Let $\Omega_1$ and $\Omega_2$ be any two Schubert matroids on $[n]$. We will show that $\IP(\Omega_2) \backslash \IP(\Omega_1) + m$ is contractible for all $m \in M$.

First, since both $\IP(\Omega_1)$ and $\IP(\Omega_2)$ are contained in $[0,1]^n$, it suffices to consider the case where $m \in [-1,1]^n \cap \ZZ^n$. Otherwise, we would have $\IP(\Omega_2) \cap \IP(\Omega_1) + m = \emptyset$ so the set difference would be the entire polytope $\IP(\Omega_2)$ which is topologically a ball.

Say that $m = e_S - e_T$ where $S$ and $T$ are disjoint subsets of $[n]$. Then,

\[\left (\IP(\Omega_1) + m\right ) \cap [0,1]^n = \operatorname{conv}\{e_I \; \lvert \;  I \in \mathcal{I}(M), S \cap I = \emptyset, T \subseteq I\} + m. \]

When the polytope on the right is non-empty, it is the independence polytope of the matroid obtained by the following sequence of operations:

* Restrict $\Omega_1$ to the complement of $S$.

* Contract by $T$.

* Trivially extend the matroid so that its ground set is again $[n]$.

* Flip loops to coloops until the elements $i \in S$ are coloops and the $i \in T$ are loops.

By Proposition <ref>, the class of Schubert matroids is closed under all these operations and therefore it suffices to restrict to the case where $m =0$.

For this, consider the restriction to $\IP(\Omega_2) \backslash \IP(\Omega_1)$ of the independence-base deformation retract $F$ for $\IP(\Omega_2)$. For any independence polytope, $P$ if $x \in P$, then $x - \lambda e_i \in P$ for $0 \leq \lambda \leq x_i$. In the contrapositive, this means that if $x \not \in \IP(\Omega_1)$, then $F(x,t) \not \in \IP(\Omega_1)$ for any $t \in [0,1]$. Therefore the independence-base deformation retraction restricts to a strong deformation retract

\[ \IP(\Omega_2)\backslash \IP(\Omega_1) \to \BP(\Omega_2) \backslash \IP(\Omega_1).\]

Let $r = \rk(\Omega_2)$. If $\rk(\Omega_1) < r$, then $\BP(\Omega_2) \cap \IP(\Omega_1) = \emptyset$, since $\BP(\Omega_2)$ is contained in the hyperplane where $\sum_{i} x_i = r$ and $\IP(\Omega_1)$ is contained in the half-space where $\sum_{i} x_i \leq \rk(\Omega_1)$. Hence this set difference is contractible. Otherwise,

\[\BP(\Omega_2) \cap \IP(\Omega_1) = \BP(\Omega_2) \cap \BP(T_r(\Omega_1)),\]

where $T_r(\Omega_1)$ is the truncation of $\Omega_1$ to rank $r$. By Proposition <ref>, we see that truncations of Schubert matroids are again Schubert matroids. Therefore, we have a strong deformation retract from

\[ \IP(\Omega_2) \backslash \IP(\Omega_1) \to \BP(\Omega_2) \backslash \BP(\Omega'),\]

where $\Omega'$ is another Schubert matroid. In Theorem <ref>, we showed that this set difference is contractible or empty for any two Schubert matroids.

As a consequence of this result together with Proposition <ref>, we have a new proof of the following result of Eur, Huh, and Larson:

The nef line bundles associated to the independence polytope of (all) Schubert matroids on $[n]$ form a basis for $K_0(\Stell_n)$. Under the Chern character, this also gives a basis of $H^*(\Stell_n)$.

§.§ Full Strongly Exceptional Sequence for the Stellahedral Variety

We now show that the independence polytopes of all Schubert matroids generate $\Def(\Sigma_n)$.

We begin by noting that the cube intersection property for base polytopes also holds for independence polytopes:

Let $P \in \Def(\Sigma_{\Stell_n})$. The intersection of $P$ with a lattice translate of $[0,1]^n$ is again in $\Def(\Sigma_{\Stell_n})$. Therefore, every $P$ has strict subdivisions by translations of polytopes in $\Def(\Sigma_{\Stell_n})$ contained in a cube.

The set of indepedence polytopes $\IP(\Omega)$ for $\Omega \in \Sch_n$ is a full collection of polytopes in $\Def(\Stell_n)$.

By Proposition <ref>, it suffices to show that $\langle \IP(\Omega) \rangle$ contains all translates of independence polytopes that are contained in a cube. Since all independence polytopes contained in the cube are independence polytopes of matroids, it suffices to generate all independence polytopes of matroids.

Let $M$ be a matroid and $\IP(M)$ be its independence polytope. Using the results of Section <ref>, there is a truncated Brianchon-Gram complex $\operatorname{TBG}_{\Delta_{k,n}}^{\bullet}$ of $\BP(M)$ corresponding to an order-preserving map $\phi$, where $\phi(\hat{0}) = \BP(M)$ and every other $\phi(x)$ is the base polytope of a Schubert matroid.

The polytope $[0,1]^n$ is in $\Def(\Sigma_{\Stell_n})$ since it is the independence polytope of the uniform matroid $U_{n,n}$, which is the matroid where every subset of $[n]$ is an independent set. This means that the translation $[-1,0]^n$ is also in $\Def(\Sigma_{\Stell_n})$. Build the inclusion-exclusion complex where

\[ \phi'(x) = \phi(x) + [-1,0]^n.\]

By Proposition <ref>, this is an exact inclusion-exclusion complex for $\Sigma_{\Stell_n}$. By Proposition <ref>, the cube $[0,1]^n$ intersects $\phi'$ properly. So, we can truncate $\phi'$ by $[0,1]^n$ and so by Corollary <ref>, we obtain an exact inclusion-exclusion complex corresponding to the poset morphism

\[ \psi(x) = \left(\phi(x) + [-1,0]^n \right ) \cap [0,1]^n.\]

By Lemma <ref>, we have that for any matroid $M$,

\[ \IP(M) = \left (\BP(M) + [-1,0]^n \right ) \cap [0,1]^n. \]

Therefore, $\psi$ is an exact inclusion-exclusion complex where $\psi(\hat{0}) = \IP(M)$ and every other $\psi(x)$ is the independence polytope of a Schubert matroid. This implies that $\IP(M)$ is in $\langle \IP(\Omega) \rangle_{\Omega \in \Sch_n}$.

Combining the results of the last two sections, we have that the independence polytopes of Schubert matroids on $[n]$ form a full strongly exceptional collection of polytopes for $\Def(\Sigma_{\Stell_n})$. By the polytopal criterion, we obtain the main result of our section.

The sequence of nef line bundles $( \mathcal{L}_{\IP(\Omega)})$ for $\Omega \in \Sch_n$ is a full strongly exceptional collection for $\Stell_n$ when ordered by non-decreasing number of lattice points.

§.§ Invariance of the Collection

Let $S_n$ act on $N_{\RR} \cong \RR^n$ by

\[w(e_i) = e_{w(i)}, \]

for $w \in S_n$ and $e_i$ the standard basis of $N$. Define an $S_n$ action on subsets of $I \subseteq [n]$ and chains of subsets $F_{\bullet}$ by

\[ \phi(I) = \{ \phi(i) \; \lvert \; i \in I\}\]

\[ \phi(F_{\bullet}): \phi(F_1) \subsetneq \phi(F_2) \subsetneq \cdots \subsetneq \phi(F_k),\]

where $\phi(i)$ is the usual action of $S_n$ on $[n]$.

The $S_n$ action on $N$ gives an action on $\Sigma_{\Stell}$.

