2.1 Matrix factorizations

We briefly review the definition of categories of matrix factorizations. For more details, see [Reference Pădurariu and TodaPT22a, § 2.6].

Consider a smooth quotient stack $\mathscr {X}=X/G$ , where $G$ is an algebraic group acting on a smooth affine scheme $X$ , with a regular function $f\colon \mathscr {X}\to \mathbb {C}$ . Consider the category of matrix factorizations by

\begin{equation*}\mathrm {MF}(\mathscr {X}, f),\end{equation*}

whose objects are tuples

(2.1) \begin{align} (\alpha \colon A \leftrightarrows B \colon \beta ), \quad \alpha \circ \beta =\cdot f, \quad \beta \circ \alpha =\cdot f, \end{align}

where $A, B \in \textrm {Coh}(\mathscr {X})$ . If $\mathbb {M}\subset D^b(\mathscr {X})$ is a subcategory, let

(2.2) \begin{align} \mathrm {MF}(\mathbb {M}, f)\subset \mathrm {MF}(\mathscr {X}, f) \end{align}

be the subcategory consisting of totalizations of tuples (2.1) with $A, B \in \mathbb {M}$ ; see [Reference Pădurariu and TodaPT22a, § 2.6] for the precise definition. If $\mathbb {M}$ is generated by a set of vector bundles $\{\mathscr {V}_i\}_{i\in I}$ on $\mathscr {X}$ , then (2.2) is generated by matrix factorizations whose factors are direct sums of vector bundles from $\{\mathscr {V}_i\}_{i\in I}$ ; see [Reference Pădurariu and TodaPT22a, Lemma 2.3].

Given an action of $\mathbb {C}^*$ on $\mathscr {X}$ for which $f$ is of weight $2$ , we also consider the category of graded matrix factorizations $\mathrm {MF}^{\mathrm {gr}}(\mathscr {X}, f)$ . Its objects consist of tuples (2.1) where $A\mbox{ and } B$ are $\mathbb {C}^{\ast }$ -equivariant and $\alpha \mbox{ and } \beta$ are of $\mathbb {C}^{\ast }$ -weight one. For a subcategory $\mathbb {M}\subset D^b(\mathscr {X})$ , we define $\mathrm {MF}^{\mathrm {gr}}(\mathbb {M}, f)\subset \mathrm {MF}^{\mathrm {gr}}(\mathscr {X}, f)$ similarly to (2.2).

Let $\mathscr {Z}\subset \mathscr {X}$ be a closed substack. A matrix factorization $F$ in $\mathrm {MF}(\mathscr {X}, f)$ has support in $\mathscr {Z}$ if its restriction to $\mathrm {MF}(\mathscr {X}\setminus \mathscr {Z}, f)$ is zero. Every matrix factorization $F$ has support included in $\mathrm {Crit}(f)\subset \mathscr {X}$ , so for any open substack $\mathscr {U} \subset \mathscr {X}$ which contains $\mathrm {Crit}(f)$ , the following restriction functor is an equivalence:

(2.3) \begin{align} \mathrm {MF}(\mathscr {X}, f) \stackrel {\sim }{\to } \mathrm {MF}(\mathscr {U}, f). \end{align}

We deduce semiorthogonal decompositions for a quiver with potential $(Q,W)$ from the case of zero potential; see for example [Reference Pădurariu and TodaPT22a, Proposition 2.5] or [Reference PădurariuPăd22, Proposition 2.1]. We extensively use the Koszul equivalence; see Theorem 2.5.

We consider either ungraded categories of matrix factorizations or graded categories which are Koszul equivalent to derived categories of bounded complexes of coherent sheaf on a quasi-smooth stack. When considering the product of two categories of matrix factorizations, which is in the context of the Thom–Sebastiani theorem, we consider the product of dg-categories over $\mathbb {C}(\!(\beta )\!)$ for $\beta$ of homological degree $-2$ in the ungraded case (see [Reference PreygelPre11, Theorem 4.1.3]) and the product of dg-categories over $\mathbb {C}$ in the graded case (see [Reference Ballard, Favero and KatzarkovBFK14, Corollary 5.18]; alternatively, in the graded case, one can use the Koszul equivalence).

2.2 Quivers, weights, and partitions

2.2.1 Basic notions

Let $Q=(I,E)$ be a quiver, i.e. a directed graph with set of vertices $I$ and set of edges $E$ . Let $d=(d^a)_{a\in I}\in \mathbb {N}^I$ be a dimension vector. Denote by

\begin{equation*}\mathscr {X}(d)=R(d)/G(d)\end{equation*}

the stack of representations of $Q$ of dimension $d$ . Here $R(d)$ and $G(d)$ are given by

\begin{align*} R(d)=\bigoplus _{(a\to b) \in E}\textrm {Hom}(V^a, V^b), \quad G(d)=\prod _{a\in I}GL(V^a). \end{align*}

We say that $Q$ is symmetric if for any $a, b \in I$ , the number of arrows from $a$ to $b$ is the same as the number from $b$ to $a$ . In this case, $R(d)$ is a self-dual $G(d)$ -representation. We have the good moduli space morphism (or GIT quotient)

\begin{align*} \pi _{X,d}=\pi _X \colon \mathscr {X}(d) \to X(d):=R(d)/\!\!/ G(d). \end{align*}

For a quiver $Q$ , let $\mathbb {C}[Q]$ be its path algebra. A potential $W$ of a quiver $Q$ is an element

\begin{align*} W \in \mathbb {C}[Q]/[\mathbb {C}[Q], \mathbb {C}[Q]]. \end{align*}

A pair $(Q, W)$ is called a quiver with potential. Given a potential $W$ , there is a regular function

(2.4) \begin{align} \mathop {\textrm {Tr}} W \colon \mathscr {X}(d) \to \mathbb {C}. \end{align}

By the property of the good moduli space, the function $\mathop {\textrm {Tr}} W$ factors through the good moduli space, $\mathop {\textrm {Tr}} W \colon \mathscr {X}(d) \xrightarrow {\pi _{X,d}} X(d) \to \mathbb {C}$ .

