In this paper, we establish that FFLV polytopes, which describe monomial bases compatible with the PBW filtration on finite-dimensional simple modules for $\lie{sl}_n, \lie{sp}_n$, are actually string polytopes as described by Littelmann and Berenstein-Zelevinsky for Demazure modules of higher rank Lie algebras.

§ INTRODUCTION

Degenerations of flag varieties have been studied in various contexts.
The PBW degenerate flag variety $\Fl^a$ was introduced by Evgeny Feigin <cit.> in the context of PBW degenerate $\lie{sl}_n(\bc)$-modules.
He demonstrated that $\Fl^a$ is a flat degeneration of the classical flag variety and provided a description in terms of sequences of subspaces similarly to the description of the classical flag variety:

\[
\Fl^a = \{ (U_i) \mid U_i \subset \bc^n, \dim U_i = i, \text{pr}_i(U_i) \subset U_{i+1}\}
\]

Subsequently, Giovanni Cerulli Irelli and Martina Lanini <cit.> showed that this variety, also defined through an abelianization of $\lie{sl}_n$ is a natural object in Lie theory, specifically isomorphic to a Schubert variety in a partial flag variety.
Precisely, they considered the element $w =  \ldots(s_3s_4s_5)(s_2s_3)s_1$ in $S_{2n}$ and the associated Schubert variety in

\[
\Fl = \{ (U_i) \mid U_i \subset \bc^{2n}, \dim U_i = 2i-1, U_i \subset U_{i+1}.\}
\]

Following this, they showed together with Littelmann in <cit.> that this isomorphism can be interpreted in terms of modules for Lie algebras.
Specifically, they established that the aforementioned PBW degenerate module for a degenerate $\lie{sl}_n$ is isomorphic to a Demazure module for $\lie{sl}_{2n}$.
To be precise, if $\lambda = \sum m_i \omega_i$ is the highest $\lie{sl}_n$-weight, then $V^a(\lambda)$ (the PBW degenerate module) is,
as a module for the degenerate Lie algebra, isomorphic to $V_w(\tilde{\lambda})$, where $\tilde{\lambda} = \sum m_i \omega_{2i-1}$.

The aim of this paper is to present a combinatorial shadow of these isomorphisms. Specifically, in <cit.>, the FFLV-polytope was introduced.
Its lattice points parametrize a monomial basis of $V^a(\lambda)$ and, consequently, of $V(\lambda)$.
This basis, particularly for $\lie{sl}_n$ with $n \geq 6$, is distinct from the well-known string bases of Littelmann <cit.> and Berenstein-Zelevinsky <cit.>, the bases constructed by Lusztig <cit.>, and those by Nakashima-Zelevinsky <cit.>.
The Gelfand-Tsetlin basis <cit.> is, in fact, a Nakashima-Zelevinsky basis for a specific reduced expression of the longest Weyl group element. All these bases can be parametrized by lattice points in convex polytopes, such as string polytopes.

Conversely, the non-degenerate object, the Demazure module, has known monomial bases, like the string bases <cit.>, derived from the canonical basis using a reduced expression of $w$. Denoting the string polytope for $\tilde{\lambda}$ and $w$ with the given reduced expression as $Q_{\underline{w}}(\tilde{\lambda})$, and the FFLV polytope for $\lambda$ $FFLV(\lambda)$, the main result of this paper is stated as follows:

The FFLV polytope $FFLV(\lambda)$ is unimodularly equivalent to the string polytope

Furthermore, in subsequent works (<cit.>), PBW degenerate objects and compatible bases were also constructed for the symplectic Lie algebra. We extend our results to this setup as well:

The symplectic FFLV-polytope is isomorphic to a string polytope for a symplectic Demazure module.

Considering the other classical types is more delicate. Something is wrong? No FFLV-Polytopes are known for an abelianization of orthogonal Lie algebras yet.
Nevertheless, Igor Makhlin has provided a polytope describing monomial bases for an almost abelian Lie algebra in type $B$, where the short root vectors are not abelianized <cit.>.
A geometric interpretation of this will be presented in the upcoming work <cit.>.
The combinatorial shadow of this is also part of an upcoming project, as our current methods do not readily apply to this setup.

In fact, we might want to change the point of view and pull back a string polytope for Demazure modules to a candidate for an FFLV polytope in type $B$ and $D$, as the results in <cit.> suggest a suitable Weyl group element and reduced expression.

Another application we are pursuing involves tableau combinatorics, such as the Littlewood-Richardson rules.
Although the FFLV polytopes are translated to PBW semistandard tableaux <cit.>, the usual tableaux combinatorics are yet to be developed.
We aim to develop a Littlewood-Richardson rule that is compatible with the PBW filtration.

In <cit.>, similar bases have been provided for type $G_2$.
Although there are existing geometric interpretations, as noted in <cit.> and <cit.>, computational experiments indicate that the polytope proposed in <cit.> does not correspond to a string polytope for a Demazure module of affine Lie algebras.

It should be mentioned that this combinatorial picture can only be a shadow of the geometric construction of desingularizations and Newton-Okounkov bodies, following the concept of birational sequences <cit.>. This still has to be explored.

The paper is structured as follows: We recall definitions and foundations on Lie algebras, monomial bases and degenerations in Section <ref>. Then we provide the unimodular equivalence for the two main theorems and the outline of their proofs in Section<ref>. In Section<ref> we give the proof while leaving out the considerations of the two cases to Section<ref> and Section <ref>.

The authors gratefully acknowledge financial support by the DFG –Project-ID 286237555–TRR 195. The authors would like to thank Xin Fang, Martina Lanini and Chrisian Steinert for helpful discussions.

§ PRELIMINARIES

We fix here definitions, notations and recall results, that will be used throughout this paper.

§.§ Notations

We introduce a few standard definitions and notations, we follow here <cit.>. We denote $\lie g = \lie n^+ \oplus \lie h \oplus \lie n^-$  a simple, finite-dimensional complex Lie algebra with a fixed triangular decomposition.
We denote the set of (positive) roots $R$ (resp. $R^+$), the set of (dominant) integral weights $P$ (resp. $P^+$).
For each positive root $\alpha$, we fix root vectors $e_\alpha \in \lie n^+, f_\alpha \in \lie n^-$ and $h_\alpha := [e_\alpha, f_\alpha]$.
If $n = \text{rk } \lie g$, then $\alpha_1, \ldots, \alpha_n$ denote the simple roots, we set $f_i := f_{\alpha_i}$, the fundamental weights are denoted $\omega_1, \ldots, \omega_n$.
The height of a positive root is the sum of the coefficients when written in terms of the simple roots.

The Weyl group is denoted $W$, the simple reflection corresponding to $\alpha_i$ is denoted $s_i$, the longest element of $W$ is denoted $w_0$.

Let $\lambda \in P^+$, the finite-dimensional simple module of highest weight $\lambda$ is denoted $V(\lambda)$ and we fix a generator of the highest weight line $v_\lambda$. Similar, we fix for each $w \in W$ a generator $v_{w(\lambda)}$ of the line of weight $w(\lambda)$.
Let $\lie b = \lie n^+ \oplus \lie h$.
With the universal enveloping algebra $U(\lie g) = U(\lie n^-) U(\lie b)$ one has $V(\lambda) = U(\lie n^-).v_\lambda$.
For fixed $\lambda \in P^+, w \in W$, we denote the Demazure module $V_w(\lambda) := U(\lie b).v_{w(\lambda)}$.

§.§ Monomial bases

Since $V(\lambda) = U(\lie n^-).v_\lambda$ and $V_w(\lambda) := U(\lie b).v_{w(\lambda)}$ one can ask for monomial bases of $V(\lambda)$ (resp. $V_w(\lambda)$) within $U(\lie n^-)$ (resp. $U(\lie n^+)$) .
A general framework on how to construct bases which are suitable within a Lie theoretic context is described in <cit.> using birational sequences.
One orders $\dim \lie n^-$ roots such that the product of their one-dimensional root subgroups is birational to $\lie n^-$, then ordered monomials in the root vectors form a spanning set of $U(\lie n^-)$. Fixing a monomial ordering provides a basis of $U(\lie n^-)$ and of course also of $V(\lambda)$.
The monomials are parametrized by their exponents and we denote $S(\lambda)$, the set of integer lattice points.
Then we have the general fact
Let $\lambda_1, \lambda_2 \in P^+$, then

\[
S(\lambda_1) + S(\lambda_2) \subseteq S(\lambda_1 + \lambda_2).
\]

The construction of monomial bases, of course, also works for Demazure modules using a birational sequence for $U(\lie n^+)$ and a fixed monomial ordering.

Famous examples of monomial bases of $V(\lambda)$ that fit into the framework of birational sequences are the Gelfand-Tsetlin bases <cit.>, the string bases of Littelmann and Berenstein-Zelevinsky <cit.>, the Lusztig bases <cit.>, the Nakashima-Zelevinsky bases <cit.>, the FFLV bases <cit.> and the Makhlin basis <cit.>. We recall here briefly two of them, the string bases and the FFLV bases.

String bases:

Let $w \in W$ and we fix a reduced expression $\underline{w}_0 = s_{i_1} \cdots s_{i_k}$ of the longest Weyl group element. Then $(\alpha_{i_1}, \ldots, \alpha_{i_k})$ is a birational sequence and the following is a spanning set of $V(\lambda)$

\{ f_{i_1}^{t_1} \cdots f_{i_k}^{t_k}. v_\lambda \mid t_j \in \N_{\geq 0} \}

For the string bases, one fixes the neglex ordering, then the bases obtained through the birational sequence and $\textit{neglex}$ are described in <cit.>.
In fact, the latter provides a convex polytope $Q_{\underline{w}_0}(\lambda)$ whose lattice points parametrize the monomials (via their exponents), e.g. $Q_{\underline{w}_0}(\lambda)^\Z = S_{\underline{w}_0}(\lambda)$.
Moreover, let $w \in W$ and $V_w(\lambda)$ be the associated Demazure module, $w = s_{i_1} \cdots s_{i_\ell}$ be a reduced expression, extended by $s_{i_{\ell+1}} \cdots s_{i_k}$ to a reduced expression $\underline{w}_0$ of $w_0$, then a monomial basis of $V_w(\lambda)$ is parametrized by the lattice points in the projection of $Q_{\underline{w}_0}(\lambda)$ to the first $\ell$ coordinates (<cit.>).
We denote this convex polytope $Q_{\underline{w}}(\lambda)$.