By definition, every map of the action is a lattice isomorphism of $N$. All we need to prove is that this function maps cones to cones. For any $\phi \in S_n$, it is immediate that if $I \leq F_{\bullet}$ is compatible, then so is $\phi(I) \leq \phi(F_{\bullet})$ where $\phi(F_{\bullet})$ is the chain $\phi(F_1) \subsetneq \phi(F_2) \subsetneq \cdots \subsetneq \phi(F_k)$. Further, we can verify that

\begin{align*}
\phi(\sigma_{I \leq F_{\bullet}}) &= \phi\left( \operatorname{cone}(e_i \; \lvert \; i \in I) + \operatorname{cone}(-e_{E\backslash F} \; \lvert \; F \in F_{\bullet}) \right) \\
&=\operatorname{cone}(e_i \; \lvert \; i \in \phi(I)) + \operatorname{cone}(-e_{E\backslash F} \; \lvert \; F \in \phi(F_{\bullet})) \\
&=\sigma_{\phi(I) \leq \phi(F_{\bullet})}.
\end{align*}

The full strongly exceptional collection of nef line bundles $\mathcal{L}_{\IP(\Sigma)}$ of $\Stell_n$ for $\Omega \in \Sch_n$ is $S_n$-invariant.

The induced action of $S_n$ on $M \cong \ZZ^n$ is given again by the permutation action on the coordinates. To show that our collection is invariant, it suffices to show that the polytopes indexing our collection are invariant under the $S_n$ action up to translation.

For two matroids $M_1$ and $M_2$ and any $g \in S_n$, we have that $g(\IP(M_1)) = \IP(M_2)$ if and only if $M_1$ and $M_2$ are isomorphic matroids. Since Schubert matroids are closed under isomorphism, the result follows.

In terms of the Hasset compactification description, this $S_n$-action corresponds to the $S_n$-action on the marked points.

§.§ Augmented Weak Maps and the Tilting Algebra

The description of the tilting quiver follows from the same arguments as for the permutahedral variety. Now, we form a quiver with vertices given by all Schubert matroids (not just the coloopless ones).

Let $M_1$ and $M_2$ be two matroids and $S$ be a subset of the loops of $M_1$. We say that we have an augmented weak map of $M_1$ into $M_2$ if for all $I \in \mathcal{I}(M_1)$, we have that $I \cup S \in \mathcal{I}(M_2)$. We denote the augmented weak map by

\[ M_1 \xlongrightarrow{S} M_2.\]

The composition of two augmented weak maps

\[M_1 \xlongrightarrow{S} M_2 \xdashrightarrow{T} M_3 \]

is the augmented weak map

\[ M_1 \xlongrightarrow{S \cup T} M_3.\]

An augmented weak map is indecomposable if it is not the composition of two other augmented weak maps.

The augmented weak map quiver $Q_n$ is the labeled quiver whose vertices are all Schubert matroids on ground set $[n]$ and the arrows are indecomposable augmented weak maps. The arrow $\Omega_1 \xlongrightarrow{S} \Omega_2$ is labeled by $S$. For any path $p: v \to u$, let $\operatorname{lab}(p)$ denote the union of the labels of the arrows in the path $p$.

Let $I$ denote the ideal

\[ I = \big \langle p - q \; \lvert \; p,q:v \to u \text{ for paths $p,q$ with $\operatorname{lab}(p) = \operatorname{lab}(q)$} \big \rangle.\]

The tilting quiver corresponding to the full strongly exceptional collection $(\mathcal{L}_{\IP(\Omega)})_{\Omega \in \Sch_n}$ is $(Q_n,I)$.

This follows from the same argument as Proposition <ref> by noting that there is an augmented weak map $M_1\xlongrightarrow{S} M_2$ if and only if $\IP(M_1) + e_S$ is contained in $\IP(M_2)$.

As before, the $S_n$-symmetry of the full strongly exceptional collection gives an admissible $S_n$ action on the quiver by $g \cdot \IP(\Omega) = \IP(g \cdot \Omega)$.

For $\Stell_2$, the polytopes in our full strongly exceptional collection are

point = 0,0
points = (0,0),status=open

point = 0,0,color=black
point = 2,0,color=black
points = (0,0),(2,0),status=open

point = 0,0,color=black
point = 0,2,color=black
points = (0,0),(0,2),status=open

point = 0,0,color=black
point = 2,0,color=black
point = 0,2,color=black
points = (0,0),(2,0),(0,2),status=open

point = 0,0,color=black
point = 2,0,color=black
point = 0,2,color=black
point = 2,2,color=black
points = (0,0),(2,0),(2,2),(0,2),status=open

The polytopes in the full strongly exceptional collection for $\Stell_2$.

The corresponding tilting quiver is

[shape = circle,draw=black] (A) at (0,0) $1$;
[shape = circle,draw=black] (B) at (3,1.4) $2$;
[shape = circle,draw=black] (C) at (3,-1.4) $3$;
[shape = circle,draw=black] (D) at (4.5,0) $4$;
[shape = circle,draw=black] (E) at (7,-0) $5$;
every node/.style=fill=white,circle,align=center,text width=5mm,inner sep = 0,
every edge/.style=draw = black, very thick]
[->] (A)  edge[bend left = 20] node[pos=0.6]1 (B);
[->](A)  edge[bend right = 20] (B);
[->] (A)  edge[bend left = 20]   (C);
[->] (A)  edge[bend right=20]  node[pos=0.6]2 (C);
[->] (B) edge  (D);
[->] (C) edge (D);
[->] (B) edge[bend left=20] node 2 (E);
[->] (C) edge[bend right=20] node 1 (E);
[->] (D) edge (E);

Tilting Quiver for $\Stell_2$. The labels of the nodes are chosen in the order of the polytopes above.

The $S_2$ symmetry of the quiver is given by reflection along the horizontal line containing three points. In other words, it swaps node $2$ with node $3$ and leaves the rest fixed.

§ TYPE $B_N$ PERMUTAHEDRAL VARIETY

We move to our final calculation which is the type $B_n$-analogue of the permutahedral variety $\Perm_n^B$.

§.§ Root Systems and Associated Toric Varieties

We set some notation in the theory of root systems. For all relevant definitions, see <cit.>.

* $\Phi$ be a finite crystallographic root system.

* $W$ be the corresponding Weyl group.

* $M$ denote the root lattice.

* $N$ denote the dual lattice to $N$ which is the coweight lattice of $\Phi$.

* $\omega_k$ denote the $k$th fundamental weight.

* $\mathcal{A}_{\Phi}$ be the Coxeter arrangement of $\Phi$.

* $\Sigma_{\Phi}$ be the fan generated by the maximal chambers of the Coxeter arrangement.

We will primarily focus on the type $B_n$ case. Let $[\pm n]$ be the set $\{-1,-2,\ldots,-n\} \cup \{1,2,\ldots,n\}$. For this root system, we can choose the data:

* The root system is

\[\Phi = \{ e_i - e_j \; \lvert \; i \not = j \in [n]\} \cup \{e_i + e_j \; \lvert \; i \not = j \in [n] \} \cup \{e_i \; \lvert \; i \in [n]\}\]

* The Weyl group is the group of signed permutations $W = S_n^{B}$ given by

\[\{f: [\pm n] \to [ \pm n] \; \lvert \; \text{$f$ is a bijection with $f(-i) = -f(i)$}\}.\]

* The root lattice $M$ is just the integer lattice $\ZZ^n$.

* The fundamental weights are

\[ \omega_k = e_1 + e_2 + \cdots + e_k,\]

for $k \leq n-1$ and

\[\omega_n = \frac{1}{2}(e_1 + e_2 + \cdots + e_n) \]

* The Coxeter arrangement $\mathcal{A}_{B_n}$ is the hyperplane arrangement given by the hyperplanes

\[ \mathcal{H}_{ij} = \{ x \in \RR^n \; \lvert \; x_i = x_j\},\]

for $i \not =j$ and

\[ \mathcal{H}^{-}_{ij} = \{ x \in \RR^n \; \lvert \; x_i = -x_j\},\]

also for $i \not = j$ and

\[ \mathcal{H}_i = \{x \in \RR^n \; \lvert \; x_i = 0\},\]

for all $i \in [n]$.

* We say that a subset $S \subseteq [\pm n]$ is admissible if $|S \cap \{i, -i\}| \leq 1$ for all $i \in [n]$. The rays of $\Sigma_{B_n}$ are indexed by admissible subsets and the primitive lattice point $v_S$ corresponding to the admissible subset $S$ is

\[ v_S = \sum_{i \in S \cap [n]} e_i - \sum_{-i \in S \cap [-n]} e_i\]

for all subsets $S$ with $|S| \leq n-1$ and

\[ v_S = \frac{1}{2} \left (\sum_{i \in S \cap [n]} e_i - \sum_{-i \in S \cap [-n]} e_i \right ),\]

when $|S| = n$.