We will consider the derived category $D^b(\mathscr {X}(d))$ of coherent sheaves on $\mathscr {X}(d)$ and the category of matrix factorizations $\mathrm {MF}(\mathscr {X}(d), \mathop {\textrm {Tr}} W)$ . Since the diagonal torus $\mathbb {C}^{\ast } \subset T(d)$ acts on $R(d)$ trivially, there are orthogonal decompositions

(2.5) \begin{align} D^b(\mathscr {X}(d))=\bigoplus _{w\in \mathbb {Z}} D^b(\mathscr {X}(d))_w, \quad \mathrm {MF}(\mathscr {X}(d), \mathop {\textrm {Tr}} W)=\bigoplus _{w\in \mathbb {Z}} \mathrm {MF}(\mathscr {X}(d), \mathop {\textrm {Tr}} W)_w, \end{align}

where each summand corresponds to the diagonal $\mathbb {C}^{\ast }$ -weight $w$ -part.

2.2.2 The weight lattice

We fix a maximal torus $T(d)$ of $G(d)$ . Let $M(d)$ be the weight lattice of $T(d)$ . For $a\in I$ and $d^a\in \mathbb {N}$ , denote by $\beta ^a_i$ for $1\leqslant i\leqslant d^a$ the weights of the standard representation of $T(d^a)$ . We have

\begin{align*} M(d)=\bigoplus _{a \in I} \bigoplus _{1\leqslant i\leqslant d^a} \mathbb {Z} \beta ^a_i. \end{align*}

With an abuse of notation, for a $T(d)$ -representation $U$ we also denote by $U \in M(d)$ the sum of weights in $U$ or, equivalently, the class of the character $\det U$ . A weight

\begin{equation*}\chi =\sum _{a\in I}\sum _{1\leqslant i\leqslant d^a} x_i^a \beta ^a_i\end{equation*}

is dominant (or antidominant) if $x_i^a \leqslant x_{i+1}^a$ (or $x_i^a \geqslant x_{i+1}^a$ ) for all $a\in I$ and $1\leqslant i\leqslant d^a$ . For a dominant weight $\chi$ , we denote by $\Gamma _{G(d)}(\chi )$ the irreducible representation of $G(d)$ with highest weight $\chi$ . The dominant or antidominant cocharacters are also defined for elements of the cocharacter lattice $N(d)=\textrm {Hom}(M(d), \mathbb {Z})$ . We denote by $1_d \in N(d)$ the diagonal cocharacter.

We denote by $M(d)_0\subset M(d)$ the hyperplane of weights with sum of coefficients equal to zero, and set

\begin{align*} M(d)_{\mathbb {R}}:=M(d)\otimes _{\mathbb {Z}}\mathbb {R}, \quad M(d)_{0,\mathbb {R}}:=M(d)_0\otimes _{\mathbb {Z}}\mathbb {R}. \end{align*}

Note that $M(d)_0$ is the weight lattice of the subtorus $ST(d) \subset T(d)$ defined by

\begin{align*} ST(d):=\text {ker}(\det \colon T(d)\to \mathbb {C}^*), \quad (g^a)_{a \in I} \stackrel {\det }{\mapsto } \prod _{a \in I} \det (g^a). \end{align*}

We denote by $\langle \,,\,\rangle \colon N(d)\times M(d)\to \mathbb {Z}$ the natural pairing, and we use the same notation for its real version. If $\lambda$ is a cocharacter of $T(d)$ and $V$ is a $T(d)$ -representation, we may abuse notation and write

\begin{equation*}\langle \lambda , V\rangle =\langle \lambda ,\quad \det (V)\rangle \end{equation*}

to ease the notation.

We denote by $W_d$ the Weyl group of $G(d)$ and by $M(d)^{W_d} \subset M(d)$ the Weyl-invariant subset. For $d=(d^a)_{a\in I}$ , let $\underline {d}=\sum _{a\in I}d^a$ be its total length. Define the Weyl-invariant weights in $M(d)_{\mathbb {R}}$ :

\begin{align*} \sigma _{d}:=\sum _{a \in I} \sum _{1\leqslant i \leqslant d^a}\beta ^a_i, \quad \tau _d:=\frac {\sigma _d}{\underline {d}}. \end{align*}

We denote by $\mathfrak {g}(d)$ the Lie algebra of $G(d)$ and by $\rho$ half the sum of the positive roots of $\mathfrak {g}(d)$ :

\begin{align*} \rho =\frac {1}{2} \sum _{a \in I}\sum _{1\leqslant i\lt j \leqslant d^a} (\beta _j^a -\beta _i^b). \end{align*}

2.2.3 Partitions

Let $(d_i)_{i=1}^k$ be a partition of $d$ . There is an identification

\begin{equation*}\bigoplus _{i=1}^k M(d_i)\cong M(d),\end{equation*}

where $\beta ^a_1, \ldots , \beta ^a_{d_1}$ correspond to the weights of the standard representation of $GL(d^a_1)$ in $M(d^a_1)$ for $a\in I$ , etc.

Definition 2.1. Let $\underline {e}=(e_i)_{i=1}^l$ and $\underline {d}=(d_i)_{i=1}^k$ be two partitions of $d\in \mathbb {N}^I$ . We write $\underline {e}\geqslant \underline {d}$ if there exist integers

\begin{equation*}a_0=0\lt a_1\lt \cdots \lt a_{k-1}\leqslant a_k=l\end{equation*}

such that for any $0\leqslant j\leqslant k-1$ , we have

\begin{equation*}\sum _{i=a_{j}+1}^{a_{j+1}} e_i=d_{j+1}.\end{equation*}

We next define a tree which is useful in decomposing dominant weights of $M(d)_{\mathbb {R}}$ .

