\wedge ds\wedge dq}{p(ps-rq)sq}. \]

One easily checks that each of these expressions is indeed equal up to a sign to the one given in (5.2): we have $d(ps-rq)=p\,ds+s\,dp-r\,dq-q\,dr$, but only one of these four terms stays non-zero after taking the wedge with $dp\wedge ds\wedge dr$ or $dp\wedge ds\wedge dq$.

Before we state the main result of this section, let us give a warm-up result on the relationship between the twist map of [Reference Marsh and ScottMS16] and the canonical form. Recall that $\tau :\operatorname {Mat}^\circ (k+\ell ,n)\to \operatorname {Mat}^\circ (k+\ell ,n)$ sends a matrix $U\in \operatorname {Mat}^\circ (k+\ell ,n)$ to another matrix ${\tilde {U}}\in \operatorname {Mat}^\circ (k+\ell ,n)$ defined in (4.1). Similarly to Lemma 4.4, one can prove that $\tau$ descends to a rational map $\operatorname {Gr}(k+\ell ,n)\dashrightarrow \operatorname {Gr}(k+\ell ,n)$. Let $\omega _{k+\ell ,n}$ denote the canonical form on $\operatorname {Gr}(k+\ell ,n)$.

Example 5.8 Let $k+\ell =2$, $n=4$, and suppose that $U$ is the row span of (5.1). After possibly switching the signs of some columns, ${\tilde {U}}:=\tau (U)$ is the row span of

\[ \begin{pmatrix} 0 & 1 & r & s\\ -1 & 0 & -p & -q \end{pmatrix} \cdot { \operatorname{diag}} \bigg(\frac1s,1,\frac1p,\frac1{ps-rq}\bigg)\simeq_{k+\ell} \begin{pmatrix} 1 & 0 & s & \frac{qs}{ps-rq}\\ 0 & 1 & \frac r p & \frac s{ps-rq} \end{pmatrix}. \]

Here ${ \operatorname {diag}}$ denotes a diagonal matrix, and the matrix on the right-hand side is obtained from the matrix on the left-hand side by multiplying the bottom row by $-s$ and switching the two rows (these operations preserve its row span). Thus a cluster of Plücker coordinates of ${\tilde {U}}$ is given by

\[ \Delta_{12}=\pm1,\quad \Delta_{23}={\pm} s,\quad \Delta_{34}={\pm} \frac s p,\quad \Delta_{14}={\pm} \frac s{ps-rq},\quad \Delta_{13}={\pm}\frac r p. \]

Applying Proposition 5.3, we see that the pullback of $\omega _{k+\ell ,n}$ under $\tau$ is equal to

(5.4)\begin{equation} {\pm}\frac{ds\wedge d \bigl(\frac{s}{p}\bigr)\wedge d\bigl(\frac{r}{p}\bigr)\wedge d\bigl(\frac{s}{ps-rq}\bigr)}{s\cdot \frac{s}{p}\cdot \frac{r}{p}\cdot \frac{s}{ps-rq}}={\pm}\frac{ds\wedge \frac{s\,dp}{p^2}\wedge \frac{dr}{p}\wedge \frac{sr\,dq}{(ps-rq)^2}}{s\cdot \frac{s}{p}\cdot \frac{r}{p}\cdot \frac{s}{ps-rq}}. \end{equation}

Here e.g. $d(\Delta _{23})=\pm ds$ and $d(\Delta _{34})=\pm ({ds}/{p}-{s\,dp}/{p^2})$, but because we are already taking the wedge with $ds$, only the ${s\,dp}/{p^2}$ term appears in the numerator of (5.4). We see that after some cancellations, (5.4) indeed yields the same result as (5.2), in agreement with Proposition 5.7.

Consider the manifold $\operatorname {Gr}(k,n)\times \operatorname {Gr}(\ell ,n)$ for some $k+\ell +m=n$ as above with $m$ even. One can consider a top form $\omega _{k,n}\wedge \omega _{\ell ,n}$ on this manifold. We are now ready to state the main result of this section.

Proposition 5.7 has a short proof using the results of [Reference Marsh and ScottMS16, Reference Muller and SpeyerMS17], see § 11.4. On the other hand, the proof of Theorem 5.9 is quite involved and is given in § 11.5. See § 6.3 for an example.

5.2 The canonical form for lower positroid cells

Given an affine permutation $f\in \tilde {S_n}(-k,n-k)$, one can consider certain collections $\mathcal {C}\subset { (\begin {smallmatrix}[n]\\ k\end {smallmatrix} )}$ of $k$-element subsets of $[n]$ called clusters, see Definition 11.3. All clusters have the same size, and give a parametrization of $\Pi ^{>0}_{k+f}$.

Proposition 5.10 [OPS15] Let $\mathcal {C}$ be a cluster for $f\in \tilde {S_n}(-k,n-k)$. Then it has size

(5.5)\begin{equation} |\mathcal{C}|=k(n-k)+1-\operatorname{inv}(f). \end{equation}

The map $\Pi ^{>0}_{k+f} \to {{\mathbb {R}}\mathbb {P}}_{>0}^\mathcal {C}$ given by $X\mapsto \{\Delta _J(X)\mid J\in \mathcal {C}\}$ is a diffeomorphism.

Here ${{\mathbb {R}}\mathbb {P}}_{>0}^{\mathcal {C}}$ denotes the subset of the $(|\mathcal {C}|-1)$-dimensional real projective space where all coordinates are positive.

Thus we have various coordinate charts on $\Pi ^{>0}_{k+f}$, which now allows us to use Proposition 5.3 to define a canonical form on this manifold.

Definition 5.11 Let $\mathcal {C}=\{J_0,J_1,\dots ,J_N\}$ be a cluster for $f\in \tilde {S_n}(-k,n-k)$, where $n=k (n-k) -\operatorname {inv}(f)$. Suppose that the values of the Plücker coordinates are rescaled so that $\Delta _{J_0}(X')=1$ for $X'$ in the neighborhood of $X\in \Pi ^{>0}_{k+f}$. Then the canonical form $\omega _{k+f}$ on $\Pi ^{>0}_{k+f}$ is defined by

\[ \omega_{k+f}(X):={\pm} \frac{d(\Delta_{J_1}(X))\wedge d(\Delta_{J_2}(X))\wedge\dots\wedge d(\Delta_{J_N}(X))}{\Delta_{J_1}(X) \cdot\Delta_{J_2}(X)\cdots\Delta_{J_N}(X)}. \]

See [Reference LamLam16b, Theorem 13.2] for a proof. In fact, we will see later that the form $\omega _{k+f}$ can be obtained from the form $\omega _{k,n}=\omega _{k+\operatorname {id}_{0}}$ by subsequently taking residues (see § 12.1 for a definition). This implies Proposition 5.12, as well as a generalization of Theorem 5.9 to lower cells, which we now explain.