In <cit.>, Batryev and Blume defined a class of smooth, projective toric varieties associated to root systems that generalize the Losev-Manin compactification of $M_{0,n}$.

For any crystallographic root system $\Phi$, the type $\Phi$ permutahedral variety $\Perm_{\Phi}$ is the toric variety associated to the fan $\Sigma_{\Phi}$ in the dual lattice $N = M^*$.

The type $A_{n-1}$-permutahedral variety is the permutahedral variety $\Perm_{n}$

We will use the notation $\Perm_n^B$ to denote the type $B_n$ permutahedral variety.

The (real) deformation cone of the fan $\Def(\Sigma_{\Phi})$ was studied in []. We will slightly adjust this definition by restricting to lattice polytopes.

A $\Phi$-generalized permutahedron is a lattice polytope in the root lattice of $\Phi$ whose normal fan coarsens $\Sigma_{\Phi}$. This is equivalently a lattice polytope whose edge directions are parallel to the roots of $\Phi$.

Every variety $\Perm_{\Phi}$ comes equipped with an action of the Weyl group $W$ induced by the action on the Coxeter arrangement. This also gives an action of $W$ on the root lattice by the usual reflection action.

In Batryev and Blume's construction, the polytopes that index the nef line bundles are polytopes in the root lattice. This is the convention we will take. In other sources, the type $\Phi$-generalized permutahedra are polytopes in the weight or coweight lattice.

We now give a description of the cones of $\Sigma_{B_n}$ in terms of signed set partitions as studied by Reiner <cit.>. A signed set partition of $[\pm n]$ is a set partition of a subset of $[\pm n]$ of the form $\{S_0, S_1, \overline{S}_1, \ldots, S_k,\overline{S}_k\}$ such that:

* If $i \in S_0$, then so is $-i$. We refer to this as the zero block of the partition. We allow $S_0$ to be empty.

* If $i \in S_{\ell}$, then $-i$ is not in $S_{\ell}$. Likewise for $S_{\ell}^{-}$. We also insist that each $S_{\ell}$ is non-empty.

* We have that $\{ -i \; \lvert \; i \in S_{\ell}\} = S_{\ell}^-$.

An ordered signed set partition is a signed set partition alongside an ordering of the subsets such that $S_i$ precedes $S_j$ if and only if $S_j^{-}$ precedes $S_i^{-}$ and $S_{0}$ is in-between $S_i$ and $S_i^{-1}$ for all $i$.

The cones of $\Sigma_{B_n}$ are in bijection with ordered sign set partitions of $[n]$ where the cone $\sigma_A$ of the ordered signed set partition $A$ is given by

* $x_i = x_j$ if $i$ and $j$ are in the same part for any $i,j \in [\pm n]$.

* $x_i = 0$ if $i \in S_0$.

* $x_i > x_j$ if $i$ appears in a block that preceeds the block containing $j$ for $i,j \in [\pm n]$.

where we use the convention that $x_{-j} = - x_j$ when $j \in [n]$.

§.§ Coxeter Matroids and Delta Matroids

Our construction of the full strongly exceptional collection for type $B_n$ will be done in analogy with the type $A_n$ case. Our main objects will come from the theory of Coxeter matroids and delta matroids.

Let $W$ be the Weyl group of $\Phi$ and let $I$ be a subset of the simple roots with corresponding parabolic subgroup $W_I$. A Coxeter matroid of type $(\Phi,I)$ is a subset $\mathcal{M} \subseteq W/W_I$ such that for any $w \in W$, there is a unique $A \in \mathcal{M}$ such that for all $B \in \mathcal{M}$

\[ w^{-1}B \leq w^{-1} A\]

where $\leq$ is the Bruhat order on the quotient $W/W_I$.

There have been various different constructions of a Coxeter matroid polytope. In the main modern definition, one usually takes the Coxeter matroid polytope to be the convex hull of the orbit of the sum of fundamental weights $\sum_{i \in I}\omega_i$ under $\mathcal{M}$ <cit.>. Unfortunately, this gives a polytope in the weight lattice (of the root system) while we are looking for polytopes in the root lattice. Instead, we will study a special class of Coxeter matroids coming from the theory of delta matroids and use their associated polytopes.

A delta matroid on ground set $E$ is a collection of subsets $\mathcal{F}(\mathcal{M})$ of $E$, called feasible sets, such that for any two $A_1,A_2 \in \mathcal{F}(\mathcal{M})$ and any $e \in A_1 \Delta A_2$ there exists an $f \in A_1 \Delta A_2$ such that $A_1 \Delta \{e,f\}$ is in $\mathcal{F}(\mathcal{M})$. Here $A_1 \Delta A_2$ represents the symmetric difference of the two sets. The feasible polytope of $\mathcal{M}$ is the polytope

\[ \FP(\mathcal{M}) = \operatorname{conv}\left (e_S \; \lvert \; S \subseteq \mathcal{F}(\mathcal{M}) \right ).\]

The set of delta matroids on $[n]$ is in bijection with the set of Coxeter matroids of type $(B_n, \omega_n)$. The bijection is the map induced by the bijection

\[ 2^{[n]} \to W/W_{I} \]

given by $S \mapsto [\pi]$ where $\pi$ is the signed permutation

\[ \pi(i) = \begin{cases}
-i & \text{if $i \in S$} \\
i & \text{if $i \not \in S$.}
\end{cases}.\]

Further $\FP(\mathcal{M})$ is a type $B_n$ Coxeter matroid polytope and hence a $B_n$-generalized permutahedron.

As with type $A_n$ matroids, we have the following classification of which $B_n$-generalized permutahedra come from delta matroids.
A polytope in $\RR^n$ is the base polytope of a delta matroid if and only if it is a lattice $B_n$-generalized permutahedra and every vertex is a vertex of the cube $[0,1]^n$.

A loop of a delta matroid is an element $i \in [n]$ that is in no feasible set and a coloop is an element that is in every feasible set. As with matroids, the feasible polytope of a delta matroid with a loop and the feasible polytope of the delta matroid obtained by changing the loop to a coloop are translations of each other.

The type $B_n$ Gale order on $2^{[n]}$ is the poset given by $A \preceq B$ for subsets $A,B \subseteq [n]$ where $A = \{a_1 < \cdots < a_j\}$ and $B = \{b_1 < \cdots < b_k\}$ if and only if $j \leq k$ and $a_{j-i} \leq b_{k - i}$ for all $i \in \{0,\ldots,j-1\}$.

By Exercise 8.5 of <cit.>, this order is isomorphic to the induced Bruhat order on the quotient maximal parabolic quotient of $W$ given by the weight $w_{n}$ of type $B_n$.

The standard Schubert delta matroid $\Delta_S$ is the delta matroid given by

\[\mathcal{F}(\Delta_S) = \{ T \subseteq [n] \; \lvert \; \text{$T \preceq S$} \}. \]

The Schubert Delta matroid $\Delta_S^w$ for $w \in S_n^B$ is the delta matroid defined by the polytope $w \cdot \BP(\Delta_S) + e_T$ where $T = \{ w(i) \in [n] \; \lvert \; w(i) < 0\}$. This translation ensures that this polytope is contained in the cube $[0,1]^n$.

Let $\operatorname{DSch}_n$ denote the set of delta Schubert matroids on $[n]$ and $\overline{\operatorname{DSch}}_n$ denote those which are loopless.

As with matroids, we have various operations on delta matroids. We only use the following:

Let $\mathcal{M}$ be a delta matroid on $[n]$.

* The restriction of $\mathcal{M}$ to $S \subseteq [n]$ is the delta matroid on $S$ given by

\[\mathcal{F}(\mathcal{M}|S) = \{ B \in \mathcal{F}(\mathcal{M}) \; \lvert \; B \subseteq S\}.\]

* If $\mathcal{M}$ is a delta matroid on $S$ and $S \subseteq T$, we say that the trivial extension of $\mathcal{M}$ to $T$ is the delta matroid with the same feasible sets but where we changed the ground set from $S$ to $T$. This amounts to viewing all $i \in T \backslash S$ as loops.