Definition 2.2. We define $\mathcal {T}$ to be the unique (oriented) tree such that:

1. (1) each vertex is indexed by a partition $(d_1, \ldots , d_k)$ of some $d \in \mathbb {N}^I$ ;

2. (2) for each $d \in \mathbb {N}^I$ , there is a unique vertex indexed by the partition $(d)$ of size one;

3. (3) if $\bullet$ is a vertex indexed by $(d_1, \ldots , d_k)$ and $d_m=(e_1, \ldots , e_s)$ is a partition of $d_m$ for some $1\leqslant m \leqslant k$ , then there is a unique vertex $\bullet '$ indexed by $(d_1, \ldots , d_{m-1}, e_1, \ldots , e_s, d_{m+1}, \ldots , d_k)$ and with an edge from $\bullet$ to $\bullet '$ ;

4. (4) all edges in $\mathcal {T}$ are as in (3).

Note that each partition $(d_1, \ldots , d_k)$ of some $d \in \mathbb {N}^I$ gives an index of some (not necessary unique) vertex. A subtree $T \subset \mathcal {T}$ is called a path of partitions if it is connected, contains a vertex indexed by $(d)$ for some $d \in \mathbb {N}^I$ , and has a unique end vertex $\bullet$ . The partition $(d_1, \ldots , d_k)$ at the end vertex $\bullet$ is called the associated partition of $T$ . We define the Levi group associated to $T$ to be

\begin{align*} L(T):=\times _{i=1}^k G(d_i). \end{align*}

2.2.4 Framed quivers

Consider a quiver $Q=(I, E)$ . Define the framed quiver

\begin{align*} Q^f=(I^f, E^f) \end{align*}

with set of vertices $I^f=I\sqcup \{\infty \}$ and set of edges $E^f=E\sqcup \{e_a\mid a\in I\}$ , where $e_a$ is an edge from $\infty$ to $a\in I$ . Let $V(d)=\bigoplus _{a\in I}V^a$ , where $V^a$ is a $\mathbb {C}$ -vector space of dimension $d^a$ . Denote by

\begin{equation*}R^f(d)=R(d)\oplus V(d)\end{equation*}

the affine space of representations of $Q^f$ of dimension $d$ and consider the moduli stack of framed representations

\begin{equation*}\mathscr {X}^f(d):=R^f(d)/G(d).\end{equation*}

We consider GIT stability on $Q^f$ given by the character $\sigma _{\underline {d}}$ . It coincides with the King stability condition on $Q^f$ such that the (semi)stable representations of dimension $(1,d)$ are the representations of $Q^f$ with no subrepresentations of dimension $(1,d')$ for $d'$ different from $d$ ; see [Reference TodaTod19, Lemma 5.1.9]. Consider the smooth variety obtained as a GIT quotient:

\begin{equation*}\mathscr {X}^f(d)^{\text {ss}}:=R^f(d)^{\text {ss}}/G(d).\end{equation*}

2.2.5 The categorical Hall product

For a cocharacter $\lambda \colon \mathbb {C}^{\ast } \to T(d)$ and a $T(d)$ -representation $V$ , let $V^{\lambda \geqslant 0}\subset V$ be the subspace generated by weights $\beta$ such that $\langle \lambda , \beta \rangle \geqslant 0$ , and let $V^\lambda \subset V$ be the subspace generated by weights $\beta$ such that $\langle \lambda , \beta \rangle =0$ . We denote by $G(d)^{\lambda } \subset G(d)^{\lambda \geqslant 0}$ the associated Levi and parabolic subgroup of $G(d)$ . If $V$ is a $G(d)$ -representation, consider the quotient stack $\mathscr {X}=V/G(d)$ and set

\begin{align*} \mathscr {X}^{\lambda \geqslant 0}=V^{\lambda \geqslant 0}/G(d)^{\lambda \geqslant 0}, \quad \mathscr {X}^{\lambda }=V^{\lambda }/G(d)^{\lambda }. \end{align*}

The projection $V^{\lambda \geqslant 0} \twoheadrightarrow V^{\lambda }$ and the inclusion $V^{\lambda \geqslant 0} \hookrightarrow V$ induce maps

(2.6) \begin{align} \mathscr {X}^{\lambda } \leftarrow \mathscr {X}^{\lambda \geqslant 0} \to \mathscr {X}. \end{align}

We apply the above construction for $V=R(d)$ to obtain maps

\begin{equation*}\mathscr {X}(d)^\lambda =\times _{i=1}^k\mathscr {X}(d_i)\xleftarrow {q_\lambda }\mathscr {X}(d)^{\lambda \geqslant 0}\xrightarrow {p_\lambda }\mathscr {X}(d).\end{equation*}

Suppose that $\lambda$ is antidominant with associated partition $(d_i)_{i=1}^k$ of $d\in \mathbb {N}^I$ , meaning that

\begin{equation*}\mathscr {X}(d)^\lambda =\times _{i=1}^k\mathscr {X}(d_i).\end{equation*}

The multiplication for the categorical Hall algebra of $(Q, 0)$ (or of $(Q,W)$ for a potential $W$ of $Q$ and possibly a grading) is defined by the following functors [Reference PădurariuPăd22], where $\bullet \in \{\emptyset , \mathrm {gr}\}$ :

(2.7) \begin{align} &m_\lambda :=p_{\lambda *}q_\lambda ^{\ast } \colon \boxtimes _{i=1}^k D^b(\mathscr {X}(d_i)) \to D^b(\mathscr {X}(d)),\nonumber \\ &m_\lambda :=p_{\lambda *}q_\lambda ^{\ast } \colon \boxtimes _{i=1}^k \mathrm {MF}^\bullet (\mathscr {X}(d_i), \mathop {\textrm {Tr}} W) \to \mathrm {MF}^\bullet (\mathscr {X}(d), \mathop {\textrm {Tr}} W). \end{align}

2.2.6 Doubled quiver

Let $Q^\circ =(I, E^\circ )$ be a quiver. Its doubled quiver is the quiver

\begin{align*} Q^{\circ , d}=(I, E^{\circ , d}) \end{align*}

with the set of edges $E^{\circ , d}=\{e, \overline {e} \mid e \in E^{\circ }\}$ , where $\overline {e}$ is the edge with the opposite orientation to $e\in E^\circ$ . Consider the following relation $\mathscr {I}$ of $\mathbb {C}[Q^{\circ , d}]$ :