By (4.6), the set $\operatorname {Gr}_{>0}^\perp (k+m,n)$ can be identified with $\operatorname {Gr}_{>0}(\ell ,n)$ by changing the sign of every second column, and since the form $\omega _{\ell ,n}$ is only defined up to a sign, we can view $\omega _{\ell ,n}$ as a top rational form on $\operatorname {Gr}_{>0}^\perp (k+m,n)$ instead. The following is our main result regarding canonical forms.

Theorem 5.13 For $f\in \tilde {S_n}(-k,\ell )$, under the stacked twist map diffeomorphism

\[ \theta:\Pi^{{>}0}_{k+f}\times \operatorname{Gr}_{{>}0}^\perp(k+m,n)\to \operatorname{Gr}_{{>}0}^\perp(\ell+m,n)\times \Pi^{{>}0}_{\ell+f^{{-}1}} \]

(see (4.7)) we have

(5.6)\begin{equation} \theta^*(\omega_{k,n}\wedge\omega_{\ell+f^{{-}1}}) ={\pm} \omega_{k+f}\wedge\omega_{\ell,n}. \end{equation}

We discuss the application of Theorem 5.13 to the amplituhedron form in § 8.

6.1 Triangulations of a pentagon

As a warm-up, we let $k=1$, $\ell =2$, $m=2$, $n=5$. In this case, the amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a pentagon in ${{\mathbb {R}}\mathbb {P}}^2$ whose vertices are the five columns of $Z\in \operatorname {Gr}_{>0}(3,5)$. We write $f\in \tilde {S_n}$ in window notation, i.e. $f=[f(1),f(2),\dots , f(n)]$. Let $P_{n,k,m}\subset \tilde {S_n}(-k,n-k)$ be the set of affine permutations $f$ with $\operatorname {inv}(f)=k\ell =2$. We say that $f,g\in \tilde {S_n}$ are the same up to rotation if for some $s\in \mathbb {Z}$ we have $f(i+s)=g(i)+s$ for all $i\in \mathbb {Z}$. There are ten affine permutations in $P_{5,1,2}$, and up to rotation, they form two classes of five affine permutations in each, shown in Figure 1 (bottom). All of these ten affine permutations are $(n,k,m)$-admissible, and the images of the corresponding positroid cells are the $ (\begin {smallmatrix}5\\ 3 \end {smallmatrix} )=10$ triangles in a pentagon. The affine permutation $[2,1,3,5,4]$ in Figure 1 (bottom left) corresponds to the convex hull of the triangle with vertex set $\{1,3,4\}$ inside a pentagon, while the affine permutation $[3,1,2,4,5]$ in Figure 1 (bottom right) corresponds to the convex hull of a triangle with vertex set $\{1,4,5\}$. These two permutations are therefore $(5,1,2)$-compatible and together with the permutation $[1,2,5,3,4]$ (corresponding to a triangle with vertex set $\{1,2,3\}$) they form a triangulation of $\mathcal {A}_{n,k,m}(Z)$.

Figure 1. Up to rotation, there are three affine permutations in $\tilde {S_n}(-k,n-k)$ for $k=2,n=5$ with two inversions (top). There are only two such affine permutations in $\tilde {S_n}(-k,n-k)$ for $k=1,n=5$ (bottom).

6.2 Triangulations of the $\ell =1$ amplituhedron

We now study the ‘dual’ case $k=2$, $\ell =1$, $m=2$, $n=5$ in detail. There are 15 affine permutations in $P_{5,2,2}$, and they form three classes (up to rotation) of five permutations in each, shown in Figure 1 (top).

The permutations $f=[2,1,3,5,4]$ and $g=[2,3,1,4,5]$ belong to $\tilde {S_n}(-k,\ell )$ while $h=[1,4, 2,3,5]$ does not belong to $\tilde {S_n}(-k,\ell )$ because $h(2)=4>2+\ell$. Thus by Lemma 3.2, $h$ is not $(n,k,m)$-admissible (and it is indeed easy to check directly that $h$ is not $Z$-admissible for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$). Note that $h^{-1}$ does not correspond to any triangle inside a pentagon.

Let us now fix $f=[2,1,3,5,4]$ and consider $V\in \Pi ^{>0}_{k+f}$ and $W\in \operatorname {Gr}_{>0}^\perp (k+m,n)$ given by

(6.1)\begin{equation} V=\begin{pmatrix} 1 & 0 & 0 & -2 & -1\\ 0 & 1 & 1 & 2 & 0 \end{pmatrix};\quad W=( 1 \quad {-1}\quad 3\quad {-2}\quad 1). \end{equation}

The fact that $V\in \Pi ^{>0}_{k+f}$ can be easily checked using (10.4) while the fact that $W\in \operatorname {Gr}_{>0}^\perp (k+m,n)$ follows since $\operatorname {alt}(W)$ is totally positive, see (4.6). We would like to compute $({\tilde {W}},{\tilde V})=\theta (V,W)$. To do so, we first need to find the matrix ${\tilde {U}}=\tau (U)$. Let us introduce a diagonal $n\times n$ matrix $D={ \operatorname {diag}}(|\Delta _{J_1}(U)|,\dots ,|\Delta _{J_n}(U)|)$, where $J_j=\{j-\ell ,j-\ell +1,\dots ,j+k-1\}$. We will consider the matrix ${\tilde {U}}\cdot D$ rather than ${\tilde {U}}$ itself in order to clear the denominators (cf. (10.1)). The matrix $U$ is given in Figure 2 (top), so

(6.2)\begin{equation} D={ \operatorname{diag}}(|{-}2|, |4|, |{-}8|, |10|,|{-}4|)={ \operatorname{diag}}(2,4,8,10,4). \end{equation}