A good ordering of the positive roots is an ordering such that $\beta \prec \alpha \Rightarrow$ the height of $\beta$ is smaller than the height of $\alpha$.
We fix a good ordering on the positive roots, starting with the highest root (and consequently ending with an arbitrary ordering of the simple roots), this provides a birational sequence and the FFLV-type bases are obtained from here by fixing the degrevlex (the homogeneous reverse lexicographic) order.
For type $A$ and $C$ they are described in <cit.> and <cit.>, while for type $G_2$ they are described in <cit.>.
In fact, in loc. cit., convex polytopes are provided whose lattice points parameterize the bases, in type $A$ and $C$ these polytopes $P(\lambda)$ are lattice polytopes for all $\lambda$.
Moreover, one has for $\lambda = \sum a_i \omega_i \in P^+$

\[
P(\lambda) = \sum a_i P(\omega_i),
\]

where $+$ denotes the Minkowski sum.

In the 1990's several families of monomial bases were constructed, mostly depending on the canonical (global) bases combinatorics. In this perspective, the upcoming remark is crucial for our interest in the topic.

In general (for type $A$, $n \geq 4$, $\lambda$ regular, other types similarly), there is no reduced expression of $w_0$ such that $P(\lambda)$ is unimodular equivalent to $Q_{\underline{w_0}}(\lambda)$. In fact, the number of facets in the FFLV polytope is in general strictly larger than the number of facets in any string polytope for $V(\lambda)$. The FFLV-type polytope does not belong to the world of string polytopes (consequently also Lusztig polytopes) obtained from canonical bases.

It should also be mentioned here, that the FFLV polytopes are instances of marked chain polytopes <cit.> while the Gelfand-Tsetlin polytopes are instances of marked order polytopes. So by <cit.> they are in general (beyond the natural representations) not unimodular equivalent. For more on this framework we refer to  <cit.>.

§.§ Degenerations

The PBW filtration on $U\lie (\lie n^-)$ induces a filtration on $V(\lambda)$, the associated graded module $V^a(\lambda)$ is then a module for the associated graded (abelian) Lie algebra $\lie n^{-,a}$. Since the PBW filtration on $V(\lambda)$ is stable under $U(\lie b)$, $V^a(\lambda)$ is a module for the semi direct product $\lie b \rtimes \lie n^{-,a}$. In type $A$ and $C$, the FFLV bases are monomial bases for this degenerate module as they are compatible with the PBW degree.

The degenerate modules we used in <cit.> to define the PBW degenerate flag variety in type $A$, later generalized in <cit.> and <cit.> to other types. Shortly after, in <cit.>, it was shown that this degenerate flag variety (in type $A$ and $C$) is actually a Schubert variety in a partial flag variety (in higher rank).
This surprising result, the variety obtained by forgetting the Lie structure on the module, is still a reasonable object in Lie theory, has also an interpretation in terms of modules.
Namely, there is an Lie algebra isomorphism (shown in<cit.> for type $A$ and $C$) between $\lie b \times \lie n^{-,a}$ and a subquotient on the Borel subalgebra in $\tilde{\lie g}$, the Lie algebra of the same type as $\lie g$ but of rank $2n-1$.
This translates to an isomorphism between $V^a(\lambda)$ and $V_{\underline{w}}(\tilde{\lambda})$ for a specific $\underline{w}$ and $\tilde{\lambda}$.
In <cit.>, this is even generalized to all classical type up to some modification on the filtration.

§ MAIN THEOREM

We are giving a combinatorial shadow of the $\lie n^{-,a}$-module isomorphism between $V^a(\lambda)$ and $V_{\underline{w}}(\tilde{\lambda})$. That is, we will show that the FFLV-type polytopes for $V(\lambda)$ are unimodularly equivalent to the string polytopes for $V_{\underline{w}}(\tilde{\lambda})$. As mentioned in Remark <ref>, the FFLV-type polytopes for $V(\lambda)$ are in general not isomorphic to string polytopes for $V(\lambda)$. The isomorphism we will construct is to string polytopes of Demazure modules.

Let $\lie g$ be of type $X_n$ (here $X$ is $A$ or $C$) and $\lambda = \sum_{i=1}^{n} a_i \omega_i \in P^+$ Denote $P(\lambda)$ the FFLV polytope. We set for $X_{2n-1}$

\[
\tilde{\lambda} = \sum_{i} a_i \omega_{2i-1}
\]

and we fix $\underline{w}$ in the various types as

$A$: $\underline{w} = \sigma_n \sigma_{n-1}\cdots \sigma_3\sigma_2\sigma_1$

$C$: $\underline{w} = \tau_{2n-1}\tau_{2n-2}\dots \tau_{n+1}\sigma_n \sigma_{n-1}\cdots \sigma_3\sigma_2\sigma_1$

\begin{align*}
\tau_j = s_j s_{j+1} \dots s_{2n-1} \quad \text{and} \quad \sigma_j= s_j s_{j+1} \dots s_{2j-1}
\end{align*}

The polytopes $P(\lambda)$ and $Q_{\underline{w}}(\tilde{\lambda})$ are isomorphic.

Restricted to the lattice points, the equivalence is weight preserving (up to a twist and shift).

There are defining inequalities for the FFLV type polytope provided in <cit.>, so using the isomorphism one obtains defining inequalities for these particular string polytopes.

We provide here the outline of the proof for both types and leave the technical differences to the next sections.

For all types $X_n$ and $\lambda \in P^+$, the rational polytopes $P(\lambda)$ <cit.> and $Q_{\underline{w}}(\tilde{\lambda})$ <cit.> are convex polytopes in $\R^N$.

There is an affine map $T_{X_n, \lambda}: \R^N \longrightarrow \R^N$ with

\[
T_{X_n, \lambda} (y) = \mathcal{X}_n(y) + t_{X_n, \lambda}
\]

where $\mathcal{X}_n$ is a linear map with determinant $\pm 1$, depending on $X_n$ but not on $\lambda$, $t_{X_n,\lambda}$ a constant vector, depending linearly on $\lambda$, e.g. $t_{X_n, -}$ is a linear assignment $\Z^n \longrightarrow \R^N$. Then

\[
T_{X_n, \lambda}(P(\lambda)) \subseteq Q_{\underline w}(\tilde{\lambda}).
\]

Proof: We prove the claim for $\lambda$ being a fundamental weight $\omega_i$ in Lemma  <ref>.

Let $\lambda \in P^+$ with a decomposition $\lambda = \sum a_i \omega_i$. Then

\[
P(\lambda) = \sum a_iP(\omega_i)
\]

and hence, by construction of the map $T_{X_n, \lambda}$

\begin{array}{rcl}
T_{X_n, \lambda}(P(\lambda)) & =& \mathcal{X}_n(P(\lambda)) + t_{X_n,\lambda} \\
&= &\sum a_i \mathcal{X}_n (P(\omega_i)) + \sum a_i t_{X_n, \omega_i} \\
& = & \sum a_i  (\mathcal{X}_n (P(\omega_i)) + \sum t_{X_n, \omega_i})
\end{array}

With Lemma <ref> one deduces

\[
T_{X_n, \lambda}(P(\lambda)) \subseteq \sum a_i Q_{\underline w}(\tilde{\omega_i}) \subseteq Q_{\underline w}(\tilde{\lambda})
\]

while the last relation is due to Proposition <ref>. This proves the claim.

For $\lambda \in P^+$ we have $T_{X_n, \lambda}(P(\lambda)) = Q_{\underline w}(\tilde{\lambda})$.

For fixed $\lambda \in P^+$, $ Q_{\underline w}(\tilde{\lambda})$ is a rational polytope. Let $k > 0$ be minimal, such that $k  Q_{\underline w}(\tilde{\lambda})$ is a lattice polytope. Then we have

\begin{eqnarray*}
T_{X_n, k\lambda}(P(k \lambda)) &=& kT_{X_n, \lambda}(P(\lambda)) \\
\text{previous claim} & \subseteq  & k  Q_{\underline w}(\tilde{\lambda}) \\
\text{Proposition}~\ref{prop:general-mink}&  \subseteq & Q_{\underline w}(\tilde{k\lambda})
\end{eqnarray*}

We can consider for all polytopes the lattice points and obtained the very same inequalities with ${}^\Z$. Since for all $\lambda$, $V(\lambda)$ has the same dimension as $V_{w}(\tilde{\lambda})$ (the degenerate module is isomorphic to the Demazure module), we have

\[
\sharp P(k \lambda)^\Z = \sharp Q_{\underline w}(\tilde{k\lambda})^\Z
\]

and since $P(\lambda)$ is a lattice polytope

\[
\sharp k(P(\lambda)^\Z) = \sharp (k  Q_{\underline w}(\tilde{\lambda}))^{\Z}
\]

This implies that all lattice points of $k Q_{\underline w}(\tilde{\lambda})$ are already in $kT_{X_n, \lambda}(P(\lambda))$. Since by assumption, $k Q_{\underline w}(\tilde{\lambda})$ is a lattice polytope, it is spanned by the lattice points in the $k$-times Minkowski sum of $Q_{\underline w}(\tilde{\lambda})$ and hence $k =1$ and $Q_{\underline w}(\tilde{\lambda})$ is a lattice polytope.

For all $\lambda$, the string polytope $ Q_{\underline w}(\tilde{\lambda})$ is a lattice polytope.

Both are convex polytopes. We provide a map $T: \R^N \longrightarrow \R^N, x \mapsto Ax + t_\lambda$ (with $t_\lambda$ being linear in $\lambda$) for every $\lambda$. We prove that this map is a bijection for the lattice points for the generators. We do this case by case.