(2.8) \begin{align} \mathscr {I}:=\sum _{e \in E^{\circ }}[e, \overline {e}] \in \mathbb {C}[Q^{\circ , d}]. \end{align}

For $d\in \mathbb {N}^I$ , consider the stack of representations of the quiver $Q^{\circ , d}$ of dimension $d$ ,

\begin{equation*}\mathscr {Y}(d):=\overline {R}(d)/G(d):=T^*R^\circ (d)/G(d),\end{equation*}

with good moduli space map

\begin{equation*}\pi _{Y,d}=\pi _Y\colon \mathscr {Y}(d)\to Y(d):=\overline {R}(d)/\!\!/ G(d).\end{equation*}

Note that

\begin{align*} \overline {R}(d):=T^*R^\circ (d)=\bigoplus _{(a\to b) \in E^{\circ }}\textrm {Hom}(V^a, V^b) \oplus \textrm {Hom}(V^b, V^a). \end{align*}

For $x \in T^*R^\circ (d)$ and an edge $(a \to b) \in E^{\circ }$ , consider the components $x(e) \in \textrm {Hom}(V^a, V^b)$ and $x(\overline {e}) \in \textrm {Hom}(V^b, V^a)$ . The relation (2.8) determines the moment map

(2.9) \begin{align} \mu \colon T^*R^\circ (d) \to \mathfrak {g}(d), \quad x \mapsto \sum _{e \in E^{\circ }}[x(e), x(\overline {e})]. \end{align}

Let $\mu ^{-1}(0)$ be the derived zero locus of $\mu$ . Define the stack of representations of $Q^{\circ , d}$ with relation $\mathscr {I}$ or, alternatively, of the preprojective algebra $\Pi _{Q^\circ }:=\mathbb {C}[Q^{\circ , d}]/(\mathscr {I})$ :

(2.10) \begin{align} j \colon \mathscr {P}(d):=\mu ^{-1}(0)/G(d) \hookrightarrow \mathscr {Y}(d). \end{align}

The stack $\mathscr {P}(d)^{\mathrm {cl}}$ has a good moduli space map

(2.11) \begin{equation} \pi _{P,d}=\pi _P\colon \mathscr {P}(d)^{\mathrm {cl}}\to P(d):=\mu ^{-1}(0)^{\textrm {cl}}/\!\!/ G(d). \end{equation}

Let $\lambda$ be an antidominant cocharacter of $T(d)$ corresponding to the decomposition $(d_i)_{i=1}^k$ of $d$ . Similarly to (2.7), there is a categorical Hall product [Reference Porta and SalaPS23, Reference Varagnolo and VasserotVV22]; see also (2.19):

(2.12) \begin{align} m_{\lambda } \colon \boxtimes _{i=1}^k D^b(\mathscr {P}(d_i)) \to D^b(\mathscr {P}). \end{align}

Remark 2.3. As mentioned in the introduction, moduli stacks of representations of preprojective algebras are interesting for at least two reasons: they describe locally the moduli stack of (Bridgeland semistable) sheaves on a Calabi–Yau surface, and their K-theory (or Borel–Moore homology) can be used to construct positive halves of quantum affine algebras (or of Yangians).

2.2.7 Tripled quivers with potential

Consider a quiver $Q^{\circ }=(I, E^{\circ })$ . Its tripled quiver

\begin{align*} Q=(I, E) \end{align*}

has set of edges $E=E^{\circ , d}\sqcup \{\omega _a \mid a\in I\}$ , where $\omega _a$ is a loop at the vertex $a \in I$ . The tripled potential $W$ (which is a potential of $Q$ ) is defined as follows:

(2.13) \begin{equation} W:=\biggl(\sum _{a\in I}\omega _a \biggr) \biggl (\sum _{e\in E^\circ }[e, \overline {e}]\biggr )\in \mathbb {C}[Q]. \end{equation}

The quiver with potential $(Q, W)$ constructed above is called the tripled quiver with potential associated to $Q^{\circ }$ .

For $d \in \mathbb {N}^I$ , let $\mathscr {X}(d)$ be the stack of representations of $Q$ . It is given by

(2.14) \begin{align} \mathscr {X}(d)=(T^*R^\circ (d) \oplus \mathfrak {g}(d))/G(d)= (\overline {R}(d) \oplus \mathfrak {g}(d))/G(d). \end{align}

Recall the function $\mathop {\textrm {Tr}} W$ on $\mathscr {X}(d)$ from (2.4). The critical locus

\begin{align*} \mathrm {Crit}(\mathop {\textrm {Tr}} W) \subset \mathscr {X}(d) \end{align*}

is the moduli stack of $(Q, W)$ -representations or, alternatively, of the Jacobi algebra $\mathbb {C}[Q]/\mathscr {J}$ , where $\mathscr {J}$ is the two-sided ideal generated by the partial derivatives $\partial W/\partial e$ for all $e\in E$ .

Consider the grading on $\mathscr {X}(d)$ which is of weight $0$ on $\overline {R}(d)$ and of weight $2$ on $\mathfrak {g}(d)$ . The Koszul equivalence (which we recall later in Theorem 2.5) says that there is an equivalence

(2.15) \begin{equation} \Theta \colon D^b(\mathscr {P}(d))\stackrel {\sim }{\to } \mathrm {MF}^{\mathrm {gr}}(\mathscr {X}(d), \mathrm {Tr}\,W). \end{equation}

Remark 2.4. Tripled quivers with potential are interesting for at least two reasons: they model the local structure of moduli stacks of semistable sheaves on an arbitrary CY3, and they can be used, in conjunction with dimensional reduction techniques (such as the Koszul equivalence (2.15)), to study (moduli of representations of) preprojective algebras.