Thus the column ${\tilde {U}}(j)$ gets rescaled by the absolute value of $\Delta _{J_j}(U)$. The matrix ${\tilde {U}}\cdot D$ is given in Figure 2 (bottom): we encourage the reader to check that the column ${\tilde {U}}(j)$ is orthogonal to the columns $U(j)$ and $U(j+1)$, and that $\langle {\tilde {U}}(j),U(j-1)\rangle =(-1)^j$. (Recall also that $U(j+n)=(-1)^{k-1}U(j)$ and ${\tilde {U}}(j+n)=(-1)^{\ell -1}{\tilde {U}}(j)$, which is consistent with Figure 2.) Thus the output of the stacked twist map in this case is given by

(6.3)\begin{equation} {\tilde{W}}\cdot D=\begin{pmatrix} -1 & 4 & -8 & 2 & 0\\ 1 & 0 & -6 & 4 & -2 \end{pmatrix};\quad {\tilde V}\cdot D=\begin{pmatrix} 1 & 0 & 2 & 2 & 0 \end{pmatrix}. \end{equation}

Figure 2. The columns of $U$ (top) and of ${\tilde {U}}\cdot D$ (bottom) for $j=-1,0,\dots ,7$.

We observe that $\operatorname {alt}({\tilde {W}})$ is a totally positive $2\times 5$ matrix, and thus ${\tilde {W}}\in \operatorname {Gr}_{>0}^\perp (\ell +m,n)$. Similarly, ${\tilde V}$ is a totally nonnegative $1\times 5$ matrix. Moreover, we have ${\tilde V}\in \Pi ^{>0}_{\ell +f'}$ for $f'=[2,1,3,5,4]=f^{-1}$, as predicted by Theorem 4.7, part (iii).

We now consider the fibers of $Z$. Let $V,W$ be given by (6.1), and consider a $4\times 5$ matrix $Z:=W^\perp$. Suppose that $V'\cdot Z^\text {t}=V\cdot Z^\text {t}$ for some $V'\in \operatorname {Gr}_{\ge 0}(k,n)$, then we must have $V'=V+A\cdot W$ for some $2\times 1$ matrix $A= (\begin {smallmatrix} a \\ b \end {smallmatrix} )$ with $a,b\in \mathbb {R}$. The matrix $U':=\operatorname {stack}(V',W)$ is therefore given by

\[ U'= \begin{pmatrix} a + 1 & -a & 3 a & -2 a - 2 & a - 1 \\ b & -b + 1 & 3 b + 1 & -2 b + 2 & b \\ 1 & -1 & 3 & -2 & 1 \end{pmatrix}. \]

We then find the matrix ${\tilde {U}}':=\tau (U')$:

\[ {\tilde{U}}'\cdot D= \begin{pmatrix} -1 & 4 & -8 & 2 & 0 \\ 1 & 0 & -6 & 4 & -2 \\ a - b + 1 & -4 a & 8 a + 6 b + 2 & -2 a - 4 b + 2 & 2 b \end{pmatrix}. \]

Here $D={ \operatorname {diag}}(2,4,8,10,4)$ is unchanged since $U'$ is obtained from $U$ by row operations. Setting $({\tilde {W}}',{\tilde V}'):=\theta (V',W)$, we get

\[ {\tilde{W}}'\cdot D\!=\!\begin{pmatrix} -1 & 4 & -8 & 2 & 0\\ 1 & 0 & -6 & 4 & -2 \end{pmatrix};\quad {\tilde V}'\cdot D=( a - b \!+\! 1 \quad -4 a \quad 8 a \!+\! 6 b \!+\! 2 \quad -2 a - 4 b +\! 2\quad 2 b ). \]

(Note that by Proposition 4.8, ${\tilde {W}}'={\tilde {W}}$. As we will later see in Lemma 10.6, we have ${\tilde V}'={\tilde V}-A^\text {t}\cdot {\tilde {W}}$.) We can now write down the fiber $\operatorname {F}(V,Z)$ as the set of pairs $(a,b)$ for which $V'=V+A\cdot W$ is totally nonnegative. There are $ (\begin {smallmatrix}5\\ 2\end {smallmatrix} )=10$ minors of $V'$, and each of them turns out to be a linear polynomial in the variables $a,b$. For example, $\Delta _{13}(V')=(a+1)(3b+1)-3ab=a+3b+1.$

The ten lines $\Delta _I(V')=0$ in the $(a,b)$-plane are shown in Figure 3. For example, the line $\Delta _{13}(V')=0$ is the black line passing through the points $(-1,0)$ and $(0,-\frac 13)$. The region where $V'$ is totally nonnegative is the shaded pentagon. The vertices of this pentagon correspond to $V'$ being in $\Pi ^{>0}_{k+f'}$ for $f'$ equal to $f$ up to rotation (in particular, the vertex $(0,0)$ corresponds to $f$ itself). Thus $\operatorname {F}(V,Z)$ intersects $\Pi ^{>0}_{k+f'}$ for all such $f'$, and hence $f$ is not $(n,k,m)$-compatible with any of its four rotations. Observe that the inverses of these five permutations correspond to the triangles in a pentagon with vertex sets $\{1,3,4\}$, $\{2,4,5\}$, $\{1,3,5\}$, $\{1,2,4\}$, and $\{2,3,5\}$. No two of these five triangles can appear together in a triangulation of the pentagon, in agreement with part (2) of Theorem 3.5.

Figure 3. Each of the ten lines represents the set of pairs $(a,b)$ for which one of the minors of $V'$ is zero. We have $V'\in \operatorname {Gr}_{\ge 0}(k,n)$ precisely when $(a,b)$ belongs to the shaded region.

On the other hand, ${\tilde V}'$ is a $1\times 5$ matrix and thus has only five minors. However, the $(a,b)$-region where all of them are nonnegative turns out to be exactly the same as the shaded pentagon in Figure 3. More precisely, for each $j\in \mathbb {Z}$, we have $\Delta _{\{j\}}({\tilde V}'\cdot D)=\Delta _{\{j,j+1\}}(V')$ (see Lemma 11.5 for a general statement). For example,

\[ \Delta_1({\tilde V}'\cdot D)=\Delta_{12}(V')=a-b+1. \]

Since the boundary of the shaded region in Figure 3 only involves circular minors of $V'$ being zero, we see that the fibers $\operatorname {F}(V,Z)$ and $\operatorname {F}({\tilde V},{\tilde Z})$ are equal as regions in the $(a,b)$-plane. However, the five vertices of the pentagon $\operatorname {F}({\tilde V},{\tilde Z})$ correspond to the five rotations of $f^{-1}$. The main idea of our proof in § 10 is based on the observations made in this example.