* If $i \in [n]$ is a loop of $\mathcal{M}$, we say that the matroid obtained by flipping the loop $i$ to a coloop is the matroid $\mathcal{M}'$

\[\mathcal{F}(\mathcal{M}') = \{ S \cup \{i\} \; \lvert \; S \in \mathcal{F}(\mathcal{M})\}. \]

The class of Schubert delta matroids is closed under restriction, trivial extensions, and flipping loops.

§.§ Feasible Polytopes of Delta Schubert Matroids are Exceptional
In this section we show the set feasible polytopes of loopless Schubert delta matroids is strongly exceptional.

Recall that for a polytope $P$ and a vertex $v$, the tangent cone $C_v$ of $v$ is the cone

\[C_v = \{x \in \RR^n \; \lvert \; v- \lambda x \in P \text{ for some $\lambda \in \RR^n_{\geq 0}$}\}. \]

It is related to the normal cone $\sigma_v$ of the vertex $v$ by $-\sigma^{\vee} = C_v$.

Let $P \in \Def(\Sigma)$ and $v$ be a vertex of $P$. Let $C$ be a cone which contains the tangent cone of $v$. Then, for any $m \in M$, the set difference $P \backslash C +m$ is contractible or empty.

Consider the straight-line deformation retract $F: P \times [0,1] \to \{v\}$ given by

\[ F(x,t) = v - t(v-x).\]

This is well-defined since if $x \in P$ then so is the line segment between $x$ and $v$ since $P$ is convex. Suppose that $P \backslash C + m$ is non-empty which happens exactly when $v  \not \in C + m$.

For any point $x \in P$, we have that $v-x \in C$ since $C$ contains the tangent cone at $v$. Therefore, if $x - t(v-x) \in C+m$ for any $t \in [0,1]$, we must have that $x$ is also in $C+m$ since $C + m$ is closed under addition. In the contrapositive, this means that if $x \in P \backslash C+m$, then $x - t(v-x) \in P \backslash C+m$ for all $t \in [0,1]$.

This implies that the straight-line deformation retract $F$ restricts to a strong deformation retract which shows that $P \backslash C+m$ is contractible.

This lemma is useful since the feasible polytopes of delta Schubert matroids polytopes can be understood as intersections of cones with the cube.

Let $S$ be an admissible subset of $[\pm n]$ of cardinality $n$ and $w \in S_n^B$. Then, we have the intersection

\[ \Delta_S^w = -\sigma_w^{\vee}+ e_{S \cap [n]} \cap [0,1]^n\]

The set of feasible polytopes of loopless Schubert delta matroids on $[n]$ forms a strongly exceptional collection.

Let $\Delta_1$ and $\Delta_2$ be any two loopless Schubert delta matroids. We show that $\FP(\Delta_2) \backslash \FP(\Delta_1) +m$ is contractible or empty for all $m \in M$.

Since $\Delta_2$ is a Schubert delta matroid, we have that $\FP(\Delta_2)$ is contained in the cube $[0,1]^n$ by Theorem <ref>. Thus,

\[ \FP(\Delta_2) \backslash (\FP(\Delta_1) + m) = \FP(\Delta_2) \backslash \left( (\FP(\Delta_1) +m) \cap [0,1]^n \right).\]

We have that $(\FP(\Delta_1) +m)$ is also contained in the cube $[0,1]^n$. So, in order for $(\FP(\Delta_1) +m) \cap [0,1]^n$ to be non-empty, we must have $m = e_S - e_T$ for disjoint subsets $S$ and $T$ of $[n]$. With this $m$, we can describe the intersection as

\[(\FP(\Delta_1) + m)\cap[0,1]^n = \{x \in \FP(\Delta_1) \;\lvert \; x_i = 0 \text{ for $i \in S$ and $x_i = 1$ for $i \in T$}\}. \]

Since every feasible polytope in a cube is the feasible polytope of a delta matroid, it suffices to specify which vectors $e_I$ are in the polytope. By this, the set on the right-hand side is the feasible polytope of the delta matroid $\FP(\Delta)$ where $\Delta$ is the delta matroid obtained by the following steps: first, restrict $\Delta_1$ to $[n] \backslash (S \cup T)$; second take the trivial extension of this restriction by $S \cup T$; and finally, flip the elements in $T$ to coloops. By Proposition <ref>, $\Delta$ is again a Schubert delta matroid. Hence, it suffices to consider the case where $m=0$.

Now consider $\FP(\Delta_2) \backslash \FP(\Delta_1)$. Say that $\Delta_1 = \Delta_S^w$, We then have

\[\FP(\Delta_2) \backslash \FP(\Delta_1) = \FP(\Delta_2) \backslash \left ( -\sigma_w^{\vee} + e_S \cap [0,1]^n \right ). \]

Since $\FP(\Delta_2)$ is also contained in $[0,1]^n$ and $e_S$ is a vertex of $[0,1]^n$, we get

\[ \FP(\Delta_2) \backslash \FP(\Delta_1) = \FP(\Delta_2) \backslash \left ( -\sigma_w^{\vee} + e_S \right ).\]

Since the normal fan of $\FP(\Delta_2)$ coarsens the type $B_n$ fan $\Sigma_{B_n}$, we see that $\sigma_w^\vee$ contains a tangent cone of $\FP(\Delta_2)$. Therefore, Lemma <ref> applies and shows that this set difference is either empty or contractible.

This result and Proposition <ref> gives a new proof of the following result of Eur, Fink, Larson, and Spink:

The classes $[\FP(\Delta)]$ for $\Delta \in \overline{\operatorname{DSch}}_n$ form a basis for $\McM(\Sigma_{B_n})$ and so the corresponding line bundles give a basis for $K_0(\Perm_n^B) \cong \McM(\Sigma_{B_n})$. Under the Chern character, this also gives a basis of $H^*(\Perm_n^B)$.

§.§ Fully Strongly Exceptional Sequence for $\Perm_n^B$

We now show that the nef line bundles of $X_{B_n}$ corresponding to loopless[As with the type $A_n$ case, the loopless condition ensures that we only choose each line bundle once since changing loops to coloops gives isomorphic line bundles.] Schubert delta matroids form a full strongly exceptional collection. The arguments here are a type $B_n$ version of the arguments of Sections <ref> and <ref>.

We have the following Coxeter extensions of the results.

The intersection of a $B_n$-generalized permutahedron $P$ with the unit cube $[0,1]^n$ is a translation of the feasible polytope of a delta matroid. Further, this gives a strict subdivision of $P$ into these feasible polytopes.

The set $\overline{\operatorname{DSch}_n}$ of base polytopes of coloopless Schubert delta matroids on ground set $[n]$ polytopally generate $\Def(\Sigma_{B_n})$.

If a Schubert delta matroid $\Delta$ contains a loop $i$, then changing the loop to a coloop gives another Schubert delta matroid. From the polytope perspective, this amounts to translating the base polytope of $\Delta$ by $e_i$. Hence, every translation of a Schubert delta matroid is contained in $\langle \overline{\operatorname{DSch}_n} \rangle_0$.

By Lemma <ref>, we see that for any type $B_n$ generalized permutahedron $P$ contained in the cube $[0,1]^n$, we obtain a truncated Brianchon-Gram complex for $P$ where every term besides $P$ consists of intersections of $[0,1]^n$ with cones of $\Sigma_n$. These intersections are exactly base polytopes of delta Schubert matroids by Proposition <ref>. Therefore, this exact sequence shows that all $P$ contained in $[0,1]^n$ are in $\langle \overline{\operatorname{DSch}_n} \rangle_1$.

By Proposition <ref>, the intersection of any lattice $B_n$-generalized permutahedron with a translation of a cube is again in $\Def(\Sigma_{B_n})$. By tiling $M_{\RR}$ by the lattice translations of $[0,1]^n$, we have a strict subdivision of $P \in \Def(\Sigma_{B_n})$ into translations of polytopes contained in $[0,1]^n$. Using the exact inclusion-exclusion complex given by this subdivision, we see that $P \in \langle \overline{\operatorname{DSch}_n} \rangle_2$ for all $P \in \Def(\Sigma)$.