2.3 The Koszul equivalence

Let $Y$ be an affine smooth scheme with an action of a reductive group $G$ . Consider the quotient stack $\mathscr {Y}=Y/G$ . Let $\mathscr {V} \to \mathscr {Y}$ be a vector bundle and let $s\in \Gamma (\mathscr {Y}, \mathscr {V})$ . Let $\mathscr {P}=s^{-1}(0)$ be the derived zero locus of $s$ , so its structure complex is the Koszul complex

\begin{equation*}\mathcal {O}_{\mathscr {P}}:= (\mathrm {Sym}_{\mathcal {O}_{\mathscr {Y}}}(\mathscr {V}^{\vee }[1]), d_{\mathscr {P}} )\end{equation*}

with differential $d_{\mathscr {P}}$ induced by $s$ . Let $\mathscr {X}=\mathrm {Tot}_{\mathscr {X}} (\mathscr {V}^{\vee })$ and define the function $f$ by

\begin{align*} f \colon \mathscr {X} \to \mathbb {C}, \; f(y, v)=\langle s(y), v \rangle \end{align*}

for $y \in \mathscr {Y}$ and $v \in \mathscr {V}^{\vee }|_{y}$ . There are maps

(2.16) \begin{align} \mathscr {P} \stackrel {j}{\hookrightarrow } \mathscr {Y} \stackrel {\eta }{\leftarrow } \mathscr {X} \stackrel {f}{\to } \mathbb {C}, \end{align}

where $j$ is the natural inclusion and $\eta$ is the natural projection. The following is the Koszul equivalence.

Theorem 2.5 [Reference IsikIsi13, Reference HiranoHir17, Reference TodaTod19]. There are equivalences

\begin{align*} \Theta \colon D^b(\mathscr {P}) \stackrel {\sim }{\to } \mathrm {MF}^{\textrm {gr}}(\mathscr {X}, f), \quad \Theta \colon \textrm {Ind}\, D^b(\mathscr {P}) \stackrel {\sim }{\to } \mathrm {MF}^{\textrm {gr}}_{\textrm {qc}}(\mathscr {X}, f). \end{align*}

The grading on the right-hand side has weight zero on $\mathscr {Y}$ and weight $2$ on the fibers of $\eta \colon \mathscr {X} \to \mathscr {Y}$ .

The equivalence $\Theta$ is given by the functor

(2.17) \begin{align} \Theta (-)=(-) \otimes _{\mathcal {O}_{\mathscr {P}}}\mathcal {K}, \end{align}

where $\mathcal {K}$ is the Koszul factorization $\mathcal {K}:= (\mathcal {O}_{\mathscr {P}} \otimes _{\mathcal {O}_{\mathscr {Y}}}\mathcal {O}_{\mathscr {X}}, d_{\mathcal {K}} )$ with $d_{\mathcal {K}}=d_{\mathscr {P}} \otimes 1 +\kappa$ , where $\kappa \in \mathcal {V}^{\vee } \otimes \mathcal {V}$ corresponds to $\textrm {id} \in \textrm {Hom}(\mathcal {V}, \mathcal {V})$ ; see [Reference TodaTod19, Theorem 2.3.3]. The following lemma is easily proved from the above description of $\Theta$ .

For the later use, we will compare internal homomorphisms under Koszul equivalence. For $\mathcal {E}_1, \mathcal {E}_2 \in D^b(\mathscr {P})$ , there exists an internal homomorphism (see [Reference Drinfeld and GaitsgoryDG13, Remark 3.4])

\begin{align*} \mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2) \in D_{\textrm {qc}}(\mathscr {P}). \end{align*}

It satisfies the following tensor-Hom adjunction: for any $A \in D_{\textrm {qc}}(\mathscr {P})$ , there are natural isomorphisms

\begin{align*} \textrm {Hom}_{D_{\textrm {qc}}(\mathscr {P})}(A, \mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2)) \cong \textrm {Hom}_{\textrm {Ind} D^b(\mathscr {P})}(A \otimes \mathcal {E}_1, \mathcal {E}_2). \end{align*}

On the other side of the Koszul equivalence, consider the internal Hom of $\mathcal {F}_1, \mathcal {F}_2 \in \mathrm {MF}^{\textrm {gr}}(\mathscr {X}, f)$ ,

\begin{align*} \mathcal {H}om(\mathcal {F}_1, \mathcal {F}_2) \in \mathrm {MF}^{\textrm {gr}}(\mathscr {X}, 0). \end{align*}

Lemma 2.7. For $\mathcal {E}_1, \mathcal {E}_2 \in D^b(\mathscr {P})$ , the equivalence $\Theta$ induces an isomorphism

(2.18) \begin{align} j_{\ast }\mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2) \stackrel {\cong }{\to } \eta _{\ast } \mathcal {H}om(\Theta (\mathcal {E}_1), \Theta (\mathcal {E}_2)) \end{align}

in $D_{\textrm {qc}}(\mathscr {Y})=\mathrm {MF}^{\textrm {gr}}_{\mathrm {qc}}(\mathscr {Y}, 0)$ .

Proof. For $A \in D_{\textrm {qc}}(\mathscr {Y})$ , we have

\begin{align*} \textrm {Hom}(A, j_{\ast }\mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2)) &\cong \textrm {Hom}(j^{\ast }A, \mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2)) \\ &\cong \textrm {Hom}(j^{\ast }A \otimes \mathcal {E}_1, \mathcal {E}_2). \end{align*}

We also have

\begin{align*} \textrm {Hom}(A, \eta _{\ast }\mathcal {H}om(\Theta (\mathcal {E}_1), \Theta (\mathcal {E}_2)) &\cong \textrm {Hom}(\eta ^{\ast }A, \mathcal {H}om(\Theta (\mathcal {E}_1), \Theta (\mathcal {E}_2))) \\ &\cong \textrm {Hom}(\eta ^{\ast }A \otimes \Theta (\mathcal {E}_1), \Theta (\mathcal {E}_2)) \\ &\cong \textrm {Hom}(\Theta (j^{\ast }A \otimes \mathcal {E}_1), \Theta (\mathcal {E}_2)), \end{align*}

where the last isomorphism follows from the explicit form of $\Theta$ in (2.17). Therefore $\Theta$ induces an isomorphism

\begin{align*} \textrm {Hom}(A, j_{\ast }\mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2)) \stackrel {\sim }{\to } \textrm {Hom}(A, \eta _{\ast }\mathcal {H}om(\Theta (\mathcal {E}_1), \Theta (\mathcal {E}_2))), \end{align*}

which implies the isomorphism (2.18).