6.3 Canonical forms for lower cells

Let $k=2$, $\ell =1$, $m=2$, $n=5$, $f=[2,3,1,5,4]$. Consider parametrizations of $\Pi ^{>0}_{k+f}$ and $\operatorname {Gr}_{>0}^\perp (k+m,n)$ by

\[ V:=\begin{pmatrix} 1 & 0 & 0 & -a & -c\\ 0 & 1 & 0 & b & 0 \end{pmatrix},\quad W:=( p \quad -q \quad 1 \quad -r \quad s ),\quad a,b,c,p,q,r,s>0. \]

We can take $\mathcal {C}:=\{\{1,2\},\{2,4\},\{4,5\},\{2,5\}\}$ to be a cluster for $V$:

\[ \Delta_{12}(V)=1,\quad \Delta_{24}(V)=a,\quad \Delta_{45}(V)=bc,\quad \Delta_{25}(V)=c, \]

and thus

\[ \omega_{k+f}={\pm}\frac{da\wedge d(bc)\wedge dc}{abc^2}={\pm}\frac{da\wedge db\wedge dc}{abc}. \]

Since $\omega _{\ell ,n}=\pm ({dp\wedge dq\wedge dr\wedge ds})/{pqrs}$, the form $\omega _{k+f}\wedge \omega _{\ell ,n}$ is given by

(6.4)\begin{equation} \omega_{k+f}\wedge \omega_{\ell,n}={\pm}\frac{da\wedge db\wedge dc\wedge dp\wedge dq\wedge dr\wedge ds}{abcpqrs}. \end{equation}

Let us now see what happens to this form when we apply the stacked twist map. Let $({\tilde {W}},{\tilde V}):=\theta (V,W)$. We find

\[ {\tilde{W}}:=\begin{pmatrix} -\frac{p}{c p + s} & 1 & -\frac{b}{a} & \frac{s}{c} & 0 \\ \frac{q}{c p + s} & 0 & -1 & \frac{c r + a s}{b c} & -\frac{1}{b} \\ \end{pmatrix},\quad {\tilde V}:=\begin{pmatrix} \frac{1}{c p + s} & 0 & 0 & 1 & 0 \end{pmatrix}. \]

Note that $f^{-1}=[3,1,2,5,4]$, and indeed ${\tilde V}\in \Pi ^{>0}_{\ell +f^{-1}}$, while ${\tilde {W}}\in \operatorname {Gr}_{>0}^\perp (\ell +m,n)$ because $\operatorname {alt}({\tilde {W}})\in \operatorname {Gr}_{>0}(k,n)$, see (4.6). Choosing a cluster $\mathcal {C}':=\{\{4\},\{1\}\}$ for $\Pi ^{>0}_{\ell +f^{-1}}$, we get that the pullback of $\omega _{\ell +f^{-1}}$ to $\Pi ^{>0}_{k+f}\times \operatorname {Gr}_{>0}^\perp (k+m,n)$ under $\theta$ is

(6.5)\begin{equation} \theta^\ast\omega_{\ell+f^{{-}1}}={\pm} \frac{d (1/(c p + s))}{1/(c p + s)} =\frac{\pm(p\, dc +c\,dp+ds)}{c p + s}. \end{equation}

Let us now compute the pullback of $\omega _{k,n}$ to $\Pi ^{>0}_{k+f}\times \operatorname {Gr}_{>0}^\perp (k+m,n)$ under $\theta$. Applying row operations to ${\tilde {W}}$, we transform it into

\[ {\tilde{W}}'=\begin{pmatrix} -\frac{a p + b q}{{(c p + s)} a} & 1 & 0 & -\frac{r}{a} & \frac{1}{a} \\ -\frac{q}{c p + s} & 0 & 1 & -\frac{c r + a s}{b c} & \frac{1}{b} \end{pmatrix}. \]

The submatrix of ${\tilde {W}}'$ with column set $\{2,3\}$ is an identity matrix, and thus we can use Definition 5.1 (together with the fact that $\omega _{k,n}$ is cyclically invariant) to calculate

(6.6)\begin{equation} \theta^\ast \omega_{k,n}={\pm}\frac{d^{2\times 3}\begin{pmatrix} -\frac{a p + b q}{{(c p + s)} a} & -\frac{r}{a} & \frac{1}{a} \\ -\frac{q}{c p + s} & -\frac{c r + a s}{b c} & \frac{1}{b} \end{pmatrix}}{\Delta_{k}^{\operatorname{circ}}({\tilde{W}}')}. \end{equation}

Computing the circular minors of ${\tilde {W}}'$, we get

\[ \Delta_{12}({\tilde{W}}')=\frac{q}{c p + s},\quad \Delta_{23}({\tilde{W}}')=1,\quad \Delta_{34}({\tilde{W}}')=\frac{r}{a},\quad \Delta_{45}({\tilde{W}}')=\frac{s}{bc},\quad \Delta_{15}({\tilde{W}}')=\frac{-p}{b(c p + s)}. \]

The denominator in the right-hand side of (6.6) is therefore

\[ \Delta_{ k}^{\operatorname{circ}}({\tilde{W}}')=\frac{-p q r s}{(c p + s)^2 a b^2 c}. \]

A longer computation is needed to find the numerator in the right-hand side of (6.6), which in the end produces the following answer (the symbol $\wedge$ is omitted):

(6.7)\begin{equation} \theta^\ast \omega_{k,n}={\pm} \bigg(\frac{d a \, d b \, d c \, d p \, d r \, d s}{a b c p r s} -\frac{d a \, d b \, d c \, d q \, d r \, d s}{a b c q r s} + \frac{s\,d a \, d b \, d c \, d p \, d q \, d r}{{ (c p + s)} a b c p q r} + \frac{d a \, d b \, d p \, d q \, d r \, d s}{{(c p + s)} a b p q r}\bigg). \end{equation}

The reader is invited to check that the form $\theta ^\ast (\omega _{k,n}\wedge \omega _{\ell +f^{-1}})=\theta ^\ast \omega _{k,n}\wedge \theta ^\ast \omega _{\ell +f^{-1}}$, obtained by taking the wedge of the right-hand sides of (6.5) and (6.7), is equal to $\pm \omega _{k+f}\wedge \omega _{\ell ,n}$, which is given on the right-hand side of (6.4).