Combining this with our previous section, we see that the set of feasible polytopes of loopless Schubert delta matroids on $[n]$ forms a full strongly exceptional collection of polytopes for $\Poly(\Sigma_{B_n})$. By the polytopal criterion, we have

The sequence of nef line bundles $( \mathcal{L}_{\FP(\Delta)})$ for $\Delta \in \overline{\operatorname{DSch}_n}$ form a full strongly exceptional collection for $\Perm_n^B$ when ordered by non-decreasing lattice points.

§.§ Invariance of the Collection

The invariance of our collection follows immediately from root theory. In particular, the Weyl group $W$ acts on the root system and the corresponding Coxeter complex.

The full strongly exceptional collection $(\mathcal{L}_{\FP(\Delta)})$ for $\Delta \in \overline{\operatorname{DSch}}_n$ is invariant under the action of $S_n^B$.

It again suffices to check that the class of polytopes indexing our line bundles is invariant up to translation under the action of $S_n^B$ on $M$. This is the reflection action and we have defined our class of polytopes as the polytopes in the $S_n^B$-orbits of feasible polytopes of standard Schubert delta matroids, so the result follows.

§.§ Augmented Inclusion Maps of Delta Matroids and the Tilting Algebra

As in the type $A_n$ case, the tilting quiver of our collection can be described by augmented inclusion. We will again focus on coloopless Schubert delta matroids which up to translation of feasible polytopes is the same set as loopless Schubert delta matroids.

Let $\Delta_1$ and $\Delta_2$ be two delta matroids on ground set $[n]$ and $S$ be a subset of loops of $\Delta_1$. We say that we have an $S$-augmented inclusion of $\Delta_1$ into $\Delta_2$ if for every $B \in \mathcal{B}(\Delta_1)$ we have $B \cup S \in \mathcal{B}(\Delta_2)$. We denote the augmented inclusion by

\[\Delta_1 \xlongrightarrow{S} \Delta_2. \]

The composition of two augmented inclusions

\[ \Delta_1 \xlongrightarrow{S} \Delta_2 \xlongrightarrow{T} \Delta_3\]

is the augmented inclusion

\[ \Delta_1 \xlongrightarrow{S \cup T} \Delta_3.\]

We say that an augmented inclusion of delta matroids is indecomposable if it is not the composition of two other augmented inclusions.

Consider the labeled quiver $Q_n$ where the vertices of $Q_n$ are the coloopless Schubert delta matroids on ground set $[n]$ and the arrows are the indecomposable augmented inclusions. The arrow corresponding to $\Delta_1 \xlongrightarrow{S} \Delta_2$ is labeled by $S$. For any path $p:v \to u$, let $\operatorname{lab}(p)$ be the union of the labels appearing in the arrows of the path $p$. Define the relations $I$ on the path algebra $\kk Q_n$ by

\[I = \big \langle p - q \; \lvert \; p,q: v \to u \text{ paths with $\operatorname{lab}(p) = \operatorname{lab}(q)$} \big \rangle. \]

The tilting quiver correpsonding the full strongly exceptional collection $(\mathcal{L}_{\FP(\Omega)})$ as $\Omega$ ranges over coloopess Schubert delta matroids is $(Q_n, I)$.

This follows from the same arguments as Proposition <ref>. We chose our definition of Schubert delta matroids so that their feasible polytopes are all contained in $[0,1]^n$. So the polytope $\Delta_1 + e_S$ is contained in $\Delta_2$ if and only if $i$ is a loop of $\Delta_1$ for all $i \in S$ and after flipping the loop to a coloop, every feasible set of $\Delta_1$ is a feasible set of $\Delta_2$.

The following polytopes index the full strongly exceptional collection of line bundles on $X_{B_2}$

point =0,0

point =0,0,color=black
point =2,0,color=black
points = (0,0),(2,0),status = open

point =0,0,color=black
point =0,2,color=black
points = (0,0),(0,2),status = open

point =0,0,color=black
point =2,0,color=black
point =0,2,color=black
points = (0,0),(2,0),(0,2),status = open

point =2,2,color=black
point =0,2,color=black
point =2,0,color=black
points = (2,2),(2,0),(0,2),status = open

point =0,0,color=black
point =2,2,color=black
point =0,2,color=black
points = (0,0),(2,2),(0,2),status = open

point =0,0,color=black
point =2,0,color=black
point =2,2,color=black
points = (0,0),(2,0),(2,2),status = open

point =0,0,color=black
point =2,0,color=black
point =0,2,color=black
point =2,2,color=black
points = (0,0),(2,0),(2,2),(0,2),status = open

Polytopes indexing the collection for $X_{B_2}$.

The corresponding quiver is below

every node/.style=draw = black,shape = circle,align=center,
every edge/.style=draw = black, very thick]
(A) at (0,0) $1$;
(B) at (3,2) $2$;
(C) at (3,-2) $3$;
(D) at (6,3) $4$;
(E) at (6,1) $5$;
(F) at (6,-1) $6$;
(G) at (6,-3) $7$;
(H) at (9,0) $8$;

every node/.style=fill=white,circle,align=center,text width=5mm,inner sep = 0,
every edge/.style=draw = black, very thick]
[->] (A)  edge[bend left = 20] node[pos=0.6]1 (B);
[->](A)  edge[bend right = 20] (B);
[->] (A)  edge[bend left = 20]   (C);
[->] (A)  edge[bend right=20]  node[pos=0.6]2 (C);
[->] (B) edge (D);
[->] (B) edge node[pos=0.3] 2(E);
[->] (B) edge node[pos=0.3] 2(F);
[->] (B) edge (G);
[->] (C) edge (D);
[->] (C) edge node[pos=0.3] 1(E);
[->] (C) edge (F);
[->] (C) edge node[pos=0.3] 1(G);
[->] (D) edge (H);
[->] (E) edge (H);
[->] (F) edge (H);
[->] (G) edge (H);

Tilting Quiver for $\Perm_2^B$. The labels of the nodes are chosen in the order of the polytopes above.

The group $S_2^B$ is generated by the cycle $(12)$ and the permutation $1 \mapsto -1$ and $2 \mapsto 2$. The first permutation swaps nodes $2$ and $3$ as well as nodes $6$ and $7$. The second permutation swaps nodes $4$ and $7$ as well as nodes $5$ and $6$.

§.§ What about the other types?

Naturally, one would hope that these results extend to the toric varieties of other root systems. However, some difficulties arise with types other than $A$ and $B$. Problems present themselves even in the type $C_n$ case.

To see this, let $\Perm_n^C$ be the toric variety corresponding to the type $C_n$ root system. The fan of type $C_{n}$ is the same as the fan of type $B_{n}$, but they have different underlying lattice. In particular, the weight lattice $M$ of $\Perm_n^C$ is the root lattice of type $C_n$ which, after choosing a standard description of the root system, consists of lattice points in $\ZZ^n$ whose coordinate sums are even.

Since the theory of Coxeter matroids is the same for type $B_n$ and type $C_n$ root systems, one could hope that the collection of Coxeter matroid polytopes corresponding to Schubert delta matroids continues to be a full strongly exceptional collection. The hope is ill-founded for two reasons.

First, the nef line bundles corresponding to these Coxeter matroids are not all acyclic. Namely, the Schubert matroids in type $C_n$ include the class of the point $\{\operatorname{pt}\}$ and the polytope below

point = 0,0,color=black
point = 2,0,color=black
point = 0,2,color=black
point = 2,2,color=black
point = 1,1,color=black
points = (0,0),(2,0),(2,2),(0,2),status = open

Type $B_n$ Schubert Coxeter matroid polytope corresponding to the uniform delta matroid.

In order for a collection of line bundles that contains the structure sheaf (the line bundle associated to the class of a point) to be strongly exceptional, every sheaf must be acyclic. By standard toric geometry arguments, we know that $H^{n-1}(X,\mathcal{L}_P^{-1})$ has dimension equal to the number of interior lattice points of $P$. The polytope above has an interior lattice point and so has nontrivial $H^1(\mathcal{L}_{P}^{-1})$. Therefore, this collection is not exceptional with respect to any order.