Let $\lambda$ be a cocharacter of $G$ . Consider the sections induced by $s$ ,

\begin{align*} s^{\lambda \geqslant 0}\in \Gamma (\mathscr {Y}^{\lambda \geqslant 0}, \mathscr {V}^{\lambda \geqslant 0}), \quad s^\lambda \in \Gamma (\mathscr {Y}^{\lambda }, \mathscr {V}^{\lambda }), \end{align*}

and their derived zero loci

\begin{equation*}\mathscr {P}^{\lambda \geqslant 0}:=(s^{\lambda \geqslant 0})^{-1}(0),\quad \mathscr {P}^{\lambda }:=(s^\lambda)^{-1}(0).\end{equation*}

Similarly to (2.6), consider the maps

\begin{align*} \mathscr {P}^{\lambda } \stackrel {q}{\leftarrow } \mathscr {P}^{\lambda \geqslant 0} \stackrel {p}{\to } \mathscr {P}, \end{align*}

where $q$ is quasi-smooth and $p$ is proper. Consider the functor, which is a generalization of the categorical Hall product for preprojective algebras [Reference Porta and SalaPS23, Reference Varagnolo and VasserotVV22]:

(2.19) \begin{align} m_{\lambda }= p_{\ast }q^{\ast } \colon D^b(\mathscr {P}^{\lambda }) \to D^b(\mathscr {P}). \end{align}

We have the following compatibility of categorical Hall products under Koszul equivalence. For the proof, see [Reference PădurariuPăd23, Proposition 3.1] or [Reference TodaTod19, Lemmas 2.4.4 and 2.4.7].

Proposition 2.8. The following diagram commutes.

The horizontal arrows are categorical Hall products for $\mathscr {P}$ and $\mathscr {X}$ . We denote by $\Theta$ the Koszul equivalence for both $\mathscr {X}$ and $\mathscr {X}^\lambda$ , and the left vertical map is the functor $\Theta '(-):=\Theta (-)\otimes \det (\mathcal {V}^{\lambda \gt 0})^{\vee }[\mathrm {rank}\, \mathcal {V}^{\lambda \gt 0}]$ .

2.4 Polytopes and categories

Let $Q=(I,E)$ be a symmetric quiver and let $d=(d^a)_{a\in I}\in \mathbb {N}^I$ be a dimension vector. Consider the multisets of $T(d)$ -weights

(2.20) \begin{align} \mathscr {A}&:=\{\beta ^a_i-\beta ^b_j \mid a,b\in I,\, (a \to b) \in E, \,1\leqslant i\leqslant d^a,\, 1\leqslant j\leqslant d^b\},\nonumber \\ \mathscr {B}&:=\{\beta ^a_i\mid a\in I,\, 1\leqslant i\leqslant d^a\}, \\ \mathscr {C}&:=\mathscr {A}\sqcup \mathscr {B}. \nonumber\end{align}

Here, $\mathscr {A}$ is the set of $T(d)$ -weights of $R(d)$ and $\mathscr {C}$ is the set of $T(d)$ -weights of $R^f(d)$ . Define the polytopes

(2.21) \begin{align} \mathbf {W}(d)&:=\tfrac {1}{2}\mathrm {sum}_{\beta \in \mathscr {A}}[0, \beta ]\subset M(d)_{0,\mathbb {R}}\subset M(d)_{\mathbb {R}},\nonumber \\ \mathbf {V}(d)&:=\tfrac {1}{2}\mathrm {sum}_{\beta \in \mathscr {C}}[0,\beta ]\subset M(d)_{\mathbb {R}}, \end{align}

where the sums above are Minkowski sums in the space of weights $M(d)_{\mathbb {R}}$ .

Definition 2.9. For a weight $\delta _d\in M(d)_{\mathbb {R}}^{W_d}$ , define

(2.22) \begin{equation} \mathbb {M}(d; \delta _d)\subset D^b(\mathscr {X}(d)) \end{equation}

to be the full subcategory of $D^b(\mathscr {X}(d))$ generated by vector bundles $\mathcal {O}_{\mathscr {X}(d)}\otimes \Gamma _{G(d)}(\chi )$ , where $\chi \in M(d)$ is a dominant weight such that

(2.23) \begin{equation} \chi +\rho -\delta _d\in \mathbf {W}(d). \end{equation}

The category (2.22) is a particular case of the noncommutative resolutions of quotient singularities introduced by Špenko and Van den Bergh [Reference Špenko and Van den BerghŠVdB17]. We may call (2.22) a `magic category’ following [Reference Halpern-Leistner and SamHLS20]. For a cocharacter $\lambda$ of $T(d)$ , define

(2.24) \begin{align} n_{\lambda }=\langle \lambda , \det (\mathbb {L}_{\mathscr {X}(d)}|_{0}^{\lambda \gt 0} ) \rangle = \langle \lambda , \det (R(d)^{\vee })^{\lambda \geqslant 0} \rangle - \langle \lambda , \det (\mathfrak {g}(d)^{\vee } )^{\lambda \gt 0}\rangle . \end{align}

The category (2.22) also has the following alternative description.