7.1 Subdivisions

Let $Z \in \operatorname {Gr}_{>0}(k+m,n)$. We denote by $\leq$ the Bruhat order on $\tilde {S_n}$, defined as follows. We say that $t\in \tilde {S_n}$ is a transposition if we have $t(i)\neq i$ for precisely two indices $i\in [n]$, and then we define the covering relation $\lessdot$ of the Bruhat order by

(7.1)\begin{equation} f\lessdot g\quad\text{if and only if}\quad f=g\circ t\text{ for some transposition }t\in\tilde{S_n} \hbox{ and } \operatorname{inv}(g) = \operatorname{inv}(f) + 1. \end{equation}

Here $\circ$ denotes the composition of bijections $\mathbb {Z}\to \mathbb {Z}$. Given two affine permutations $f,g\in \tilde {S_n}(-k,n-k)$, we have $f \leq g$ if and only if $\Pi ^{\ge 0}_{k+f} \subseteq \Pi ^{\ge 0}_{k+g}$, see e.g. [Reference LamLam16b, Theorem 8.1]. In particular, this implies the following property that was used in the proof of Theorem 3.5: for $f,g\in \tilde {S_n}(-k,\ell )$, we have

(7.2)\begin{equation} \Pi^{{>}0}_{k+f} \subset \Pi^{\ge0}_{k+g} \quad\text{if and only if}\quad \Pi^{{>}0}_{\ell+f^{{-}1}} \subset \Pi^{\ge0}_{\ell+g^{{-}1}}. \end{equation}

It thus follows that a ($Z$- or $(n,k,m)$-) triangulation of the amplituhedron is a ($Z$- or $(n,k,m)$-) subdivision $f_1,\dots ,f_N\in \tilde {S_n}(-k,n-k)$ of $\mathcal {A}_{n,k,m}(Z)$ such that the permutations $f_1,\dots ,f_N$ are ($Z$- or $(n,k,m)$-)admissible.

The following result gives an analog to part (3) of Theorem 3.5, with the same proof, which we omit.

7.2 The universal amplituhedron

Let $\textrm {Fl}(\ell ,k+\ell ;n)$ be the two-step flag variety consisting of pairs of subspaces $W \subset U \subset \mathbb {R}^n$ such that $\operatorname {dim}(W) =\ell$ and $\operatorname {dim}(U) = k+\ell$. Define the universal amplituhedron $\mathcal {A}_{n,k,m} \subset \textrm {Fl}(\ell ,k+\ell ;n)$ to be the closure (in the analytic topology)

\[ \mathcal{A}_{n,k,m}:= \overline{ \{(W,U) \mid W\in \operatorname{Gr}_{{>}0}^\perp(k+m,n) \text{ and } \operatorname{span}(V,W) = U \hbox{ for some } V \in \operatorname{Gr}_{\ge 0}(k,n)\}}. \]

Let

\[ \pi: \textrm{Fl}(\ell,k+\ell;n) \to \operatorname{Gr}(\ell,n) \]

be the projection $(W,U)\mapsto W$. Let us fix $Z \in \operatorname {Gr}_{>0}(k+m,n)$ and let $W:=Z^\perp$. The fiber $\pi ^{-1}(W) \cap \mathcal {A}_{n,k,m}$ can be identified with the $Z$-amplituhedron $\mathcal {A}_{n,k,m}(Z)$ via the map sending $U$ to the row span of the $(k+m)\times (k+m)$ matrix $U\cdot Z^\text {t}$ (which has rank $k$). Thus we have

\[ \operatorname{dim}(\mathcal{A}_{n,k,m}) = (k+m)(n-k-m) + km = km + k\ell + \ell m. \]

We have a map

\[ \mathcal{Z}: \operatorname{Gr}_{\ge 0}(k,n)\times\operatorname{Gr}_{{>}0}^\perp(k+m,n) \to \mathcal{A}_{n,k,m} \]

given by $(V,W) \mapsto (W,\operatorname {span}(V,W))$. Since the matrix $U\cdot Z^\text {t}$ has rank $k$, and the subspace $U$ contains the kernel $W$ of $Z$, it follows that the subspace $\operatorname {span}(V,W)$ always has dimension $k+\ell$. For $f \in \tilde {S_n}(-k,n-k)$, define

\begin{align*} \mathcal{A}^{{>}0}_{k+f} &:= \mathcal{Z}(\Pi^{{>}0}_{k+f} \times \operatorname{Gr}_{{>}0}^\perp(k+m,n)) \subset \mathcal{A}_{n,k,m}, \\ \mathcal{A}^{\ge0}_{k+f} &:= \overline{\mathcal{A}^{{>}0}_{k+f}} \subset \mathcal{A}_{n,k,m}. \end{align*}

Define $\mathcal {A}_{n,k,\ge m} \subset \mathcal {A}_{n,k,m}$ by

\[ \mathcal{A}_{n,k,\ge m}:= \{(W,U) \mid W\in \operatorname{Gr}_{{>}0}^\perp(k+m,n) \text{ and } \operatorname{span}(V,W) = U \hbox{ for some } V \in \operatorname{Gr}_{\ge m}(k,n)\}. \]

The map $\mathcal {Z}$ restricts to a map $\mathcal {Z}: \operatorname {Gr}_{\ge m}(k,n)\times \operatorname {Gr}_{>0}^\perp (k+m,n)\to \mathcal {A}_{n,k,\ge m}$. Let $\mathcal {Z}^{\operatorname {op}}:\operatorname {Gr}_{>0}^\perp (\ell +m,n) \times \operatorname {Gr}_{\ge m}(k,n) \to \mathcal {A}_{n,\ell ,\ge m}$ denote the similar map sending $({\tilde {W}},{\tilde V})$ to $({\tilde {W}},\operatorname {span}({\tilde {W}}, {\tilde V}))$.