Since the Minkowski sum of the polytopes is contained in the polytope of the sum and the FFLV type polytopes are the Minkowski sum of the generators, we deduce that the image of the FFLV type polytope is contained in the string polytope.
The string polytope is a rational polytope, we fix $\lambda$ big enough, such that the polytope is a lattice polytope. Because of the module isomorphism, the lattice points are all in the image of the map.

We deduce an isomorphism for all $\lambda$ of lattice polytopes but then this is an isomorphism for all $\lambda$.

§ GENERAL SETUP

We fix a type and rank $X_n$. Recall, that $N = |R^+|$, the number of positive roots and we label the standard basis of $\R^N$ by $\alpha \in R^+$. We denote

\[
e_{\alpha} = \begin{cases}  e_{\ell, j} & \text{ if } \alpha = \alpha_\ell + \ldots + \alpha_j, 1 \leq \ell \leq j \leq n  \\ e_{\ell, \bar{j}} & \text { if } \alpha = \alpha_\ell + \ldots + 2 \alpha_j + \ldots \end{cases}
\]

We set $1 < 2 < \dots < n = \overline{n} < \bar{n-1} < \dots < \bar 1$ and define for $X_n$ a linear map $\mathcal{X}_n: \R^N \longrightarrow \R^N$

\begin{align*}
\text{if } X=A: \quad e_{a, b} &\mapsto -\sum_{n \geq c \geq b } e_{a,c} - \sum_{1 \leq c < a} e_{c,b} \\
\text{if } X=C: \quad e_{a, b} &\mapsto - \begin{cases}
e_{a,\overline{a}} + 2\sum\limits_{1 \leq c < a} e_{c,b}, & \text{if } \overline{a} = b \\
\sum\limits_{\overline{a} \geq c \geq b} e_{a,c} + \sum\limits_{1 \leq c<a} (e_{c,b} + e_{c,\overline{a}}), &  \text{else.}
\end{cases} %\\
% \text{if } X=B: \quad e_{a, b} &\mapsto -\begin{cases}
%     \sum\limits_{1 \leq c \leq a} e_{c,b}, & \text{if } \overline{a} = b, \\
%     e_{a,\overline{a}} + \sum\limits_{\overline{a} \geq c \geq b} e_{a,c} + \sum\limits_{1 \leq c < a} (e_{c,b} + e_{c,\overline{a}}) , & \text{else.}
% \end{cases}
\end{align*}

For $A_3$, we fix the basis $ A = (e_{1,3},e_{2,3},e_{3,3},e_{1,2},e_{2,2},e_{1,1})$ and for $C_2$, we fix the basis $C= (e_{1,\overline{1}},e_{1,2},e_{2,2},e_{1,1})$, then:

\[
\tensor[^A]{{\cA_3}}{^A} = -\left(\begin{matrix}
1 & 1 & 1 & 1 & 0 & 1\\
0 & 1 & 1 & 0 & 1 & 0\\
0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{matrix}
\right) \quad \tensor[^C]{{\mathcal{C}_2}}{^C} = -\left(\begin{matrix}
1 & 1 & 0 & 1 \\
0 & 1 & 2 & 1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{matrix}
\right)
\]

We set $H(X_n):=\{(\ell,j) \ | \ \alpha_{\ell,j} \in R^+\}$ i.e. indices of the positive roots.

More explicitly we have

\begin{equation*}
H(X_n) = \begin{cases}
\{(\ell,j) \ | \ 1 \leq \ell \leq j \leq n\}, & \text{ if } X = A \\
\{(\ell,\overline{j}) \ | \ 1 \leq \ell \leq j \leq n-1\} \cup H(A_n), & \text{ if } X=C
\end{cases}
\end{equation*}

The map $\mathcal{X}_n$ is unimodular, i.e. we have $|\det(\mathcal{X}_n)| = 1$ for $X \in \{A,C\}$. Moreover, the only non-zero coefficients are $-1,-2$ with respect to any basis given by some ordering of the $e_{a,b}$ for $(a,b) \in H(X_n)$.

We prove the first part of this proposition in the sections corresponding to types $A,C$ respectively. The second part follows immediately from the definition.

For a fixed $X_n$ and $\lambda \in P^+$ we introduce the translation vector $t_{X_n, \lambda}$ as follows

Fix $1 \leq i \leq n$. We set

\begin{equation*}
t_{X_n,\omega_i} := \sum_{(\ell,j) \in H(X_n)} \alpha^i_{\ell,j}(X_n) e_{\ell,j}
\end{equation*}

\begin{equation*}
\alpha^i_{\ell,j}(A_n) =
\begin{cases}
1, & \text{if } j \geq i \land \ell \leq i \\
0, & \text{else.}
\end{cases}
\begin{comment}
\alpha^i_{\ell,j}(C_n) &=
\left\{ \begin{array}{rl}
\alpha^i_{\ell,j}(A_n), & \text{if } (\ell,j) \in H(A_n) \\
2, & \text{if } \overline{j} \leq n-1 \land \ell \neq \overline{j} \\
1, & \text{if } \overline{j} \leq n-1 \land \ell = \overline{j} \\
0, & \text{else.}
\end{array}\right. \\
% \alpha^i_{\ell,j}(B_n) &=
%     \left\{ \begin{array}{rl}
%         \alpha^i_{\ell,j}(A_n), & \text{if } (\ell,j) \in H(A_n) \\
%         2, & \text{if } \overline{j} \leq n-1 \\
%         0, & \text{else.}
%     \end{array}\right.
\end{comment}
\end{equation*}

\begin{equation*}
\alpha^i_{\ell,j}(C_n) = \begin{cases}
1 & \text{if } i\leq j < \overline i \land \ell \leq i\\
1 & \text{if } \overline \ell = j \land \ell \leq i\\
2 & \text{if } \overline \ell > j \land \overline i \leq j\\
0 & \text{else}
\end{cases}
\end{equation*}

and for $\lambda = \sum_{j=1}^n a_j \omega_j$, we set $t_\lambda(X_n) = \sum_{j=1}^n a_j t_{\omega_j}(X_n)$ as outlined in the proof of <ref>

This translation vector is the string sequence that maps the highest weight vector $v_{\tilde{\lambda}}$ to the extremal weight vector $v_{w(\tilde{\lambda})}$. Thus, the computation of the coefficients is straightforward, as they correspond to the coefficients of $w'(\tilde{\lambda})$ for each left subword $w'$ of $w$.

For $n=3$, we have

\begin{align*}
t_{A_3, \omega_2} &= e_{1,2} + e_{2,2} + e_{1,3} +e_{2,3} \\
t_{C_3,\omega_2} &= e_{1,2} + e_{2,2} + e_{1,3} + e_{2,3} + 2e_{1,\overline{2}} + e_{2,\overline{2}} + e_{1,\overline{1}} \\
%t_{B_2, \omega_2} &= e_{1,2} + e_{2,2} + e_{1,3} + e_{2,3} + 2e_{1,\overline{2}} + 2e_{2,\overline{2}} + 2e_{1,\overline{1}}
\end{align*}

This introduces the affine map $T_{X_n, \lambda} : \R^N \longrightarrow \R^N$ for the proof of <ref>. As outlined in the proof of <ref> we are left to prove, that $T_{X_n, \lambda}$ maps lattice points of the FFLV-type polytope to lattice points of the string polytope. And again, as outlined in the proof, it is enough to prove this for $\lambda$ being a fundamental weight.

For $X \in \{A,C\}$ and $1 \leq i \leq n$ set

\begin{equation*}
\begin{split}
T_{X_n,\omega_i}: \R^N &\to \R^N \\
v &\mapsto -\mathcal X_n(v) + t_{A_n, \omega_i}
\end{split}
\end{equation*}

Then $T_{X_n,\omega_i}(P(\omega_i)^\Z) \subseteq Q_{\underline{w}}(\tilde \omega_i)^\Z$.

We set $\Pi_w$ the set of operators

\[
\{ f_{i_1}^{\ell_{1}} \cdots f_{i_N}^{\ell_{N}} \mid \underline w = s_{i_1} \cdots s_{i_N} \}.
\]

It is shown in <cit.> and <cit.> that, despite the construction of Demazure modules as $U(\lie b)$-modules, $\Pi_{\underline{w}}.v_{\lambda}$ is a generating set for $V_w(\lambda)$ for all $\lambda$.
We will use this in the following and find within $\Pi_w$ the subset that gives the string bases of
$V_w(\lambda)$. As explained above, we can restrict ourselves to fundamental weights.

For $1 \leq i \leq 2n-1$ and $f,g \in \Pi_{\underline{w}}$. We define

\begin{equation*}
f \sim_{i} g :\Leftrightarrow \exists r \in \Q_+ \forall v \in \Lambda^{i}\C^d : r \cdot f(v) = g(v) \quad \text{for } d=\left\{ \begin{array}{lr}
n+1, & \text{if } X=A \\
%2n+1, & \text{if } X=B \\
2n, & \text{if } X=C
\end{array}\right.
\end{equation*}

For $a, b \in \Z^N$, we say $a \sim_i b$ if $f^a \sim_i f^b$. It follows from the definition, that $\sim_i$ is an equivalence relation.

Since $V_w(\omega_i)$ can be realized as a submodule of $\Lambda^{i}\C^d$, if $f\sim_i g$, then $f$ and $g$ operate the same (up to a nonzero scalar) on $V_w(\omega_i)$.

For $1 \leq i,l,j, \leq n$ and $d$ as above the following are equivalent:

* $|l-j| \neq 1$

* $f_l f_j (v) = f_j f_l (v)$ for all $v \in \C^{d}$

* $f_l f_j \sim_i f_j f_l$

The implication $i) \Rightarrow ii)$ follows from general theory, if two nodes are not linked in the Dynkin diagram, their corresponding $\mathfrak{sl}_2$-subalgebras form direct sums, hence all elements commute. In type $A,C$, two nodes are linked iff they are consecutive.

$ii) \Rightarrow iii)$ is clear.