Second, even if it were exceptional, our argument that the type $B_n$ Coxeter matroid polytopes corresponding to Schubert delta matroids is full no longer works. In particular, we no longer have a truncated Brianchon-Gram complex to reduce the arbitrary Coxeter matroid polytope case to the Schubert case. See Remark 3.15 in <cit.> for more discussion.

As discussed in <cit.>, the type $C_n$ permutahedral variety is a quotient of the type $B_n$ permutahedral variety. It might be possible to give a description of the derived category of $\Perm_n^C$ through this quotient but we do not pursue that in this paper.

§ COMPATABILITY OF OUR COLLECTIONS WITH THE CUSPIDAL DECOMPOSITION

We now move to our final section where we show that our full strongly exceptional collections and their induced semi-orthogonal decompositions of the derived category are compatible with a recursively defined semi-orthogonal decomposition of varieties studied by Castatev and Tevelev.

In the process of studying the moduli space $\overline{M}_{0,n}$, they defined the following category and the related semi-orthogonal decomposition:

Given a collection of smooth maps $\pi_i: X \to Y_i$ for $i \in I$, we say that a sheaf $\mathcal{E}$ of $X$ is cuspidal if $R\pi_{i*}(\mathcal{E}) = 0$ for all $i \in I$. The cusipdal derived category $\operatorname{D}^b_{cusp}(X)$ is the subcategory of $\D(X)$ consisting of all cuspidal coherent sheaves.

Let $X$ be a contravariant functor from the category of subsets of $[n]$ with morphisms given by inclusions to smooth projective varieties. For $S \subseteq T$, let $\pi_{S/T}$ be the map assigned to $X_T \to X_S$. We call these the forgetful maps. Suppose the following three conditions hold:

* We have

\[ R\pi_{i*}(\mathcal{O}_{X_S}) = \mathcal{O}_{X_{S \backslash i}},\]

for all $i \in S$.

* For all $i,k \in S$, with $i \not = k$, the morphisms

\[\pi_i:X_{S\backslash k} \to X_{S \backslash \{i,k\}} \quad \quad \text{and} \quad \quad \pi_{k}: X_{S \backslash i} \to X_{S \backslash \{i,k\}} \]

are Tor-independent.

* If we let

\[Y := X_{S \backslash i} \times_{X_{S \backslash \{i,k\}}} X_{S \backslash k}\]

and $\alpha_{i,k}: X_S \to Y$ be the map induced by $\pi_i$ and $\pi_k$, then

\[R\alpha_{i,k*} \mathcal{O}_{X_S} = \mathcal{O}_Y. \]

Under these assumptions, we have a semi-orthogonal decomposition

\[\operatorname{D}^b(X_S) = \left \langle \D_{cusp}(X_S), \left \{ \pi_{K}^* \D_{cusp}(X_{S \backslash K}) \right \}_{K \subset S}, \pi_S^*\D_{cusp}(X_{\emptyset}) \right \rangle, \]

as $K$ runs over all proper subsets of $S$ in order of increasing cardinality. We refer to this as the cuspidal decomposition of the category.

§.§ Forgetful Species of Toric Varieties and their Cuspidal Decomposition

We begin by introducing a special class of contravariant functors for which we always have a cuspidal decomposition. This class is based on Joyal's theory of species <cit.> and Aguiar and Mahajan's theory of Hopf monoids <cit.> as applied to generalized permutahedra by Ardila and Aguiar <cit.>.

For any finite set $S$, let $\RR^S$ denote the vector space of real-valued functions on $S$ and let $e_i$ be the standard basis where $i \in S$. Let $M_S$ be the integer lattice in $\RR^S$ and $N_S$ be the dual lattice.

Let $\mathbf{F}$ be a contravariant functor $\mathbf{F}$ from the category of finite sets with inclusions to the category of toric varieties with toric morphisms such that the weight lattice of $\mathbf{F}[S]$ is $M_S$ and the map between $\mathbf{F}[T]$ and $\mathbf{F}[S]$ for $S \subseteq T$ is induced by the inclusion $M_S \hookrightarrow M_T$.  Let $\pi_{S,T}$ denote both the dual map $N_T \mapsto N_S$ and the induced map of toric varieties. We denote the fan of $\mathbf{F}[S]$ by $\Sigma_S$.

We say that $\mathbf{F}$ is a forgetful species of toric varieties if for every $S \subsetneq T$, we have that

\[\Sigma_S = \{ \pi_{S,T}(\sigma) \; \lvert \; \sigma \in \Sigma_T\}.\]

The forgetful maps of a forgetful species satisfy the following nice property:

Let $f: X_{\Sigma_1} \to X_{\Sigma_2}$ be a toric morphism of toric varieties. We say that $f$ is weakly semistable if

* Every cone of $\Sigma_1$ surjects onto a cone of $\Sigma_2$ under the induced map $N_1 \to N_2$.

* Whenever $f(\sigma_1) = \sigma_2$, then we have an equality of monoids $f(\sigma_1 \cap N_1) = \sigma_2 \cap N_2$.

We say that $f$ is semistable if the varieties $X_{\Sigma_1}$ and $X_{\Sigma_2}$ are smooth.

Let $\mathbf{F}$ be a forgetful species of toric varieties. The forgetful maps $\pi_{S,T}: \mathbf{F}[T] \to \mathbf{F}[S]$ are weakly semistable.

The main consequences of this property that we will need are the following two results:

If $f: X_{\Sigma_1} \to X_{\Sigma_2}$ is weakly semistable, then $f$ is flat and has reduced fibers.

If $f_1: X_{\Sigma_1} \to X_{\Sigma_3}$ is a semistable map, then for any toric morphism $f_2: X_{\Sigma_2} \to X_{\Sigma_3}$, we have that the fiber product $X_{\Sigma_1}\times_{X_{\Sigma_3}}X_{\Sigma_2}$ is a toric variety and hence is the fiber product in the category of toric varieties with toric morphisms.

Let $\mathbf{F}$ be a forgetful species of smooth projective toric varieties. Then, the conditions of Theorem <ref> hold. Therefore, every $\mathbf{F}[S]$ has a cuspidal decomposition.

The conditions follow from the flatness and properness of our forgetful maps.

* For this, we only have to verify that $\operatorname{R}^p\pi_{i*}(\mathcal{O}_{\mathbf{F}[S]})$ vanishes for $p \geq 1$. It is proven in <cit.> that if $f: X \to Y$ is a proper toric morphism then the higher direct images of the structure sheaves vanish. A map between projective varieties is projective and hence proper which implies the condition.

* Two maps are Tor-independent when one of them is flat. So it suffices to show that the maps $\pi_{i}$ are flat. This follows from Theorem <ref> since our forgetful maps are semistable.

* By Theorem <ref>, we know that $Y$ is a toric variety since our forgetful maps are semistable. Since each $\pi_i$ is proper, we know that $\alpha_{i,k}$ is also proper by base change and now the argument for part (1) works again.

Let $\mathcal{P}_E$ be a full strongly exceptional collection of lattice polytopes for the fan of $\mathbf{F}[E]$. We say that the collections $\{\mathcal{P}_E\}$ are compatible with the forgetful maps if

* For any $S \subseteq E$, the inclusion map $i: M_S \to M_E$ gives a bijection between $\mathcal{P}_S$ and the polytopes $P \in \mathcal{P}_E$ such that some translation of $P$ is contained in $M_S \subseteq M_E$.

We say that a polytope $P \subseteq \RR^T$ is generic if $P+m \not \subseteq \RR^S$ for any $m \in M$ and any $S \subset T$. For instance, if $P$ is full-dimensional then it is generic.

Let $\mathbf{F}$ be a forgetful species of smooth varieties and $P_S$ be full strongly exceptional collections of polytopes for the fan of the variety $\mathbf{F}[S]$ such that $\{P_S\}$ is compatible with the forgetful maps. Let $\mathcal{P}_T^{gen}$ be the generic polytopes in $\mathcal{P}_T$.