Theorem 7.3 The stacked twist map

\[ \theta: \operatorname{Gr}_{\ge m}(k,n)\times\operatorname{Gr}_{{>}0}^\perp(k+m,n) \to \operatorname{Gr}_{{>}0}^\perp(\ell+m,n)\times \operatorname{Gr}_{\ge m}(\ell,n) \]

descends to a homeomorphism ${\tilde \theta }: \mathcal {A}_{n,k,\ge m} \to \mathcal {A}_{n,\ell ,\ge m}$, forming the following commutative diagram.

(7.3)

The map ${\tilde \theta }$ restricts to homeomorphisms ${\tilde \theta }:\mathcal {A}^{>0}_{k+f} \to \mathcal {A}^{>0}_{\ell +f^{-1}}$ for each $f\in \tilde {S_n}(-k,\ell )$.

7.3 Subdivisions of the universal amplituhedron

We recast the notions of $(n,k,m)$-triangulations/subdivisions in the language of the universal amplituhedron.

The following is an analog of Definition 7.1.

Proof. Let

\[ \mathcal{A}'_{n,k,m}:= \{(W,U) \mid W\in \operatorname{Gr}_{{>}0}^\perp(k+m,n) \text{ and } \operatorname{span}(V,W) = U \hbox{ for some } V \in \operatorname{Gr}_{\ge 0}(k,n)\} \]

so that by definition $\mathcal {A}_{n,k,m}$ is the closure of $\mathcal {A}'_{n,k,m}$. Suppose that $f_1,\dots ,f_N$ form an $(n,k,m)$-subdivision of the amplituhedron. Thus the images $Z(\Pi ^{>0}_{k+f_1}),\dots ,Z(\Pi ^{>0}_{k+f_N})$ form a dense subset of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$. We claim that $\mathcal {A}^{>0}_{k+f_1},\dots ,\mathcal {A}^{>0}_{k+f_N}$ form a dense subset of $\mathcal {A}'_{n,k,m}$, from which it follows that their union is dense in $\mathcal {A}_{n,k,m}$. Indeed, take a pair $(W,U)\in \mathcal {A}'_{n,k,m}$ and let $Z:=W^\perp \in \operatorname {Gr}_{>0}(k+m,n)$. There exists $1\leq s\leq N$ such that $U\cdot Z^\text {t}$ belongs $Z(\Pi ^{\ge 0}_{k+f_s})$, and thus $(W,U)$ belongs to the closure of $\mathcal {A}^{>0}_{k+f_s}$.

Conversely, suppose that $\mathcal {A}^{>0}_{k+f_1},\dots ,\mathcal {A}^{>0}_{k+f_N}$ form a dense subset of $\mathcal {A}_{n,k,m}$. Fix $Z\in \operatorname {Gr}_{>0} (k+m,n)$, let $W:=Z^\perp$, $V\in \operatorname {Gr}_{\ge 0}(k,n)$, $U:=\operatorname {span}(V,W)$, and consider the point $Y:=V\cdot Z^\text {t}$ of the amplituhedron $\mathcal {A}_{n,k,m}(Z)$. We want to show that $Y$ belongs to the closure of $Z(\Pi ^{>0}_{k+f_s})$ for some $1\leq s\leq N$. We know that $(W,U)$ belongs to the closure of $\mathcal {A}^{>0}_{k+f_s}$ for some $1\leq s\leq N$. By assumption, there exists a sequence $(W_1,U_1),(W_2,U_2),\dots$ converging to $(W,U)$ in $\textrm {Fl}(\ell ,k+\ell ;n)$ such that $(W_i,U_i)\in \mathcal {A}^{>0}_{k+f_s}$ for all $i\geq 1$. By the definition of $\mathcal {A}^{>0}_{k+f_s}$, for each $i\geq 1$ there exists $V_i\in \Pi ^{>0}_{k+f_s}$ such that $U_i = \operatorname {span}(V_i,W_i)$. Let us now consider the sequence $(W,\operatorname {span}(V_i,W))\in \mathcal {A}^{>0}_{k+f_s}$ for $i\geq 1$. We claim that a subsequence of the points $\operatorname {span}(V_i,W)$ converges to $U$ in $\operatorname {Gr}(k+\ell ,n)$, and thus $(W,U)$ belongs to the closure of $\mathcal {A}^{>0}_{k+f_s}$. The points $V_i$ all belong to the compact set $\Pi ^{\ge 0}_{k+f_s}$ and therefore a subsequence $V_{i_1},V_{i_2},\ldots$ converges to $V' \in \Pi ^{\ge 0}_{k+f_s}$. Any subspace $V''\in \operatorname {Gr}(k,n)$ in the neighborhood of $V'$ satisfies $V''\cap W=\{0\}$, and thus the map $V''\mapsto \operatorname {span}(V'',W)$ is continuous at $V'$. We get that $\operatorname {span}(V_{i_1},W) = U_{i_1}, \operatorname {span}(V_{i_2},W) = U_{i_2}, \ldots$ converges to $\operatorname {span}(W,V')$ in $\operatorname {Gr}(k+\ell ,n)$, and we must also have $\operatorname {span}(W,V') = U$.

We have shown the first part of the lemma, and the proof of the second part is completely analogous.

The following result follows from Lemmas 7.4 and 7.6 together with Theorem 3.5.

A similar result applies to triangulations of $g \in \tilde {S_n}(-k,n-k)$.

8.1 Degree $1$ cells

Thus in order for $f$ to have $Z$-degree $1$, we must have $\operatorname {inv}(f)=k\ell$. We note that if $f$ satisfies $\operatorname {dim} \Pi _{k+f}^\mathbb {C}=\operatorname {dim} Z(\Pi _{k+f}^\mathbb {C})=km$ for a generic $Z\in \operatorname {Gr}_\mathbb {C}(k,k+m)$ then $f$ is said to have kinematical support, see [Reference Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov and TrnkaABC16, § 10].

Thus if $f$ has $Z$-degree $1$ then $Z$ is injective on an open dense subset of the positroid cell $\Pi ^{>0}_{k+f}$, but not necessarily on the whole positroid cell.