$iii) \Rightarrow  i)$:

Let $r \in \R_+$. Without loss of generality, let $j = l+1 \leq n$, and choose

\begin{equation*}
w:= \begin{cases}
e_1 \wedge \dots \wedge e_{l+1} \wedge e_{l+3} \wedge \dots \wedge e_{i+1} & \text{if } i > l \\
e_1 \wedge \dots \wedge e_{i-2} \wedge e_l \wedge e_{l+1} & \text{else}
\end{cases} \in \Lambda^{i} \C^{d}
\end{equation*}

we have

\begin{equation*}
r \cdot f_{l+1}f_l(w) = 0 \neq f_l f_{l+1}(w) = \begin{cases}
e_1 \wedge \dots \wedge e_{l-1} \wedge e_{l+1} \wedge \dots \wedge e_{i+1} & \text{if } i > l \\
e_1 \wedge \dots \wedge e_{i-2} \wedge e_{l+1} \wedge e_{l+2} & \text{ else}
\end{cases}
\end{equation*}

Let $x = \sum \alpha_{a,b} e_{a,b} \in \Z^{N}_{\geq 0}$, if $X = A$, we introduce $\alpha_{\ell,j} = 0$ for $\overline{n-1} \leq j \leq \overline{1}$ such that in what follows we need not make a distinction between types.

\begin{align*}
\underline{f}^x := &f_{2n-1}^{\alpha_{1,\overline{1}}} (f_{2n-2}^{\alpha_{1,\overline 2}} f_{2n-1}^{\alpha_{2,\overline{2}}}) \dots (f_{n+1}^{\alpha_{1,\overline{n-1}}} \dots f_{2n-1}^{\alpha_{n-1,\overline{n-1}}}) \\
&(f_{n}^{\alpha_{1,n}} f_{n+1}^{\alpha_{2,n}}\dots f_{2n-1}^{\alpha_{n,n}}) (f_{n-1}^{\alpha_{1,n-1}} \dots f_{2n-3}^{\alpha_{n-1,n-1}}) \dots (f_2^{\alpha_{1,2}} f_3^{\alpha_{2,2}}) f_1^{\alpha_{1,1}}
\end{align*}

and for $i,m\in \N$, $v\in \Lambda^i\C^m$ we set $x.v := \underline{f}^x(v)$.

We define an ordering on $H(X_n)$, as follows:

\begin{align*}
(a,b) \preceq (c,d) :\Leftrightarrow b < d \lor (b = d \land a \geq c)
\end{align*}

This is the reverse-lex order where in the last component we have the usual $<$ and in the first component it is reversed i.e. $>$.
For $A_3$ the ordering reads as follows:

The ordering on $H(A_3) = \{e_{1,1},e_{1,2},e_{2,2},e_{1,3},e_{2,3},e_{3,3}\}$ is

\begin{align*}
e_{1,1} \preceq e_{2,2} \preceq e_{1,2} \preceq e_{3,3} \preceq e_{2,3} \preceq e_{1,3}
\end{align*}

The points in $H(X_n) \subseteq \Z^N$ correspond to simple root operators, the ordering represents the structure of the reduced expression $\underline w$ on the level of points:

\begin{align*}
\underline w = s_3s_4s_5s_2s_3s_1 \quad f_3 f_4 f_5 f_2 f_3 f_1 = \underline f^{e_{1,3}} \underline f^{e_{2,3}} \underline f^{e_{3,3}} \underline f^{e_{1,2}} \underline f^{e_{2,2}}\underline f^{e_{1,1}}
\end{align*}

§ TYPE A

For $n \in \N$, let

\begin{align*}
B_n := (e_{1,n},\dots,e_{n,n},e_{1,n-1},\dots,e_{n-1,n-1},\dots,e_{1,2},e_{2,2},e_{1,1})
\end{align*}

This ordering is a decreasing chain with respect to $\preceq$.
We show by induction on $n \in \N$ that $\tensor[^{B_n}]{(-\cA_n)}{^{B_n}}$ is upper triangular with determinant $\pm 1$.

If $n=1$, then $\tensor[^{B_1}]{(-\cA_1)}{^{B_1}} = (-1)$.

For the induction step note that $B_n = (e_{1,n},\dots,e_{n,n},B_{n-1})$, hence:

\begin{equation*}
\tensor[^{B_n}]{(-\cA_n)}{^{B_n}} = \left(\begin{array}{c|c}
\star & \star \\ \hline
\text{\raisebox{-1.5mm}{$\star$}} & \text{ \raisebox{-2mm}{$\tensor[^{B_{n-1}}]{(-\cA_{n-1})}{^{B_{n-1}}}$} }
\end{array}\right)
\end{equation*}

to determine the left side of the matrix, fix $1 \leq j \leq n$ such that $e_{j,n} \in H(A_n)$. Then

\begin{equation*}
\cA_n(e_{j,n}) = -\sum_{c=1}^j e_{c,n}
\end{equation*}

and thus $ \tensor[^{B_n}]{(-\cA_n)}{^{B_n}}$ is upper triangular with the first block having $-1$s on the diagonal.

thus we arrive at the desired form

\begin{equation*}
\tensor[^{B_n}]{(-\cA_n)}{^{B_n}} = \left(\begin{array}{ccc|ccc}
1 & \dots & 1 & \\
& \ddots & \vdots & & \text{\raisebox{1.5mm}{$\star$}}\\
0 & & 1 \\  \hline
& \text{\raisebox{-2mm}{$0$}} &  & & \text{ \raisebox{-2mm}{$\tensor[^{B_{n-1}}]{(-\cA_{n-1})}{^{B_{n-1}}}$} }
\end{array}\right)
\end{equation*}

By induction we deduce $|\det(-\cA_n)| = |\det (-\cA_{n-1})| = 1$.

Let $B = (e_1,\dots,e_{2n})$ be the standard basis of $\R^{2n}$.

$i) \Leftrightarrow ii)$:
We have $\tensor[^B]{f}{_j^B} = E_{j+1,j}$, hence

\begin{align*}
\tensor[^B]{{(f_jf_l - f_lf_j)}}{^B} = E_{j+1,j}E_{\ell+1,l} - E_{\ell+1,l}E_{j+1,j} = \delta_{j,l+1} E_{j+1,l} - \delta_{\ell,j+1}E_{\ell+1,j}
\end{align*}

which is $\neq 0$ if and only if $j = l+1$ or $l = j+1$. Hence it is $0$ if and only if $|l-j| \neq 1$.

We make a few general observations that will aid us in the next proof.

If we fix the weight $\omega_{2i-1}$, i.e. we fix some $i$, then we can restrict all computations to a subword of $\underline{w}$, namely

\begin{align*}
w := s_n \cdots s_{n+i-1}\cdots s_{i+1} \cdots s_{2i}s_i \cdots s_{2i-2}s_{2i-1}
\end{align*}

and $Q_{\underline w}(\tilde \omega_i) = Q_{w}(\tilde \omega_i) \times \{0\}^{m} \subseteq \{0,1\}^N$, where $m = N - i(n-i+1)$.

If $x = \sum \alpha_{a,b} e_{a,b} \in \Z^N_{\geq 0}$ such that $\underline{f}^x.(e_1 \wedge \dots \wedge e_{2i-1}) \neq 0$ for some $i$, then $\alpha_{a,b} \leq 1$ for all $a,b$, since $f_j^2$ acts by $0$ on $V(\omega_{2i-1})$. Furthermore, we need $\alpha_{\ell,j} = 0$ for all $j < i$, since we first have to operate with $f_{2i-1}$ on $e_1 \wedge \dots \wedge e_{2i-1}$. The first (with respect to the order <ref>) non-zero action is at the index $(i,i)$, hence at all $(\ell,j) \prec (i,i)$ the coefficients $\alpha_{\ell,j}$ have to be $0$. Analogously, we see that $\alpha_{\ell,j} = 0$ if $\ell > i$.

Each point of $Q_{\underline w}(\tilde\omega_{i})$ can therefore be written in a tableau of format $i \times (n-i+1)$ with entries $0,1$, where the $\ell,j$'th entry of the tableau corresponds to the positive root $\alpha_{\ell,j+i-1}$.

Let $n=4, i=2$ and $x = e_{1,2} + e_{2,2} + e_{1,3} + e_{2,3} + e_{2,4}$. Then the tableau of $x$ is given by

\[
\begin{blockarray}{ccc}
\begin{block}{|ccc|} \hline
1 & 1 & 0 \\
1 & 1 & 1 \\ \hline
\end{block}
\end{blockarray}
\]

We can also interpret the tableau as some monomial in the root operators. The index of the root operator for an entry in the tableau is given as follows:

\begin{blockarray}{cccc}
\begin{block}{|cccc|} \hline
i & i+1 & \cdots & n \\ %\hline
i+1 & i+2 & \cdots & n+1   \\ %\hline
\vdots & \vdots & \ddots & \vdots \\ %\hline
2i-1 & 2i & \cdots & n+i-1 \\ \hline
\end{block}
\end{blockarray}
\begin{array}{c|c|c|c}

\end{array}

It should be noted that this diagram is supposed to be read column-wise from bottom to top and left to right to get the correct order of the root operators given in <ref>.

The operator corresponding to the tableau in the previous example is $f_5f_3f_4f_2f_3$.

We turn to the proof of Lemma <ref> for type $A$.

We show two parts: For all $p \in P(\omega_i)^\Z$, the point $T_{A_n,\omega_i}(p) \in \Z^N$

* does not annihilate $e_1 \wedge \dots \wedge e_{2i-1}$,

* is minimal, i.e. the smallest among all $\underline{f} \in \Pi_w$ with $\underline f \sim_{2i-1} T_{A_n,\omega_i}(p)$.

