\section{The G-amplituhedron and Parke--Taylor functions}\label{sec:G}
In this section, we first give the definition of $w$-chambers in $\Ank$. Then we
describe the interpretation of some of our results in the \emph{G-amplituhedron} $\mathcal{G}_{n,k,2}$, which is a ``$Z$-independent" version of $\Ank$ introduced in \cite{LPW}. In this context, the Parke--Taylor functions $\PT(w)$ have a geometric interpretation as \emph{canonical functions} of certain \emph{positive geometries}. 

Recall the definition of twistor coordinates $\llrr{Yab}$ from \Cref{def:tw_coords}.
\begin{definition}[{\cite[Definition 10.7]{LPW}}]\label{def:ampchamber}
	Fix $Z \in \Mat_{n, k+2}^{>0}$. Let $w \in D_{k+1,n}$, $I_a=\cdes(w^{(a)})$, and define
	\begin{equation*}
	(\cham{w})^{\circ}:=\bigl\{ Y \in \Gr_{k,k+2}\colon	\sgn \llrr{Y ab}= (-1)^{|I_a \cap [a, b-1]|-1} \mbox{ for all } a<b  \bigr\} \label{eq:sgnCond}
	\end{equation*}
	The closure $\cham{w}:= \overline{(\cham{w})^{\circ}}$ in $\Gr_{k,k+2}$ is a \emph{$w$-chamber} of $\Ank$.
\end{definition}

Depending on the choice of $Z$, $\cham{w}$ may be empty \cite[Section 11.3]{PSW}, though there always exists a $Z$ such that $\cham{w}$ is nonempty \cite[Theorem 11.5]{PSW}. To avoid this irregularity, one may pass to the \emph{G-amplituhedron} and define $w$-chambers there.


	Given $v\in \R^n$, let
	$\mbox{var}(v)$ be the number of times the entries of $v$ changes sign 
	when we read the entries from left to right and ignore any
	zeros. For example, if $v:=(4,-1,0,-2)$ then $\mbox{var}(v)=1$.
\begin{definition}[{\cite[Definition 11.9]{LPW}}]
	Fix $k<n$ and let 
	\begin{align*}
		\mathcal{G}_{n,k,2}^{\circ}:= \{z\in \Gr_{2,n} \ &\colon \  
		P_{i,i+1}(z)>0 \text{ for }1 \leq i \leq n-1,
		(-1)^kP_{1n}(z) >0,\\
		&\text{ and } 
		\var((P_{12}(z), 
		P_{13}(z),  \dots 
		P_{1n}(z))=k\}.
	\end{align*}
	The closure $\mathcal{G}_{n,k,2}:=\overline{\mathcal{G}_{n,k,2}^{\circ}}$ in $\Gr_{2,n}$ is
	the \emph{G-amplituhedron}. The \emph{total G-amplituhedron} is the union
 \[\mathcal{G}_n := \bigcup_{k=0}^{n-2}\mathcal{G}_{n,k,2} \subset \Gr_{2,n}.\]
\end{definition}
One should think of $\mathcal{G}_n$ as an analogue of the hypercube: just as the hypercube is the union of all (projected) hypersimplices, the total G-amplituhedron is the union of all G-amplituhedra.

Motivated by the decomposition of $\Ank$ into $w$-chambers, we analogously define $w$-chambers for $\mathcal{G}_{n,k,2}$ which are (closures of) certain uniform oriented matroid strata in $\mbox{Gr}_{2,n}$.
\begin{definition}[{\cite[Definition 11.13]{LPW}}]\label{def:ampchamber2}
	Let $w \in D_{k+1,n}$, $I_a=\cdes(w^{(a)})$, and define
	\begin{equation*}
	(\gcham{w})^{\circ}:=\bigl\{ z \in \Gr_{2,n}\colon	\sgn P_{ ab}(z)= (-1)^{|I_a \cap [a, b-1]|-1} \mbox{ for all } a<b  \bigr\} \label{eq:sgnCond}
	\end{equation*}
	The closure $\gcham{w}:= \overline{(\gcham{w})^{\circ}}$ in $\Gr_{2,n}$ is a \emph{$w$-chamber} of the G-amplituhedron $\mathcal{G}_{n,k,2}$.
\end{definition}

\begin{remark}
 In analogy with the amplituhedron $\mathcal{A}_{n,k,2}(Z)$, the G-amplituhedron $\mathcal{G}_{n,k,2}$ is the union of the $w$-chambers $\gcham{w}$, with $w$ ranging over  $D_{k+1,n}$ \cite[Theorem 11.21]{PSW}. The amplituhedron $\mathcal{A}_{n,k,2}(Z)$ can be seen as a $2k$-dimensional linear slice of $\mathcal{G}_{n,k,2}$ and the $w$-chamber $\cham{w}$ in $\mathcal{A}_{n,k,2}(Z)$ as a $2k$-dimensional linear slice of the $w$-chamber $\gcham{w}$ (see \cite[Proposition 11.11, Remark 11.17]{PSW}).   
\end{remark}