It is clear that (c) implies (b). In what follows, we restrict ourselves to only $(n,k,m)$-admissible affine permutations that have degree $1$. It is not difficult to see that Conjecture 8.4 holds for $k =1$, since in this case every affine permutation $f\in \tilde {S_n}(-k,n-k)$ with $\operatorname {inv}(f)=k\ell$ satisfies (a), (b), and (c). Let us now explain why Conjecture 8.4 also holds for $\ell = 1$. Let $\ell =1$, and consider an affine permutation $f\in \tilde {S_n}(-k,\ell )$ with $\operatorname {inv}(f)=k\ell$. Then $f$ satisfies (a) by Theorem 3.5, part (1). Next, $f$ satisfies (b) by Proposition 8.5 below. Finally, we have already noted above that $f^{-1}$ satisfies (c), i.e. that $f^{-1}\in \tilde {S_n}(-\ell ,k)$ has ${\tilde Z}$-degree $1$ for all ${\tilde Z} \in \operatorname {Gr}_{>0}(\ell +m,n)$. In particular, it is easy to see that for any ${\tilde V}\in \operatorname {Gr}_{>0}(\ell ,n)$, there exists a unique ${\tilde V}'\in \Pi _{\ell +f^{-1}}^\mathbb {C}$ such that ${\tilde V}\cdot {\tilde Z}^\text {t}={\tilde V}'\cdot {\tilde Z}^\text {t}$. By Lemma 10.6, it follows that for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ and $V\in \operatorname {Gr}_{>0}(k,n)$, there exists a unique $V'\in \Pi _{k+f}^\mathbb {C}$ such that $V\cdot Z^\text {t}=V'\cdot Z^\text {t}$. Since this holds for all $V\in \operatorname {Gr}_{>0}(k,n)$, it must hold for $V$ belonging to a Zariski dense subset of $\operatorname {Gr}_\mathbb {C}(k,n)$, and thus $f$ indeed has $Z$-degree $1$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$. This shows that Conjecture 8.4 holds for $\ell =1$.

We will see in Lemma 9.3 that if $f\in \tilde {S_n}(-k,n-k)$ has degree $1$ then we must have $f\in \tilde {S_n}(-k,\ell )$. By a result of [Reference LamLam16a], $f$ has degree $1$ if and only if the coefficient of a certain rectangular Schur function in the corresponding affine Stanley symmetric function is equal to $1$. An important corollary of this is the following result that we prove in § 12.3.

Note that (without knowing Conjecture 8.4) it may not be true that an $(n,k,m)$-triangulation of degree $1$ is a $Z$-triangulation of degree $1$ for all $Z\in \operatorname {Gr}_{>0}(k+m,n)$, but only for all $Z$ belonging to an open dense subset of $\operatorname {Gr}_{>0}(k+m,n)$.

8.2 The amplituhedron form

For us, the crucial property of a degree $1$ map is that it allows one to define the pushforward form $\omega _{Z(\Pi _{k+f}^{\mathbb {C}})}$ on $\operatorname {Gr}_\mathbb {C}(k,k+m)$ via a pullback. Suppose that $f\in \tilde {S_n}(-k,n-k)$ has $Z$-degree $1$ and $\operatorname {inv}(f) = k\ell$. Thus $Z$ restricts to an isomorphism between Zariski open subsets $A\subset \Pi _{k+f}^\mathbb {C}$ and $B\subset \operatorname {Gr}_\mathbb {C}(k,k+m)$. We define the rational top form $\omega _{Z(\Pi _{k+f}^{\mathbb {C}})}$ on $B$, and by extensionFootnote ⁴ on $\operatorname {Gr}_\mathbb {C}(k,k+m)$, as the pullback

(8.1)\begin{equation} \omega_{Z(\Pi_{k+f}^{\mathbb{C}})}=Z_\ast\omega_{k+f}:=(Z^{{-}1})^\ast \omega_{k+f}, \end{equation}

where $Z^{-1}:B\to A$ denotes the inverse of the isomorphism $Z:A\to B$.

Recall from § 5.2 that for $f\in \tilde {S_n}(-k,n-k)$, the canonical form $\omega _{k+f}$ on $\Pi ^{>0}_{k+f}$ is defined up to a sign. For the purposes of the amplituhedron form, we need to add up the forms $\omega _{Z(\Pi _{k+f}^{\mathbb {C}})}$ for various $f$. Thus it is necessary pick a particular sign of $\omega _{k+f}$ for each $f\in \tilde {S_n}(-k,n-k)$.

Thus for an $(n,k,m)$-admissible affine permutation of degree $1$, we can distinguish between $Z$ being orientation reversing or orientation preserving on the real manifold $\Pi ^{>0}_{k+f}$.

Let us choose some rational top form $\omega ^{\operatorname {ref}}_{k,k+m}$ on $\operatorname {Gr}(k,k+m)$ that is non-vanishing on $\mathcal {A}_{n,k,m}(Z)$. This is always possible since $\mathcal {A}_{n,k,m}(Z)$ lies completely within some open Schubert cell (diffeomorphic to the orientable manifold $\mathbb {R}^{km})$ of $\operatorname {Gr}(k,k+m)$, see [Reference LamLam16b, Proposition 15.2 and Lemma 15.6]. We call $\omega ^{\operatorname {ref}}_{k,k+m}$ the reference form.

We note that $\omega _{k+f}$ is non-vanishing on $\Pi ^{>0}_{k+f}$. By Proposition 8.8, we may and will pick a sign for $\omega _{k+f}$ so that the pushforward form $\omega _{Z(\Pi _{k+f}^{\mathbb {C}})}$ is positive with respect to the fixed reference form $\omega ^{\operatorname {ref}}_{k,k+m}$ on $\operatorname {Gr}(k,k+m)$.

Definition 8.10 For $Z\in \operatorname {Gr}_{>0}(k+m,n)$ and a $Z$-triangulation $\mathcal {T}=\{f_1,\dots ,f_N\}$ of degree $1$, define the amplituhedron form $\omega ^{{({\mathcal {T}})}}_{\mathcal {A}_{n,k,m}(Z)}$ on $\operatorname {Gr}_\mathbb {C}(k,k+m)$ by

(8.2)\begin{equation} \omega^{{({\mathcal{T}})}}_{\mathcal{A}_{n,k,m}(Z)}:=\sum_{s=1}^N\omega_{Z(\Pi_{k+f_s}^\mathbb{C})}. \end{equation}

We emphasize again that the sign of every summand in the right-hand side of the above equation is chosen so that the signs of $\omega _{Z(\Pi _{k+f_s}^\mathbb {C})}$ and $\omega ^{\operatorname {ref}}_{k,k+m}$ coincide on $Z(\Pi ^{>0}_{k+f_s})$ for each $1\leq s\leq N$. The following has been numerically verified in the physics literature in a number of cases when $m = 4$.