Set $\tau = t_{A_n, \omega_i}$ and $T = T_{A_n,\omega_i}$.
Let $e_{\alpha_{\ell_1,j_1}} + \dots + e_{\alpha_{\ell_s,j_s}} =: p \in P(\omega_i)^{\Z}$ with

$$\ell_s < \ell_{s-1} < \ldots < \ell_1 \leq i \leq j_1 < \ldots < j_s$$

Proof of (i):
We first show by induction on $s$, that for any $p\in P(\omega_i)^{\Z}$ there exists $q_p \sim_{2i-1} \tau + \cA_n(p)$ where

\begin{align*}
q_p = \sum_{k=1}^i \sum_{l=i}^{c^p_k} e_{k,l}
\end{align*}
for $c_k^p\in \N$ with $j_1=c^p_1\leq c^p_2 \leq \dots \leq c^p_i$, where we set $j_1:=n$ if $s=0$.

The case $s=0$ is trivial, here $T(p)=\tau$ and $c^p_k=n$ for all $1\leq k\leq i$. Now consider $s>0$. Set $\alpha:=\alpha_{\ell_1,j_1}$. By the induction hypothesis, the statement holds for $p-\alpha$.
We introduce a linear map

\begin{align*}
\text{shift}: \R^N \to \R^N:
e_{a,b} \mapsto \begin{cases}e_{a+1, b-1} & \text{if }a<\ell_1 \land b>j_1\\ e_{a,b} & \text{else}\end{cases}
\end{align*}

Then by the induction hypothesis, there are $c_k^{p-\alpha} \in \N$ such that $T(p-\alpha) \sim_{2i-1} q_{p-\alpha}$
and we have

\begin{align*}
T(p)&= \tau + \cA_n(p-\alpha) + \cA_n(\alpha)\\
&= T(p-\alpha) + \cA_n(\alpha)\\
&\sim_{2i-1} q_{p-\alpha} + \cA_n(\alpha) \\
&\overset{\ref{comm}}\sim_{2i-1}\text{shift}(q_{p-\alpha} + \cA_n(\alpha))= q_p
\end{align*}

The last $\sim_{2i-1}$-step can be visualized using the tableau-notation from before:

\[
\begin{blockarray}{cccccccccccccc}
\begin{block}{|cccccccccccccc|} \hline
1 & \cdots & 1 & \tikzmark{j1}{$0$} & \tikzmark{shiftul}{1} & \cdots & \tikzmark{c1}{1} & 0 & & 0 & & 0 & \cdots & \tikzmark{shiftur}{0} \\
1 & \cdots & 1 & \tikzmark{img_ar_ul}{$0$} & 1 & \cdots & 1 & \cdots & \tikzmark{c2}{$1$} & 0 & & 0 & \cdots & 0\\
\vdots & \ddots  & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots  & \vdots & & & \vdots & \ddots & \vdots \\
1 & \cdots & 1 & 0 & \tikzmark{shiftdl}{$1$} & \cdots & 1 & \cdots & 1 & \cdots & \tikzmark{ci}{$1$} & 0 & \cdots & \tikzmark{shiftdr}{$0$} \\
1 & \cdots & 1 & \tikzmark{img_ar_dl}{$0$} & 0 & \cdots & 0 & \cdots & 0 & \cdots & 0 & 0 & \tikzmark{img_ar_dr}{$\cdots$} & \tikzmark{i}{$0$}\\
1 & \cdots & 1 & 1 & 1 & \cdots & 1 & \cdots & 1 & \cdots & 1 & 1 & \cdots & 1 \\
\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
1 & \cdots & 1 & 1 & 1 & \cdots & 1 & \cdots & 1 & \cdots & 1 & 1 & \cdots & 1 \\ \hline
\end{block}
\end{blockarray}
\]

[overlay,remember picture]
[above=(.3cm of j1), font=, name=free1] $j_1$;
[above=(.3cm of c1), font=, name=freec1]$ \quad c_1^{p-\alpha}$;
[above=(.73cm of c2), font=, name=freec2]$ \quad c_2^{p-\alpha}$;
[above=(1.73cm of ci), font=, name=freeci]$ \quad c_{\ell_1-1}^{p-\alpha}$;
[right=(.3cm of i), font=, name=freei]$\ell_1$;
[left=(-.14cm of shiftul), yshift=.14cm, name=ul];
[left=(-.14cm of shiftdl), yshift=-.14cm, name=dl];
[right=(-.14cm of shiftur), yshift=.14cm, name=ur];
[right=(-.14cm of shiftdr), yshift=-.14cm, name=dr];
[->] (free1) to (j1);
[->] (freec1) to (c1);
[->] (freec2) to (c2);
[->] (freeci) to (ci);
[->] (freei)  to (i);
[blue] (dl) rectangle (ur);
[->,blue] (shiftul) to (img_ar_ul);
[->,blue] (shiftdl) to (img_ar_dl);
[->,blue] (shiftdr) to (img_ar_dr);

The blue block will be shifted in the direction of the arrows:

\[
\begin{blockarray}{cccccccccccc}
\begin{block}{|cccccccccccc|} \hline
1 & \cdots & \tikzmark{c1}{1} & \tikzmark{j1}{0} & \cdots & & 0 & & 0 & \cdots & 0 & 0 \\
1 & \cdots & 1 & \tikzmark{shiftul}{1} & \cdots & \tikzmark{c2}{1} & 0 & & 0 & \cdots & \tikzmark{shiftur}{0} & 0 \\
\vdots & \ddots  & \vdots & \vdots & \ddots & \vdots & & & \vdots & \ddots & \vdots & \vdots \\
1 & \cdots & 1 & \tikzmark{shiftdl}{1} & \cdots & 1 & \cdots & \tikzmark{ci}{1} & 0 & \cdots & \tikzmark{shiftdr}{0} & \tikzmark{i}{0}\\
1 & \cdots & 1 & 1 & 1 & \cdots & 1 & \cdots & 1 & \cdots & 1 & 1 \\
\vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots & \vdots \\
1 & \cdots & 1 & 1 & 1 & \cdots & 1 & \cdots & 1 & \cdots & 1 & 1\\ \hline
\end{block}
\end{blockarray}
\]

[overlay,remember picture]
[above=(.3cm of j1), font=, name=free1] $j_1$;
[above=(.6cm of c1), font=, name=freec1]$c_1^{p}= j_1-1$;
[above=(.4cm of c2), font=, name=freec2]$c_2^{p}=c_1^{p-\alpha}-1$;
[above=(2.03cm of ci), font=, name=freeci]$ c_{\ell_1}^{p}=c^{p-\alpha}_{\ell_1-1}-1$;
[right=(.3cm of i), font=, name=freei]$\ell_1$;
[left=(-.14cm of shiftul), yshift=.14cm, name=ul];
[left=(-.14cm of shiftdl), yshift=-.14cm, name=dl];
[right=(-.14cm of shiftur), yshift=.14cm, name=ur];
[right=(-.14cm of shiftdr), yshift=-.14cm, name=dr];
[->] (free1) to (j1);
[->] (freec1) to (c1);
[->] (freec2) to (c2);
[->] (freeci) to (ci);
[->] (freei)  to (i);
[blue] (dl) rectangle (ur);

As we saw, the $\text{shift}$ map moves the upper right $\ell_1 \times j_1$-block by one to the bottom left, so we set

\begin{align*}
c_k^p = \begin{cases}
j_1, & \text{if } k = 1 \\
c_{k-1}^{p-\alpha}-1 & \text{if } 2 \leq k \leq \ell_1 \\
n & \text{else.}
\end{cases}
\end{align*}

The inequalities from the induction hypothesis get carried over. With $\ref{comm}$ we can reorder each monomial into a monomial in the word

\begin{align*}
w^t:=(s_n \cdots  s_i) \cdots (s_{n+i-1} \cdots s_{2i-1})
\end{align*}

which translates to reading the tableau row-wise (left to right and bottom to top) instead.

In our setup, each row only acts on one of the wedge-components, that is the $m$-th row sends $e_{i+m-1}$ to $e_{m+c_m^p}$ and fixes the rest.
Since $m+c_m^p \leq m+c_{m+1}^p < m+1+c_{m+1}^p$, the resulting vector $f^{\tau+\cA_n(p)}.e_1 \wedge \dots \wedge e_{2i-1} = e_1 \wedge \dots \wedge e_{i-1} \wedge e_{1+c_1^p} \wedge \dots \wedge e_{i+c_i^p}$ does not vanish.

Proof of (ii):
All weight spaces in $V_{w}(\omega_{2i-1})$ are one-dimensional as all fundamental modules in type $A$ are minuscule, so we are left to check that $T_{A_n,\omega_i}(p)$ is minimal among all monomial of this given weight that act non-zero on $v_{\omega_{2i-1}}$.

For $s=0$ the minimality follows immediately, since $T(p)=x$ is the only vector with the corresponding weight, that does not annihilate $e_1\wedge \cdots \wedge e_{2i-1}$, which follows from <ref>.

We first consider the case $s=1$, say $\ell_1 \leq n \leq j_1$. We are then considering the weight space $w(\omega_{2i-1}) + \alpha_{\ell_1} + \alpha_{\ell_1 +1} +\ldots + \alpha_{j_1}$.
This implies that for all $s_k$ in $\underline w$ with $k \geq \ell_1 + n$, the coordinate in any non-zero acting monomial is equal to $1$.