Then the line bundles $\mathcal{L}_{P_i}^{-1}$ for $P_i \in \mathcal{P}_T^{gen}$ form a full strongly exceptional collection for $\D_{cusp}(\mathbf{F}[T])$ when they are ordered by non-increasing number of lattice points.

First, we show that these line bundles are in the cuspidal category. Consider a map $\pi_{S,T}: \mathbf{F}[T] \to \mathbf{F}[S]$ for $S \subseteq T$. Let $P \in \mathcal{P}_T^{gen}$.
To show that $R\pi_{S,T*}(\mathcal{L}_P^{-1}) = 0$, it suffices to show that $\operatorname{RHom}^p_{\mathbf{F}[S]}(\mathcal{L}_Q^{-1}, R\pi_{S,T*}(\mathcal{L}_P^{-1})) = 0$ for all $p \geq 0$ and for all $Q \in \mathcal{P}_S$. This is true since the collection $\mathcal{L}_{Q}^{-1}$ is a full strongly exceptional collection for $\mathbf{F}[S]$.

Consider $Q \in \mathcal{P}_S$. Let $Q'$ denote the polytope in $\mathcal{P}_T$ that indexes the line bundle $\pi_{S,T}^*(\mathcal{L}_Q)$. This is just the same polytope as $Q \subseteq M_S$ viewed as a polytope in $M_T$. Then, using the push-pull adjunction, exactness of pullbacks for locally free sheaves, and the fact that duality commutes with pullbacks for locally free sheaves, we have

\begin{align*}
\operatorname{RHom}^p_{\mathbf{F}[S]}( \mathcal{L}_Q^{-1}, R\pi_{S,T*}(\mathcal{L}_P^{-1}))
&= \operatorname{RHom}^p_{\mathbf{F}[T]}( \pi_{S,T}^*(\mathcal{L}_Q)^{-1}, \mathcal{L}_P^{-1})\\
&= \operatorname{RHom}^p_{\mathbf{F}[T]}(\mathcal{L}_{Q'}^{-1}, \mathcal{L}_{P}^{-1}) \\
&= \operatorname{RHom}^p_{\mathbf{F}[T]}(\mathcal{L}_P, \mathcal{L}_{Q'}).
\end{align*}

Since $P$ and $Q'$ are both in the full strongly exceptional collection $\mathcal{P}_T$, we know that this expression vanishes for $p \geq 1$. All that remains is to compute this for $p = 0$. For this, we have the usual toric-geometric interpretation of Hom-spaces given by

\[\operatorname{Hom}_{\mathbf{F}[T]}(\mathcal{L}_P, \mathcal{L}_{Q'}) \cong \ZZ[m \in M \; \lvert \; P + m \subseteq Q]. \]

By construction, $Q'$ is contained in $\RR^S$ and since $P$ is generic there is no $m$ such that $P+m \subseteq Q' \subseteq \RR^S$. This shows that $\operatorname{RHom}^p_{\mathbf{F}[S]}(\mathcal{L}_{Q'}^{-1}, R\pi_{S,T*}(\mathcal{L}_P^{-1})) = 0$ for all $p \geq 0 $ and for all $Q \in \mathcal{P_S}$. Thus $R\pi_{S,T*}(\mathcal{L}_P^{-1}) = 0$. This argument holds for all $S \subset T$ and so $\mathcal{L}_P^{-1}$ is in $\D_{cusp}(\mathbf{F}[T])$.

Next, we show that $\{\mathcal{L}_{P}^{-1}\}_{P \in \mathcal{P}_T^{gen}}$ generates the cupsidal category. To see this, we show that if $\mathcal{E} \in \D_{cusp}(\mathbf{F}[T])$ and $\mathcal{E} \in \langle \mathcal{L}_{P}^{-1} \rangle_{P \in \mathcal{P}_T^{gen}}^{\perp}$, then it must be $0$. Let $Q'$ be a non-generic polytope in $\mathcal{P}_T$. Suppose $S$ is a subset such that $Q' \subseteq \RR^S$. Then by the compatibility condition, we have that there is some $Q \in \mathcal{P}_S$ such that $\mathcal{L}_{Q'} = \pi_{S,T}^*(\mathcal{L}_Q)$. Since $\mathcal{E}$ is in $\D_{cusp}(\mathbf{F}[T])$, this implies

\[\operatorname{RHom}_{\mathbf{F}[S]}^p(\mathcal{L}_{Q'}^{-1}, R\pi_{S,T*}(\mathcal{E})) = \operatorname{RHom}_{\mathbf{F}[T]}^{p}(\mathcal{L}_{Q}^{-1}, \mathcal{E}) = 0, \]

following the same manipulations as earlier in the proof. Therefore $\mathcal{E}$ is orthogonal to all $\mathcal{L}_{P}^{-1}$ with $P \in \mathcal{P}_T$ which means $\mathcal{E} = 0$.

Finally, there are no extensions between the line bundles $\mathcal{L}_{Q}^{-1}$ for $Q \in \mathcal{P}_{T}^{gen}$ since they were part of a full strongly exceptional collection for $\D(\mathbf{F}[T])$.

This implies the semi-orthogonal decomposition for $\D(\mathbf{F}[T])$ induced by the full strongly exceptional collection $\{\mathcal{L}_P^{-1}\}_{P \in \mathcal{P}_T}$ refines the cuspidal decomposition of $\mathbf{F}[T]$.

§.§ The Permutahedral Varieties

We now construct forgetful species of smooth varieties out of the permutahedral varieties and show that the full strongly exceptional collections we constructed for them are compatible with the forgetful maps.

For any finite set $E$, let $\Perm_E$ be the variety defined on the fan $\Sigma_E \subseteq \{x \in \RR^E\; \lvert \; \sum x_i = 0\}$ whose top-dimensional cones are the chambers of the hyperplane arrangement given by the hyperplanes

\[ A_{i,j} = \{ x \in \RR^E \; \lvert \; x_i = x_j\},\]

for $i \not = j \in E$ all intersected with $\{x \in \RR^E\; \lvert \; \sum x_i = 0\}$. Let $\mathbf{Perm}[E] = \Perm_E$. It is clear that $\Perm_E$ is isomorphic to $\Perm_n$ where $n = |E|$. We can similarly extend the definitions of $\Stell_n$ and $\Perm_n^{B}$ to get varieties $\mathbf{Stell}[E]$ and $\mathbf{Perm}^B[E]$.

For the following proof, we will use the cone descriptions of the fans of our varieties from Propositions <ref>, <ref>, and <ref>. Those statements were only made when $E = [n]$ but they naturally extend for any finite set $E$.

For the functors $\mathbf{Perm},\mathbf{Stell}$, and $\mathbf{Perm}^B$, the maps $\pi_S: N_E\to N_S$ for $S \subseteq E$ given by projecting onto the coordinates labeled by $S$ make these into forgetful species.

We do this case by case. All we need to show is that if we have a cone $\sigma \in \Sigma_{E}$, then the cone obtained by projecting onto the coordinates indexed by $S$ is a cone in $\Sigma_S$ and that all cones of $\Sigma_S$ arise in this way. This is not too hard since we have exact descriptions of all the cones in these cases.

* We begin with $\mathbf{Perm}[E]$. Let $\sigma_A$ be the cone indexed by the ordered set partition $(A_1,\ldots, A_k)$. This is the cone where $x_i = x_j$ for two coordinates in the same part and $x_i < x_j$ when $i$ appears in a part that comes before the part containing $j$. When we remove the coordinates indexed by $i \in S$, then we will get the cone indexed by the set partition $A|S$ which is obtained by removing all elements in $E \backslash S$ and then removing all empty parts of the ordered set partition.

It is clear that every ordered set partition of $S$ is obtained in this way from an ordered set partition of $E$.

* Now consider $\mathbf{Stell}[E]$. Let $I \leq F_{\bullet}$ be a compatible pair and let $\sigma_{I \leq F_{\bullet}}$ be the cone indexed by it. That is,

\[\sigma_{I \leq F_{\bullet}} = \operatorname{cone}(e_i \; \lvert \; i \in I) + \operatorname{cone}(-e_{E\backslash F} \; \lvert \; F \in F_{\bullet}). \]

If we project onto $S$, we get the cone

\[\operatorname{cone}(e_i \; \lvert \; i \in I \cap S) + \operatorname{cone}(-e_{S\backslash F} \; \lvert \; F \in F_{\bullet}).  \]

This is the cone of $\Sigma_S$ indexed by the pair $S \cap I \leq F_{\bullet}'$ where $F_{k}' = F_k \cap S$. If $I \leq F_{\bullet}$ is compatible then so is $I \cap S \leq F_{\bullet}'$.