Lemma 2.10 [Reference Halpern-Leistner and SamHLS20, Lemma 2.9]. The subcategory $\mathbb{M}(d; \delta_d)$ of $D^b(\mathscr{X}(d))$ is generated by vector bundles $\mathcal{O}_{\mathscr{X}(d)} \otimes \Gamma$ for a $G(d)$ -representation $\Gamma$ such that for any $T(d)$ -weight $\chi$ of $\Gamma$ and any cocharacter $\lambda$ of $T(d)$ , we have that

(2.25) \begin{align} \langle \lambda , \chi -\delta _d\rangle \in [{-}\tfrac {1}{2}n_{\lambda }, \tfrac {1}{2}n_{\lambda } ]. \end{align}

We also define a larger subcategory corresponding to the polytope $\mathbf {V}(d)$ . Let

(2.26) \begin{align} \mathbb {D}(d; \delta _d) \subset D^b(\mathscr {X}(d)) \end{align}

be generated by vector bundles $\mathcal {O}_{\mathscr {X}(d)}\otimes \Gamma _{G(d)}(\chi )$ , where $\chi$ is a dominant weight of $G(d)$ such that

\begin{equation*}\chi +\rho -\delta _d\in \mathbf {V}(d).\end{equation*}

The following definition will be used later in decomposing $\mathbb {D}(d; \delta _d)$ into categorical Hall products of $\mathbb {M}(d; \delta _d)$ .

Next, for a quiver with potential $(Q, W)$ , we define quasi-BPS categories as follows.

Definition 2.13. Define the quasi-BPS category to be

(2.27) \begin{equation} \mathbb {S}(d; \delta _d):=\mathrm {MF}(\mathbb {M}(d; \delta _d), \mathrm {Tr}\,W)\subset \mathrm {MF}(\mathscr {X}(d), \mathop {\textrm {Tr}} W). \end{equation}

Suppose that $(Q, W)$ is a tripled quiver of a quiver $Q^{\circ }=(I, E^{\circ })$ . In this case, the graded version is similarly defined as

(2.28) \begin{equation} \mathbb {S}^{\mathrm {gr}}(d; \delta _d):=\mathrm {MF}^{\mathrm {gr}}(\mathbb {M}(d; \delta _d), \mathop {\textrm {Tr}} W) \subset \mathrm {MF}^{\textrm {gr}}(\mathscr {X}(d), \mathop {\textrm {Tr}} W). \end{equation}

Next, we define quasi-BPS categories for preprojective algebras. Let $Q^\circ$ be a quiver, consider the moduli stack $\mathscr {P}(d)$ of dimension- $d$ representations of the preprojective algebra of $Q^\circ$ , and recall the closed immersion $j \colon \mathscr {P}(d) \hookrightarrow \mathscr {Y}(d)$ . The quasi-BPS category for $\mathscr {P}(d)$ is defined as follows.

Definition 2.14. Let $\widetilde {\mathbb {T}}(d; \delta _d)\subset D^b(\mathscr {Y}(d))$ be the subcategory generated by vector bundles $\mathcal {O}_{\mathscr {Y}(d)} \otimes \Gamma _{G(d)}(\chi )$ , where $\chi$ is a dominant weight of $G(d)$ satisfying

\begin{equation*}\chi +\rho -\delta _d\in \tfrac {1}{2}\text {sum}_{\beta \in \mathscr {A}}\,[0,\beta ],\end{equation*}

where $\mathscr {A}$ is the set of $T(d)$ -weights of $\overline {R}(d)\oplus \mathfrak {g}(d)$ . Define the preprojective quasi-BPS category (also called the quasi-BPS category of preprojective algebra)

\begin{align*} \mathbb {T}(d; \delta _d) \subset D^b(\mathscr {P}(d)) \end{align*}

as the full subcategory of $D^b(\mathscr {P}(d))$ with objects $\mathcal {E}$ such that $j_{\ast }\mathcal {E}\in \widetilde {\mathbb {T}}(d; \delta _d)$ .

By Lemma 2.6, the Koszul equivalence (2.15) restricts to the equivalence

(2.29) \begin{align} \Theta \colon \mathbb {T}(d; \delta _d) \stackrel {\sim }{\to } \mathbb {S}^{\textrm {gr}}(d; \delta _d). \end{align}

For $v\in \mathbb {Z}$ and $\bullet \in \{\emptyset , \mathrm {gr}\}$ , we will use the following shorthand notation:

\begin{equation*}\mathbb {M}(d)_v:=\mathbb {M}(d; \delta _d),\quad \mathbb {S}^\bullet (d)_v:=\mathbb {S}^\bullet (d; v\tau _d),\quad \mathbb {T}(d)_v:=\mathbb {T}(d; v\tau _d).\end{equation*}

3.1 Preliminaries

Let $Q=(I, E)$ be a symmetric quiver. For $d \in \mathbb {N}^I$ , recall from § 2.4 that $\mathbf {W}(d) \subset M(d)_{0, \mathbb {R}}$ is the polytope given by the Minkowski sum

\begin{align*} \mathbf {W}(d)=\tfrac {1}{2}\mathrm {sum}_{\beta \in \mathscr {A}}\,[0, \beta ] \subset M(d)_{0, \mathbb {R}}, \end{align*}

where

\begin{equation*}\mathscr {A}:=\{\beta ^a_i-\beta ^b_j \mid a,b\in I, \,(a \to b) \in E,\, 1\leqslant i\leqslant d^a,\, 1\leqslant j\leqslant d^b\}.\end{equation*}

We denote by

\begin{equation*}H_1, \ldots , H_m \subset \mathbf {W}(d)\end{equation*}

the codimension-one faces of $\mathbf {W}(d)$ and by

(3.1) \begin{equation} \lambda , \ldots , \lambda _m\colon \mathbb {C}^*\to ST(d) \end{equation}

the cocharacters of $ST(d)$ such that $\lambda _i^{\perp } \subset M(d)_{0, \mathbb {R}}$ is parallel to $H_i$ for all $1\leqslant i\leqslant m$ .