Now let us consider $s_{n+i-1}$, then $f_{n+i-1}$ should act non-zero, enforcing an operator $f_{n+i-1} \cdots f_{2i-1}$.
Thus the bottom line in the tableau is filled with $1$. Inductively, all bottom rows, containing $f_{\ell_1+n}$ are filled with $1$.
The operator $f_{\ell_1}+n-1$ appears in the remaining tableau only once, in the last column. Due to weight reasons, the entry in the tableau is $0$.
The operator $f_{\ell_1+n-2}$ appears twice, to the top of the $0$ and to the left of the $0$.
Since we are considering the minimal monomial with respect to the negative lexicographic order (which orders the operators first by column from bottom to top), we put $1$ in the last column. Inductively we obtain the tableau

\begin{align*}
\begin{blockarray}{cccccccc}
\begin{block}{|cccccccc|} \hline
1 & 1 & \cdots &1 & 1 & 1& \cdots &1 \\
1 & 1 & \cdots &1 & 1 & 1& \cdots &1 \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots& \ddots &\vdots \\
1 & 1 & \cdots &1 & 1 & 1& \cdots &1 \\
1 & 1 & \cdots &1 & 0 & 0& \cdots &0 \\
1 & 1 & \cdots &1 & 1 & 1& \cdots &1 \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots& \ddots &\vdots \\
1 & 1 & \cdots &1 & 1 & 1& \cdots &1 \\ \hline
\end{block}
\end{blockarray}
\end{align*}

where the leftmost 0 is in the $j_1$-th column.
We are dualizing our argument to the column by considering the order given by $w^t$ and get

\begin{blockarray}{cccccccc}
\begin{block}{|cccccccc|} \hline
1 & 1 & \cdots &1 & 0 & 1& \cdots &1 \\
1 & 1 & \cdots &1 & 0 & 1& \cdots &1 \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots& \ddots &\vdots \\
1 & 1 & \cdots &1 & 0 & 1& \cdots &1 \\
1 & 1 & \cdots &1 & 0 & 0& \cdots &0 \\
1 & 1 & \cdots &1 & 1 & 1& \cdots &1 \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots& \ddots &\vdots \\
1 & 1 & \cdots &1 & 1 & 1& \cdots &1 \\ \hline
\end{block}
\end{blockarray}

which corresponds to a minimal monomial.

We proceed by induction on $s$ to  $1 \leq \ell_1 < \ell_2 < \cdots < \ell_s \leq i \leq j_s < \ldots < j_1 \leq n$. By induction, we can assume that the minimal monomial for the sequence $1 \leq \ell_1 < \ell_2 < \cdots < \ell_{s-1} \leq i \leq j_{s-1} < \ldots < j_1 \leq n$ corresponds to the tableau with $0$-hooks (as for the induction start) defined by the pairs $(\ell_1, j_1), \ldots, (\ell_{s-1}, j_{s-1})$.

The operator $f_{\ell_s+n-1}$ may appear in several rows while all entries in the last rows are already fixed to be $1$, so the corresponding entry in the last column equals $0$ and the rest of the induction step is similar to the induction start.

As before, we are dualizing our argument to the column, and as a result, we obtain that the predicted monomial, $\underline f^{T(p)}$, is in fact minimal of the given weight in $\Pi_w$.

§ TYPE C

This proof will be very similar to the variant for type A.

For $n \in \N$, let

\begin{align*}
B_n := (\underbrace{e_{1,\bar 1}, e_{1, \bar 2}, e_{2, \bar 2}, \dots e_{n-1, \bar {n-1}}}_{=:B_{n,1}}, \underbrace{e_{1,n},\dots,e_{1,2},e_{2,2},e_{1,1}}_{=:B_{n,2}})
\end{align*}

We show by induction over $n \in \N$ that $\tensor[^{B_n}]{(-\cC)}{_n^{B_n}}$ is upper triangular with determinant $\pm 1$.
If $n=1$, then $\tensor[^{B_1}]{(-\cC_1)}{^{B_1}} = (-1)$.

For the induction step note that
$$B_n = (B_{n-1, 1},e_{1,\bar{n-1}}, \dots, e_{n-1, \bar{n-1}}, e_{1,n},\dots,e_{n,n},B_{n-1,2})$$

\begin{equation*}
\tensor[^{B_n}]{\cC}{_n^{B_n}} = \left(\begin{array}{c|c|c}
\cC^{1,1}_{n-1} & \star & \cC^{1,2}_{n-1} \\ \hline
A & B & \star \\ \hline
\cC^{2,1}_{n-1} & \Gamma & \cC^{2,2}_{n-1}
\end{array}\right)
\end{equation*}

Where for the sake of readability we use

\begin{align*}
\tensor[^{B_{n-1}}]{\cC}{_n^{B_{n-1}}} = \round{\begin{array}{c|c}
\cC^{1,1}_{n-1} & \cC^{1,2}_{n-1} \\ \hline
\cC^{2,1}_{n-1} & \cC^{2,2}_{n-1}
\end{array}}
\end{align*}

where each $\cC^{j,k}_{n-1}$ corresponds to the induced map
$$\tensor[^{B_{n-1}}]{\cC}{_n^{B_{n-1}}}: \spitz{B_{n-1, k}} \to \C^{(n-1)^2}/\spitz{B_{n-1, 3-j}}$$

Considering <ref>, we see that $A, \Gamma$ are zero matrices and
$$B = \round{\begin{array}{ccc|c|ccc}
1 & \dots & 1 & 2 & & & \\
0 & \ddots & \vdots & \vdots & &I_{n-1} & \\
0 & 0 & 1 & 2 & & & \\\hline
0 & \dots & 0 & 1 & 0 & \dots & 0 \\\hline
& & & 0 & 1 & \dots & 1\\
& 0 & & \vdots & &\ddots & \vdots \\
& & & 0 & 0 & 0 & 1
\end{array}}$$

The key idea for this section is to realize the string polytopes for $C_n$ inside of the string polytopes for $A_{2n-1}$ (that is for fundamental weights) and use the results from the previous section.

We follow here <cit.> for the embedding. As a reminder, we are considering modules for the Lie algebra of type $C_{2n-1}$ and considering the Lie algebra embedding into a Lie algebra of type $A_{4n-3}$

\begin{align*}
f_j &\mapsto \begin{cases}
f_j - e_{2n-1+j}, & \text{if } j < 2n-1\\
f_{2n-1,4n-3}, & \text{else.}
\end{cases}
\end{align*}

We change from the standard basis of $\C^{4n-2}$ to
$$(e_1, \dots, e_{2n-1}, e_{4n-2}, -e_{4n-3}, e_{4n-4}, \dots,  e_{2n})$$

For the new basis the operators and points are translated as follows:

\begin{align*}
f_j &\mapsto \text{unfold}(f_j) := \begin{cases}
f_j + f_{4n-2-j}, & \text{if } 1 \leq j<2n-1 \\
f_{2n-1}, & \text{if } j=2n-1
\end{cases} \\
% \text{unfold}(e_{\ell,j})&= \begin{cases}
%     e_{\ell,j} + e_{2n-j,2n-\ell}, & \text{if } (\ell,j) \neq (k,\overline{k}) \text{ for all }1 \leq k \leq n \\
%     e_{\ell,j}, & \text{else}
% \end{cases} \\
\\
e_{\ell,j}&\mapsto \text{fold}(e_{\ell,j}) := \begin{cases}
e_{\ell,j}, & \text{if } (\ell,j) \in H(C_n) \\
e_{4n-2-j,4n-2-\ell}, & \text{else}
\end{cases}
\end{align*}

Here unfold maps operators from $C_{2n-1}$ to operators from $A_{4n-3}$ and fold maps points from the string polytope for $A_{4n-3}$ and weight $\omega_{2i-1}$ to the string polytope for $C_{2n-1}$ and weight $\omega_{2i-1}$.

Let $n=2$, $i=2$, hence we consider $C_3$ into $A_5$. Then $t_{C_2,\omega_2} = 2e_{1,2} + e_{2,2} + e_{1,\overline{1}}$, the image representing monomials in the Lie algebra of type $C_3$ acting on $V(\omega_3)$. Thus

\begin{align*}
\text{unfold}(\underline f^{t_{C_2,\omega_2}}) &= \text{unfold}(f_3 f_2^2 f_{3}) = f_3 (f_2 + f_4)^2 f_3 = f_3(f_2^2 + f_2f_4 + f_4f_2 +f_4^2)f_3 \\
&\overset{\ref{restrA}}{\sim}_3 f_3f_2f_4f_3 + f_3f_4f_2f_3 \overset{\ref{comm}}{\sim}_3  2f_3f_2f_4f_3
\end{align*}

As an example for $\text{fold}$:

\begin{align*}
\text{fold}(t_{A_3,\omega_2}) &= \text{fold}(e_{1,2} + e_{2,2} + e_{1,3} + e_{2,3}) = e_{1,2} + e_{2,2} + e_{1,\overline 1} + e_{1,2} = t_{C_2,\omega_2}
\end{align*}

The example actually generalizes to

If $a \in \curly{0,1}^{\binom{2n}{2}}$, then $\underline f^a$ is a summand in $\mathrm{unfold}\left(\underline f^{\mathrm{fold}(a)}\right)$, and $\text{fold}(t_{A_{2n-1},\omega_i}) = t_{C_n,\omega_i}$.

If suffices to show this for $a = e_{\ell,j}$. If $(\ell,j) \in H(C_n)$, then

\begin{align*}
\mathrm{unfold}(\underline f^{e_{\ell,j}}) = \mathrm{unfold}(f_{\ell+j-1})
\end{align*}

of which $f_{\ell+j-1}$ clearly is a summand.
Otherwise $j + \ell > 2n$ and $\fold(a)=e_{2n-j,2n-\ell}$. Then

\begin{align*}
\mathrm{unfold}\left(\underline f ^{e_{2n-j,2n-\ell}}\right) = \mathrm{unfold}(f_{4n-j-\ell-1}) = f_{4n-j-\ell-1} + f_{j+\ell-1}
\end{align*}

where the latter is $\underline f^{a}$.

The second statement follows from definition <ref>.

We have to show for all $p \in P(\omega_i)^\Z$, $t_{C_n,\omega_i} + \cC_n(p)$. Again, we show two parts:

* does not annihilate $e_1 \wedge \dots \wedge e_{2i-1}$,

* is minimal.

Proof of (i):
First we note, that due to the action with simple root operators, all summands in $\text{unfold}(\underline{f}^m).e_1 \wedge  \ldots \wedge e_{2i-1}$ have strictly increasing wedges with non-negative integer coefficients.

We fix $p \in P(\omega_i)^\Z$  and set $m = t_{C_n,\omega_i} + \cC_n(p)$. Clearly, showing that $\underline{f}^{m}$ does not annihilate $e_1 \wedge \ldots \wedge e_{2i-1}$ is equivalent to show that $\text{unfold}(\underline{f}^{m})$ does not annihilate $e_1 \wedge \ldots \wedge e_{2i-1}$.
Due to the introductory remark, it suffices to show that there is one summand in $\text{unfold}(\underline{f}^m)$ that does not annihilate $e_1 \wedge \ldots \wedge e_{2i-1}$.