\begin{proposition}[{\cite[Proposition 11.15]{PSW}}] \label{proposition:GchamberNonempty}
	Let $w \in D_{k+1, n}$. Then 
	$(\gcham{w})^{\circ}$ 
	is nonempty and is contractible.
\end{proposition}
From the proof of \cite[Proposition 11.15]{PSW}, we can find points in $(\gcham{w})^{\circ}$ explicitly. Consider $n$ vectors $v_1, v_2, \dots, v_n$ in $\mathbb{R}^2$ so that the matrix
\begin{equation}\label{eq:app_matrix}
  \begin{pmatrix}
		v_1 & v_{w_1^{(1)}}& v_{w_2^{(1)}} &\dots& v_{w_{n-1}^{(1)}}
	\end{pmatrix} \in \mbox{Mat}^{>0}_{2,n},  
\end{equation}
 equivalently, the vectors $v_1, v_{w_1^{(1)}}, v_{w_2^{(1)}}, \dots, v_{w_{n-1}^{(1)}}$ drawn in a plane are ordered counterclockwise. (Recall that $w^{(1)}$ denotes the rotation of $w$ ending in $1$, and hence $w^{(1)}_n=1$.)
	Now, set $z_1:=v_1$ and $z_b := (-1)^{|I_1 \cap [1, b-1]|-1}v_b$, for $b>2$. Then
 \begin{equation}\label{eq:ptInChamber}
		z=\begin{pmatrix}
		z_1 & z_2& z_3 &\dots& z_n
	\end{pmatrix}\end{equation}
	represents a point in $(\gcham{w})^{\circ}$. Moreover, all points in $(\gcham{w})^{\circ}$ arise this way.

\begin{example} \label{ex:realizable}
	Let $w=2564137\in D_{4,7}$, so $w^{(1)}=3725641$. 
	We have $I_1 = \{1,2,4,6\}$ and
	\begin{align*}
		(v_1, v_{w_1^{(1)}}, v_{w_2^{(1)}}, v_{w_3^{(1)}}, v_{w_4^{(1)}}, v_{w_5^{(1)}}, v_{w_6^{(1)}})
		&= (v_1, v_3, v_7, v_2, v_5, v_6, v_4)  
		= \begin{pmatrix}
			1 & 1 & 1 & 1 & 1 & 1 & 1 \\
			1 & 2 & 3 & 4 & 5 & 6 & 7
		\end{pmatrix}.
	\end{align*}
	We then get
	\begin{equation*}
		z = \begin{pmatrix}
			1 & 1 & -1 & -1 & 1 & 1 & -1 \\
			1 & 4 & -2 & -7 & 5 & 6 & -3
		\end{pmatrix} \in (\gcham{w})^{\circ}.
	\end{equation*}
\end{example}
\begin{remark}
 The positive Grassmannian $\Gr^{\geq 0}_{2,n}$ is a \emph{positive geometry} in the sense of \cite{ABL,Lam:2022yly}. Furthermore, the Parke--Taylor function $\PT(\mathbf{I}_n)$ is the \emph{canonical function} of $\Gr^{\geq 0}_{2,n}$; that is 
multiplying $\PT(\mathbf{I}_n)$ by the standard top form of $\Gr_{2,n}$ (cf. \cite[Appendix C.2]{ABL}) gives the \emph{canonical form} of $\Gr^{\geq 0}_{2,n}$. The poles $P_{i,i+1}=0$ of $\PT(\mathbf{I}_n)$ correspond to the facets of $\Gr^{\geq 0}_{2,n}$. 

More generally, $\Gr_{k,n}^{\ge 0}$ is a positive geometry \cite{ABL,Lam:2022yly} and the amplituhedron $\A_{n, k, m}(Z)$ is conjectured to be a positive geometry. This was recently proved for $k=2, m=2$ \cite{Ranestad:2024svp}.
\end{remark}
\begin{remark}
 Each $w$-chamber $\gcham{w}$ of the G-amplituhedron is isomorphic to $\Gr^{\geq 0}_{2,n}$ by a map induced by permuting and rescaling columns \cite[Proposition 11.15]{PSW}.
Hence $\gcham{w}$ is also a positive geometry and the weight function $\Omega(\simp{w})=\PT (w)$ of the $w$-simplex $\simp{w}$ is the canonical function of $\gcham{w}$. Indeed, the poles $P_{w_i, w_i+1}=0$ of $\PT (w)$ correspond to the facets of $\gcham{w}$. From the definition of $z_a$ and \eqref{eq:app_matrix}, the facets of $\gcham{w}$ are in the locus where $v_{w^{(1)}_i},v_{w^{(1)}_{i+1}}$ are parallel and are therefore cut out by the equation $P_{w_i, w_i+1}=0$ in the Pl\"uckers of the matrix $z$ of \eqref{eq:ptInChamber}.   
\end{remark}



Given a $(k,n)$-bicolored subdivision $\sigma$, one can define \emph{tiles} in the G-amplituhedron by considering the union of the $w$-chambers $\gcham{w}$ over $(w) \in \Ext(C_\sigma)$. One can use these tiles to tile the G-amplituhedron $\mathcal{G}_{n,k,2}$ and such tilings of $\mathcal{G}_{n,k,2}$ are in bijection with tilings of $\Delta_{k+1,n}$ (and with all-$Z$ tilings of $\Ank$). 

The total G-amplituhedron $\mathcal{G}_{n}$ is given by the union of the $w$-chambers $\cham{w}$ over $w \in D_n$ \cite[Theorem 11.21]{PSW}.
Then one can consider tricolored subdivision $\tdiv$ and define the region in $\mathcal{G}_{n}$ which is the union of the $w$-chambers $\cham{w}$ with $w \in \Ext(C_\tdiv)$. This is the analogue of the Parke--Taylor polytope $\ptile{\tdiv}$. As the weight function $\Omega(\simp{w})$ of $w$-simplices gives the canonical function of $w$-chambers $\Delta^\mathcal{G}_w$, it would be interesting to interpret the weight function of Parke--Taylor polytopes $\Omega(\ptile{\tdiv})$ as canonical functions of some positive geometries inside the (total) G-amplituhedron. 