Further, every compatible pair of subsets and flags of subsets of $S$ is a compatible pair for $E$ by just considering every subset of $S$ as a subset of $E$. So, every cone arises is this way.

* Finally, consider $\mathbf{Perm}^B[E]$. Let $A$ be an ordered signed set partition and let $\sigma_A$ be the corresponding cone of $\Sigma_E$ as in Proposition <ref>. Then, by a similar argument to the one in the permutahedral case, the projection onto the coordinates indexed by $S$ gives the cone indexed by the ordered signed set partition $A|S$ where we delete all indices not in $S$ and delete all non-empty subsets except the zero block. Further, any ordered signed set partition of $E$ arises in this way from a signed set partition of $S$.

This shows that all of these functors are forgetful species with these maps. Notice that in the process, we showed that these maps are compatible with the fan and hence give toric morphisms.

Let $\overline{\Sch}_E, \Sch_E$ and $\overline{\operatorname{DSch}}_E$ denote the sets of loopless Schubert matroids on ground set $E$, all Schubert matroids on ground set $E$, and loopless Schubert delta matroids on ground set $E$, respectively. It is clear that our calculation of full strongly exceptional collections never depended on the ground set. So, we have full strongly exceptional collection of nef line bundles for $\D(\mathbf{Perm}[E]), \D(\mathbf{Stell}[E]),$ and $\D(\mathbf{Perm}^B[E])$ labeled by the polytopes in $\overline{\Sch}_E, \Sch_E$ and $\overline{\operatorname{DSch}}_E$, respectively.

The collections $\{\BP(\Omega)\}_{\overline{\Sch}_E}, \{\IP(\Omega)\}_{\Sch_E}$ and $\{\FP(\Omega)\}_{\overline{\operatorname{DSch}}_E}$ are compatible with the forgetful maps of the forgetful species $\mathbf{Perm}[E]$, $\mathbf{Stell}[E]$, and $\mathbf{Perm}^B[E]$, respectively.

We only do this for $\mathbf{Perm}[E]$ since all the cases are essentially identical. We need to show that base polytopes of loopless Schubert matroids on ground set $E$ where some translation is contained in $M_S \subseteq M_E$ are in bijection with the polytopes obtained by embedding base polytopes of loopless Schubert matroids on ground set $S$ under the map $M_S \hookrightarrow M_E$.

This follows immediately from the fact that embedding the base polytope of a loopless Schubert matroid on ground set $S$ under the map $M_S \hookrightarrow M_E$ is exactly the operation of trivially extending from $S$ to $E$ and flipping loops to coloops which corresponds to translations. We also have the observation that if the base polytope of a matroid on ground set $[E]$ is contained in a translation of this hyperplane, then each $i \in S$ must be a loop or coloop. Therefore the polytopes in $\mathcal{P}_S$ are in bijection with the base polytopes of loopless Schubert matroids on $E$ such that each $i \in E \backslash S$ is a coloop, which are exactly the polytopes in $\mathcal{P}_E$ which have a translation contained in $M_S$

By noting that the generic polytopes in each of the families above are exactly given by loopless and coloopless Schubert matroids and Schubert delta matroids, we obtain a calculation for the cuspidal part of all of our varieties.

Let $E$ be a finite set.

* The line bundles $(\mathcal{L}_{\BP(\Omega)}^{-1})$ indexed by loopless and coloopless Schubert matroids give a full strongly exceptional collection for $\D_{cusp}(\mathbf{Perm}[E])$ when ordered by non-increasing number of lattice points.

* The line bundles $(\mathcal{L}_{\IP(\Omega)}^{-1})$ indexed by loopless and coloopless Schubert matroids give a full strongly exceptional collection for $\D_{cusp}(\mathbf{Stell}[E])$ when ordered by non-increasing number of lattice points.

* The line bundles $(\mathcal{L}_{\FP(\Delta)}^{-1})$ indexed by loopless and coloopless Schubert delta matroids give a full strongly exceptional collection for $\D_{cusp}(\mathbf{Perm}^B[E])$ when ordered by non-increasing number of lattice points.

§.§ Relation to Castavet and Tevelev's Collection
Castravet and Tevelev constructed a full exceptional collection of sheaves for $\D(\Perm_n)$ by constructing one for $\D_{cusp}(\Perm[S])$, for all finite sets $S$, and then using the cuspidal decomposition from Theorem <ref>. They described their sheaves in terms of external tensor products of line bundles coming from the wonderful compactification[This is the wonderful compactification of groups which is different than the wonderful compactification of hyperplane arrangements.] of $\operatorname{PGL}_n$. There is a different way to describe their collection that relates it to our collection.

The permutahedral variety $\Perm_n$ embeds into the flag variety $G/B$ of type $A_n$ as the torus orbit closure of a generic point. Let $i: \Perm_n \hookrightarrow G/B$ be the inclusion map. By the Borel-Weil theorem, we know that the nef line bundles of $G/B$ are parameterized by the dominant weights.

For any general dominant weight $\lambda$, let $\mathcal{L}_{\lambda}$ be the line bundle associated to $\lambda$. Let $G_i = i^*(\mathcal{L}_{\omega_i})$ for $i =1, \ldots, n-1$ where $\omega_i = e_1 + e_2 + \cdots + e_i$ is the $i$th fundamental weight. In this language, their result is:

The derived category $\D_{cusp}(\Perm_n)$ has a full strongly exceptional collection of sheaves of cardinality equal to the number of derangements. It contains the $n-1$ line bundles $G_i$ for $i=1,\ldots,n-1$.

The nef line bundle  $i^*(\mathcal{L}_{\lambda})$ on $\Perm_n$ is the nef line bundle $\mathcal{L}_P$ given by the polytope

\[ P = \operatorname{conv}( \sigma \cdot \lambda \; \lvert \; \sigma \in S_n).\]

These are the weight polytopes studied for instance by Postnikov in <cit.>. When applied to the fundamental weights, we see that these polytopes are exactly the base polytopes of the uniform matroid of rank $1, \ldots, n-1$. These are coloopless and loopless Schubert matroids. Therefore, our collection of line bundles includes the line bundles $G_i$ from the collection of Castravet and Tevelev. It would be interesting to find a relationship between the other sheaves in their collection and matroids.

Another interesting aspect of this comparison is that Castravet and Tevelev's showed that their collection has cardinality equal to the number of derangements of $n$. These are the permutations of $[n]$ which have no fixed points. This gives a curious equicardinality.

The number of loopless and coloopless Schubert matroids on $[n]$ is equal to the number of derangements on $[n]$.

We have a type $B_n$-version of this calculation. A signed derangement is a signed permutation $\pi: [\pm n] \to [\pm n]$ such that $\pi(i) \not = i$ for all $i \in [\pm n]$.

The number of loopless and coloopless Schubert delta matroids on $[n]$ is equal to the number of signed derangements of $[\pm n]$. Hence, the rank of $K(\D_{cusp}(\Perm_n^{B}))$ is the number of signed derangements of $[\pm n]$.

Let $Sch(n)$ be the number of loopless and coloopless Schubert delta matroids on $[n]$. By the cuspidal decomposition of $\D(X_{B_n})$ and Theorem <ref>, we get the following recurrence for $|S_n^{B}|$

\[|S_n^{B}| = \sum_{A \subseteq [n]} Sch(n-|A|), \]

where $Sch(0) = 1$.

Let $Der_B(n)$ be the number of signed derangements of size $n$. Then, by keeping track of the subset $A$ of fixed points, we have the recurrence

\[ |S_n^{B}| = \sum_{A \subseteq [n]} Der_B(n -|A|).\]

Since $Sch(n)$ and $Der_B(n)$ satisfy the same recurrence, they are the same numbers.