$p = 0$ : This case follows from the fact that the point defined in definition <ref> provides the unique sequence for the extremal weight vector in $V(\omega_{2i-1})$.

$p \neq  0$ : We show that there is a summand of $ \mathrm{unfold}(\underline{f}^m)$ that can be reordered to be an operator for type $A_{2n-1}$ of the form $\underline f^x$, where $x = t_{A_{2n-1},\omega_i} + \cA_{2n-1}(p)$ and we embed $p$ canonically (not using fold) into the FFLV-polytope for type $A$. Then we apply our result for type $A$ and with the above consideration the proof is finished.

For simplicity, we identify this summand for $p = 0$ (thereby providing another proof for this case). The $p > 0$ case is similar but involves more complex notations.

\begin{align*}
\text{unfold}(\underline f^{t_{C_n,\omega_i}}) = &2^{k} [f_{2n-1}][f_{2n-2} f_{2n} f_{2n-1}] \cdots [f_{2n-i} f_{2n-2+i} \cdots f_{2n-2} f_{2n} f_{2n-1}] \\
&[(f_{2n-i-1} + f_{2n-1+i})(f_{2n-i-2} + f_{2n+i}) \cdots (f_{2n-2} + f_{2n})] \cdots \\
&[(f_i+ f_{4n-2-i})(f_{i+1} + f_{4n-3-i}) \cdots (f_{2i-1} + f_{4n-1-2i})]
\end{align*}

\begin{align*}
\text{unfold}(\underline f^{t_{C_n,\omega_i}}) = &2^{k} [f_{2n-1}][f_{2n-2} \hat{f}_{2n-2} f_{2n-1}] \cdots [f_{2n-i} \hat{f}_{2n-i} \cdots f_{2n-2} \hat{f}_{2n-2} f_{2n-1}] \\
&[(f_{2n-i-1} + \hat{f}_{2n-i-1})(f_{2n-i-2} + \hat f_{2n-i-2}) \cdots (f_{2n-2} + \hat{f}_{2n-2})] \cdots \\
&[(f_i+\hat{f}_i)(f_{i+1} + \hat{f}_{i+1}) \cdots (f_{2i-1} + \hat{f}_{2i-1})]
\end{align*}

for some $k \in \N$, but we will ignore the scalars here. Each column of the tableau is written in rectangular brackets for better readability.

Because of the argument above we may choose in each factor with sums the summand with the minimal index.

The summand

\begin{align*}
f := &[f_{2n-1}][f_{2n-2} f_{2n} f_{2n-1}] \cdots [f_{2n-i} f_{2n-2+i} \cdots f_{2n-2} f_{2n} f_{2n-1}] \\ &[f_{2n-i-1}f_{2n-i-2} \cdots f_{2n-2}]
\cdots [f_i f_{i+1} \cdots f_{2i-1}]
\end{align*}

\begin{align*}
f := &[f_{2n-1}][f_{2n-2} \hat{f}_{2n-2} f_{2n-1}] \cdots [f_{2n-i} \hat{f}_{2n-i} \cdots f_{2n-2} \hat{f}_{2n-2} f_{2n-1}] \\ &[f_{2n-i-1}f_{2n-i-2} \cdots f_{2n-2}]
\cdots [f_i f_{i+1} \cdots f_{2i-1}] \\
=  &[f_{2n-1}][f_{2n-2} f_{2n} f_{2n-1}] \cdots [f_{2n-i} f_{2n-2+i} \cdots f_{2n-2} f_{2n} f_{2n-1}] \\ &[f_{2n-i-1}f_{2n-i-2} \cdots f_{2n-2}]
\cdots [f_i f_{i+1} \cdots f_{2i-1}]
\end{align*}

can be reordered using <ref> to be a string in $A_{2n-1}$:

\begin{align*}
%f = g &[f_{2n-i-1}f_{2n-i-2} \cdots f_{2n-2}] \cdots [f_i f_{i+1} \cdots f_{2i-1}] \\
%f \sim_{2i-1} &[f_{2n-1}f_{2n} \cdots f_{2n-2+i}][f_{2n-2}f_{2n-1} \cdots f_{2n-3+i}] \cdots [f_{2n-i}f_{2n-i-1} \cdots f_{2n-2}f_{2n-1}] \\
%&[f_{2n-i-1}f_{2n-i-2} \cdots f_{2n-2}]
f \sim_{2i-1} f_{\rho_{2n-1}} f_{\rho_{2n-2}} \cdots f_{\rho_i} = \underline f^{t_{A_{2n-1},\omega_i}}
\end{align*}

where $\rho_j = s_{j}s_{j+1}\cdots s_{j+i-1}$ and $f_{\rho_j} := f_{j} f_{j+1} \cdots f_{j+i-1}$.

Let $n=4$, $i=3$, then $f^{t_{C_4,\omega_3}} = [f_7][f_6^2f_7][f_5^2 f_6^2 f_7][f_4f_5f_6][f_3f_4f_5]$ and

\begin{align*}
\mathrm{unfold}(f^{t_{C_4,\omega_3}}) &= [f_7][(f_6 + f_8)^2f_7]\cdots [(f_3 + f_{11}) (f_4 + f_{10}) (f_5 + f_9)] \\
&= 8 \cdot [f_7] [f_6 f_8 f_7] \cdots [(f_3 + f_{11})(f_4 + f_{10})(f_5 + f_9)]
\end{align*}

Then the summand we are interested in is $f:=[f_7] [f_6 f_8 f_7][f_5 f_9 f_6 f_8 f_7] [f_4 f_5 f_6] [f_3 f_4 f_5]$ and reordering provides obviously:

\begin{align*}
f %&= [f_7] [f_6 f_8 f_7][f_5 f_9 f_6 f_8 f_7] [f_4 f_5 f_6] [f_3 f_4 f_5] \\
%&\overset{\ref{comm}}{\sim}_{2i-1} [f_7 f_8 f_9] [f_6 f_7][f_5 f_6 f_8 f_7] [f_4 f_5 f_6] [f_3 f_4 f_5] \\
&\overset{\ref{comm}}{\sim}_{5} [f_7 f_8 f_9] [f_6 f_7 f_8][f_5 f_6 f_7] [f_4 f_5 f_6] [f_3 f_4 f_5] = f_{\rho_7} f_{\rho_6} f_{\rho_5} f_{\rho_4}f_{\rho_3} = \underline f^{t_{A_7,\omega_3}}
\end{align*}

We continue with the proof and $p > 0 $. Here, we obtain a subword of $\text{unfold}(\underline f^{t_{C_n,\omega_i}})$, so we can just repeat the procedure above to get an operator for type $A_{2n-1}$,
we claim, that $t_{\cA_{2n-1},\omega_i} - \cA_{2n-1}(p)$ yields this operator.

Indeed, let $\delta=1-\delta_{n,\ell+j-1}$, then one checks that

(see <ref>) and

(see <ref>). Thus
\begin{align*}
(t_{\cC_{n},\omega_i} - \cC_{n}(p))_{\ell, j} &=(t_{\cA_{2n-1},\omega_i} - \cA_{2n-1}(p))_{\ell, j}+\delta(t_{\cA_{2n-1},\omega_i} - \cA_{2n-1}(p))_{2n-j,2n-\ell} \\
\Rightarrow t_{\cC_{n},\omega_i} - \cC_{n}(p) &= \mathrm{fold}\left(t_{\cA_{2n-1},\omega_i} - \cA_{2n-1}(p)\right)
\end{align*}

Therefore, by Proposition <ref> we can obtain the summand $\underline f^{t_{\cA_{2n-1},\omega_i} - \cA_{2n-1}(p)}$ by choice of operators as described above. Since we already proved, that this string does not annihilate the highest weight vector in the preceding section, we get with the arguments above that neither does $t_{\cC_{n},\omega_i} - \cC_{n}(p)$.

Proof of (ii):

Let $\tilde f\sim x+\cC(p)$.

Let $f_\min= f^{t_{A_{2n},\omega_i}-\cA_{2n}(p)}$. We know from type $A$, that $f_\min$ does not annihilate the highest weight vector, therefore there has to exist a summand $\tilde f_\min$ of $\tilde f$ with $f_\min \sim_{2i-1} \tilde f_\min$. From Type $A$ we know, that $f_\min$ is minimal, therefore $f_\min \leq_A \tilde f_\min$.

Now $f_\min \leq_A \tilde f_\min \Leftrightarrow f^{x+\cC(p)} \leq_C \tilde f$, because the folding does not change the order of operators with the same weight and if we fold $\tilde f_\min$ and $f_\min$ we get $\tilde f$ and $x+\cC(p)$ back respectively.

Let $v:=t_{C_n,\omega_i}-\cC_n(p)$ and $\tilde v \sim_{2i-1} v$. It suffices to show, that $v$ is smaller than $\tilde v$ in type $C_{2n-1}$.

We define $v_\min=t_{A_{2n-1},\omega_i} + \cA_{2n-1}(p)$.
Since $\fold(v_\min)=v$, we have that $\underline f^{v_\min}.e_1\wedge \dots \wedge e_{2n-1}$ is a summand of $\underline f^v.e_1\wedge \dots \wedge e_{2n-1}$ by Proposition <ref>.
We showed in the proof above, that $v$ does not annihilate the highest weight vector, therefore there has to exist a minimal $A_{4n-3}$-string  $\tilde v_\min$ with $\fold(\tilde v_\min)=\tilde{v}$ and $\tilde v_\min\sim_{2i-1} v_\min$ as $A_{4n-3}$-strings.
Since $v_\min$ is minimal as an $A_{4n-3}$-string (with <ref> a), we know that $v_\min$ is smaller than $\tilde v_\min$ in type $A_{4n-3}$. Since the ordering is preserved under $\text{fold}$ for $\sim_{2i-1}$ equivalent monomials, we deduce $v < \tilde{v}$ for type $C_{2n-1}$ and we are done. We now show the latter:

Since $v_\min$ is minimal as $A$-string, we have $(v_\min)_q > (\tilde v_\min)_q$ (because the corresponding entry was shifted down along the anti-diagonal and fold does not change the order among operators of equal weight).