When Conjecture 8.11 holds, we denote by $\omega _{\mathcal {A}_{n,k,m}(Z)}$ the triangulation independent amplituhedron form. This conjectural amplituhedron form is the one presented in the original definition of the amplituhedron [Reference Arkani-Hamed and TrnkaAT14]. A different conjectural characterization of the amplituhedron form via ‘positive geometries’ is given in [Reference Arkani-Hamed, Bai and LamABL17]. It is an important open problem to give a formula for $\omega _{\mathcal {A}_{n,k,m}(Z)}$ without referring to triangulations. See Conjecture 13.5 for a related discussion.

8.3 The universal amplituhedron form

The form $\omega ^{{({\mathcal {T}})}}_{\mathcal {A}_{n,k,m}(Z)}$ is a top form on $\operatorname {Gr}_\mathbb {C}(k,k+m)$ that depends on $Z$. We now let $Z$ vary over $\operatorname {Gr}_{>0}(k+m,n)$, and define a closely related top form $\omega ^{{({\mathcal {T}})}}_{\mathcal {A}_{n,k,\ge m}}$ on $\textrm {Fl}_\mathbb {C}(\ell ,k+\ell ;n)$.

As in the construction of $\omega ^{\operatorname {ref}}_{k,k+m}$, each $Z$-amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is contained in an open Schubert cell $C(Z)$ of $\operatorname {Gr}(k,k+m)$. Thus the interior of the universal amplituhedron is contained in the orientable submanifold of $\textrm {Fl}(\ell ,k+\ell ;n)$, which maps to $\operatorname {Gr}_{>0}(k+m,n)$ with fiber $C(Z)$ over each $Z \in \operatorname {Gr}_{>0}(k+m,n)$. (Note that $\operatorname {Gr}_{>0}(k+m,n)$ is diffeomorphic to an open ball and is in particular orientable and contractible.) We may thus choose a reference top form $\omega ^{\operatorname {ref}}_{\textrm {Fl}(\ell ,k+\ell ;n)}$ on $\textrm {Fl}(\ell ,k+\ell ;n)$ so that it is non-vanishing on the interior of $\mathcal {A}_{n,k,m}$.

Proposition 8.12 Suppose that $f\in \tilde {S_n}(-k,n-k)$ has degree $1$. Then the rational map

\[ \mathcal{Z}:\Pi_{k+f}^\mathbb{C} \times\operatorname{Gr}_\mathbb{C}(\ell,n)\dashrightarrow \textrm{Fl}_\mathbb{C}(\ell,k+\ell;n) \]

sending $(V,W)$ to $(W, \operatorname {span}(V,W))$ is a birational map.

By Proposition 8.12, if $f$ has degree $1$ then the pushforward form $\mathcal {Z}_* (\omega _{k+f} \wedge \omega _{\ell ,n})$ on $\textrm {Fl}_\mathbb {C}(\ell ,k+\ell ;n)$ can be defined as the pullback via the inverse of $\mathcal {Z}$ (see (8.1)). By Proposition 8.13, if $f$ is furthermore $(n,k,m)$-admissible we may choose a sign for $\omega _{k+f}$ such that the form $\mathcal {Z}_* (\omega _{k+f} \wedge \omega _{\ell ,n})$ has the same sign as $\omega ^{\operatorname {ref}}_{\textrm {Fl}(\ell ,k+\ell ;n)}$ when restricted to $\mathcal {A}^{>0}_{k+f}$.

Definition 8.14 Given an $(n,k,m)$-triangulation $\mathcal {T}=\{f_1,\dots ,f_N\}\subset \tilde {S_n}(-k,\ell )$ of degree $1$, the universal amplituhedron form $\omega ^{{({\mathcal {T}})}}_{\mathcal {A}_{n,k,\ge m}}$ on $\textrm {Fl}_\mathbb {C}(\ell ,k+\ell ;n)$ is defined by

(8.3)\begin{equation} \omega^{{({\mathcal{T}})}}_{\mathcal{A}_{n,k,\ge m}}=\sum_{s=1}^N \mathcal{Z}_* (\omega_{k+f_s} \wedge \omega_{\ell,n}). \end{equation}

Here the signs of the terms in the right-hand side are chosen in the same fashion as in (8.2).

It follows from Proposition 8.19 below that Conjecture 8.15 is equivalent to Conjecture 8.11 for all $Z$ such that the triangulations in question have $Z$-degree $1$. Also, Conjectures 8.11 and 8.15 are known for $k = 1$: a triangulation independent construction of the polytope form is given in [Reference Arkani-Hamed, Bai and LamABL17]. It follows from Theorem 8.16 that the two conjectures also hold for $\ell =1$.

We state our main result on the amplituhedron form.

Theorem 8.16 Suppose that we are given an $(n,k,m)$-triangulation $\mathcal {T}=\{f_1,\dots ,f_N\}\subset \tilde {S_n}(-k,\ell )$ of degree $1$, and let $\mathcal {T}':=\{f_1^{-1},\dots ,f_N^{-1}\}\subset \tilde {S_n}(-\ell ,k)$ be the corresponding $(n,\ell ,m)$-triangulation of degree $1$ (cf. Corollary 8.7). Then the stacked twist map preserves the universal amplituhedron form:

(8.4)\begin{equation} {\tilde\theta}^\ast\omega^{{({\mathcal{T}'})}}_{\mathcal{A}_{n,\ell,\ge m}}={\pm}\omega^{{({\mathcal{T}})}}_{\mathcal{A}_{n,k,\ge m}}. \end{equation}

In (8.4), the map ${\tilde \theta }$ can be considered either as a diffeomorphism between the interiors of $\mathcal {A}_{n,k,\ge m}$ and $\mathcal {A}_{n,\ell ,\ge m}$, or as a rational map ${\tilde \theta }: \textrm {Fl}_\mathbb {C}(\ell ,k+\ell ,n) \dashrightarrow \textrm {Fl}_\mathbb {C}(k,k+\ell ;n)$.