Therefore $\text{fold}(v_\min)_{\text{fold}(e_q)}>\text{fold}(\tilde v_\min)_{\text{fold}(e_q)}$. By the minimality of $q$, it follows that $v = \text{fold}(v_\min) <_C \text{fold}(\tilde v_\min) = \tilde v$.

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11institutetext: UNICAMP 22institutetext: CERN

# Charm and multi-charm baryon measurements via strangeness tracking with the
upgraded ALICE detector

David Dobrigkeit Chinellato for the ALICE Collaboration 1122
<EMAIL_ADDRESS>
###### Abstract

We present a new method for detection of multiply charmed baryons via their
decays into strange baryons, using ‘strangeness tracking’. This method makes
use of the state-of-the-art upgraded silicon detectors in ALICE during Runs 3,
4 and beyond will enable the novel possibility of tracking strange hadrons
directly before they decay, leading to a very significant improvement in
impact-parameter resolution. In this work, we will discuss how this new
technique will be crucial to distinguish secondary strange baryons originating
from charm decays from primary strange baryons. This is a particularly
interesting possibility for the $\Omega^{-}$ baryon coming from
$\Omega_{\rm{c}}^{0}\rightarrow\Omega^{-}\pi^{+}$ decays, since there is no
other relevant feeddown source for $\Omega^{-}$. This, in turn, means that the
main $\Omega^{-}$ background for the $\Omega_{\rm{c}}$ measurement will point
most accurately to the primary vertex, unlike pions or protons from other
charm baryon decays.

We will illustrate the achievable performance of strangeness tracking for the
Run 3 configuration of ALICE with the upgraded Inner Tracking System, which is
fully instrumented with silicon pixel detectors. Moreover, we will discuss the
potential of this technique in a future experiment with an extensive silicon
tracking detector with a first layer very close to the interaction point.
Finally, we will also cover other potential major applications of strangeness
tracking, including measurements of hypernuclei such as the
${}^{3}_{\Lambda}\rm{H}$.

##  1 Introduction

A fundamental ingredient of the ALICE physics programme for the new decade is
a comprehensive study of charm and multi-charm baryon production. Because
charm is exclusively produced in initial hard scatterings, such measurements
may provide unique insight into the QGP medium as well as hadronization from
proton-proton to lead-lead collisions. However, the reconstruction of charmed
hadrons poses a significant challenge. In particular, charm particles decay
into strange particles, in which case topological reconstruction can serve as
a powerful particle identification method. However, due to the large
extrapolation distances, weak decay reconstruction is usually also associated
with less than optimal pointing resolution to the primary vertex.

We propose to overcome this difficulty using a novel reconstruction technique
called ‘strangeness tracking’, which is made possible by the next-generation
detectors of the experiment. In this method, we use high-resolution silicon
detectors very close to the primary vertex to measure the hits of a weakly
decaying hadron prior to its decay and combine them with the information from
the decay daughters to significantly improve pointing resolution. In this
context, the upgraded ALICE inner tracking system, the ITS2 its2tdr , is ideal
for a first implementation of such a method. This is because the ITS2 has an
innermost silicon pixel layer at a radius of 22 mm, as opposed to 39 mm for
the previous silicon pixel detectors. As a consequence, the two multi-strange
baryons $\Xi^{-}$ and $\Omega^{-}$, reconstructed in the channels
$\Xi^{-}\rightarrow\pi^{-}+\rm{p}$ and
$\Omega^{-}\rightarrow\rm{K}^{-}+\rm{p}$ channels and having a proper lifetime
of 4.9 and 2.5 cm, respectively, are significantly more likely to leave hits
in the innermost layers of the ITS2. An example of a reconstructed
$\Omega^{-}$ decay obtained with a simulation of the full geometry of the
upgraded ALICE detector and using GEANT3 propagation is shown in Fig. 1.

Figure 1: Event display of an $\Omega^{-}\rightarrow K^{-}+\Lambda$ decay in
the upgraded ALICE detector in Run 3. The $\Omega^{-}$ hits in the inner
layers of the ALICE ITS2 are highlighted in red and are used in the
reconstruction of the weak decay when the strangeness tracking technique is
employed.

##  2 Feasibility of strangeness tracking

Algorithmically, the task of performing strangeness tracking involves a
precise enough reconstruction of the decay daughters such that the back
propagation of a $\Xi^{-}$ or $\Omega^{-}$ candidate is sufficiently
constraining for the association of the correct inner layer hits to this
trajectory. This establishes a crucial performance constraint: in order to
ensure a low fake hit association rate, the hit density in a given layer,
$\rho_{hits}$, should obey the relation
$1/\rho_{hits}\gg\delta_{\rm{search}}$, where $\delta_{\rm{search}}$ is the
search window that is used when propagating the cascade trajectory inward to
that layer. The parameter $\delta_{\rm{search}}$ has been calculated in
simulations and is below 0.05 $\rm{mm}^{2}$ for typical $\Omega^{-}$ and
$\Xi^{-}$ inwards propagation in the innermost layer, while hit densities are
not expected to exceed 50 $\rm{cm}^{-2}$ even in central Pb–Pb collisions.
Because $1/\rho_{hits}=2$ $\rm{mm}^{2}\gg 0.05$ $\rm{mm}^{2}$, no major
difficulties are expected when implementing strangeness tracking for Run 3
data. In this study, Monte Carlo association of the cascade hits will be
employed to illustrate the expected gains of the strangeness tracking method
in an idealised scenario.

##  3 Reconstruction improvements

The direct tracking of weakly decaying hadrons is expected to have a large
impact in the pointing resolution, which can, for instance, be readily
quantified by investigating the effect on the distribution of distance of
closest approach (DCA) of $\Omega^{-}$ baryons to the primary vertex. This is
shown in Fig. 2 for the $\rm{DCA}_{\rm xy}$ and the $\rm{DCA}_{\rm z}$, which
are measured in the transverse plane and in the longitudinal direction,
respectively. Three distributions are shown: (a) one in which the $\Omega^{-}$
decayed prior to the first layer and has therefore only been reconstructed
using daughter information, (b) a second curve in which the $\Omega^{-}$
decayed after the first layer but no direct measurement was performed,
illustrating the significant worsening of the DCA resolution and (c) a third
curve in which $\Omega^{-}$ baryons decay after the first layer and direct
detection information is added to the daughter information. The resulting
improvement in DCA is of a factor of approximately 4 in both transverse and
longitudinal directions, showcasing the potential of the method.

Figure 2: Distance of closest approach to the primary vertex for $\Omega^{-}$
baryons reconstructed in proton-proton collisions at 13 TeV with the upgraded
ALICE detector in Run 3 in three different configurations, as obtained with a
full detector simulation.

When used for the detection of $\Omega^{-}$ from $\Omega_{\rm{c}}^{0}$ decays,
strangeness tracking is crucial to isolate differences in the pointing to the
primary vertex between these two sources, which will be due to the lifetime of
the $\Omega_{\rm{c}}^{0}$, which has $c\tau$ = 80 $\mu\rm{m}/c$, as
illustrated in Fig. 3. While for the ITS2 the differences between primary and
secondary $\Omega^{-}$ baryons are still not too evident, this is expected to
improve significantly with further upgrades to the ALICE apparatus, which will
enable a better DCA to primary vertex resolution for primary $\Omega^{-}$;
such upgrades will be discussed in the last section of this document.

Figure 3: Distance of closest approach to the primary vertex for primary
$\Omega^{-}$ baryons and $\Omega^{-}$ from
$\Omega_{c}^{0}\rightarrow\Omega^{-}+\pi^{+}$ decays as reconstructed in
proton-proton collisions at 13 TeV with the upgraded ALICE detector in Run 3.

Similar tests were also performed for the $\Xi^{-}$ baryon. The DCA resolution
increase is comparable to the $\Omega^{-}$ one and strangeness tracking also
outperforms the situation in which the $\Xi^{-}$ decays before the first
silicon layer and therefore all daughters are tracked as best as possible.
Moreover, early tests with hypertriton reconstruction also indicate a
significant improvement in pointing performance that may also serve to greatly
reduce the background in hypertriton analyses, as DCA selections help in
removing background from secondary deuterons.

##  4 Outlook

As studied in this work, strangeness tracking promises to dramatically improve
the pointing resolution of weakly decaying particles such as the $\Xi^{-}$,
$\Omega^{-}$ and even hypernuclei. However, further upgrades will enable even
better performance. Notably, the upgrade of the ITS scheduled for the third
long shutdown of the LHC will bring the innermost layers even closer to the
interaction vertex with the ITS3 its3loi  and reduce material budget
significantly, further improving both the performance and the expected
fiducial for strangeness tracking. Internal testing of strangeness tracking in
that setup has already begun and shows significant gains.

Further in the future, a next-generation, all-silicon apparatus has been
proposed as a replacement for the TPC-based ALICE experiment alice3 . For this
new setup, a design is being considered that has a retractable silicon layer
that is positioned only 5 mm away from the beam axis during datataking and
would serve as the ultimate strangeness tracking tool. A cornerstone of the
physics programme of this setup would be the detection of the multi-charm
baryons $\Xi_{\rm{cc}}^{++}$, $\Omega_{\rm{cc}}^{+}$ and
$\Omega_{\rm{ccc}}^{++}$, all of which have decay channels into multi-strange
baryons. Also in this case, strangeness tracking studies are ongoing and the
ability to separate signal from background will be reported very soon.

## References

  * (1) B. Abelev, J. Phys. G. 41, 087002 (2014)
  * (2) L. Musa, W. Riegler, CERN-LHCC-2019-018
  * (3) D. AdamovÃ¡, arXiv:1902.01211