Note that there is a natural pairing (we abuse notation and use the same notation as for the pairing in § 2.2.2)

(3.2) \begin{align} \langle -, -\rangle \colon \mathbb {R}^I\times M(d)_{\mathbb {R}}^{W_d} \to \mathbb {R} \end{align}

defined by

\begin{equation*}\langle e^b, \det V^a\rangle =\delta ^{ab},\end{equation*}

where $e^b$ is the basis element of $\mathbb {R}^I$ corresponding to $b \in I$ . Note that an element $\ell \in M(d)_{0, \mathbb {R}}^{W_d}$ is written as

(3.3) \begin{align} \ell =\sum _{a}\ell ^{a}\det V^{a}, \quad \langle d, \ell \rangle =\sum _{a}d^a \ell ^a=0, \end{align}

i.e. $\ell$ is a $\mathbb {R}$ -character of $G(d)$ which is trivial on the diagonal torus $\mathbb {C}^{\ast } \subset G(d)$ . Note the following lemma.

Proof. For simplicity, suppose that $\lambda$ corresponds to a decomposition $d=d_1+d_2$ . Since $\lambda ^{\perp }$ is spanned by a subset of weights in $R(d)$ , for fixed vertices $a, b \in I$ we can write

(3.4) \begin{align} \ell =\sum _{i, j, p, q}{c_{ij}^{pq}}(\beta _i^{p}-\beta _j^{q}) \end{align}

for some $c_{ij}^{pq}\in \mathbb {R}$ , where the sum on the right-hand side is over all $p, q \in I$ and all $1 \leqslant i \leqslant d_1^{p}\mbox{ and } 1\leqslant j \leqslant d_1^{q}$ or $d_1^{p}\lt i \leqslant d^{p}\mbox{ and } d_1^{q}\lt j \leqslant d^{q}$ . On the other hand, as $l$ is Weyl-invariant, we can write

(3.5) \begin{align} \ell =\sum _{a \in I}\ell ^a(\beta _1^a+\cdots +\beta _{d^a}^a) \end{align}

for some $\ell ^a \in \mathbb {R}$ ; see (3.3).

In the right-hand side of (3.4), the sum of the coefficients of $\beta _i^{p}$ for all $p \in I$ and $1\leqslant i \leqslant d_1^{p}$ is zero. Therefore, by taking the sum of coefficients of such $\beta _i^p$ in the right-hand side of (3.5), we conclude that

\begin{align*} \sum _{a \in I}\ell ^a d_1^a=\langle d_1, \ell \rangle =0. \end{align*}

If $M(d)_{0, \mathbb {R}}^{W_d} \subset \lambda ^{\perp }$ , we have $\langle d_j, \ell \rangle =0$ for all $\ell \in M(d)_{0, \mathbb {R}}^{W_d}$ , and hence $d_j$ is proportional to $d$ .

For $\ell \in M(d)^{W_d}_{0, \mathbb {R}}$ , consider the open substack $\mathscr {X}(d)^{\ell \text {-ss}} \subset \mathscr {X}(d)$ of $\ell$ -semistable points. By [Reference KingKin94], the $\ell$ -semistable locus consists of $Q$ -representations $R$ such that for any subrepresentation $R' \subset R$ of dimension vector $d'$ , we have that $\langle d', \ell \rangle \geqslant 0$ . Consider the good moduli space morphism

\begin{align*} \mathscr {X}(d)^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}. \end{align*}

A closed point of $X(d)^{\ell \text {-ss}}$ corresponds to a direct sum

\begin{align*} \bigoplus _{i=1}^k V^{(i)} \otimes R^{(i)}, \end{align*}

where $V^{(i)}$ is a finite-dimensional $\mathbb {C}$ -vector space and $R^{(i)}$ is an $\ell$ -stable $Q$ -representation whose dimension vector $d^{(i)}$ satisfies $\langle d^{(i)}, \ell \rangle =0$ for each $1\leqslant i \leqslant k$ .

3.2 Quasi-BPS categories for semistable stacks

For each $\delta _d \in M(d)_{\mathbb {R}}^{W_d}$ , recall from Definition (2.9) the quasi-BPS category

(3.6) \begin{align} \mathbb {M}(d; \delta _d) \subset D^b(\mathscr {X}(d)). \end{align}

By Lemma 2.10, it is the subcategory of objects $P$ such that for any cocharacter $\lambda \colon \mathbb {C}^{\ast } \to T(d)$ , we have that

(3.7) \begin{align} \mathrm {wt}_{\lambda }(P) \subset [{-}\tfrac {1}{2}n_{\lambda }, \tfrac {1}{2}n_{\lambda } ]+\langle \lambda , \delta \rangle , \end{align}

where $n_{\lambda }:=\langle \lambda , \det (\mathbb {L}_{\mathscr {X}(d)}^{\lambda \gt 0}|_{0}) \rangle$ ; see (2.24).

Consider a complex $A\in D^b(B\mathbb {C}^*)$ . Write $A=\bigoplus _{w\in \mathbb {Z}}A_w$ , where $\mathbb {C}^*$ acts with weight $w$ on $A_w$ . Define the set of weights

(3.8) \begin{equation} \mathrm {wt}(A):=\{w\mid A_w\neq 0\}\subset \mathbb {Z}. \end{equation}

Define also the integers

(3.9) \begin{equation} \mathrm {wt}^{\mathrm {max}}(A):=\mathrm {max}(\mathrm {wt}(A)),\quad \mathrm {wt}^{\mathrm {min}}(A):=\mathrm {min}(\mathrm {wt}(A)). \end{equation}

We define a version of quasi-BPS categories for semistable loci. Let

(3.10) \begin{align} \mathbb {M}^{\ell }(d; \delta _d) \subset D^b(\mathscr {X}(d)^{\ell \text {-ss}}) \end{align}

be the subcategory of objects $P$ such that for any map $\nu \colon B\mathbb {C}^{\ast } \to \mathscr {X}(d)^{\ell \text {-ss}}$ , we have

(3.11) \begin{align} \mathrm {wt}(\nu ^{\ast }P) \subset [{-}\tfrac {1}{2}n_{\nu }, \tfrac {1}{2}n_{\nu } ]+\mathrm {wt}(\nu ^{\ast }\delta _d), \end{align}

where $n_{\nu }:=\mathrm {wt}(\det ((\nu ^{\ast }\mathbb {L}_{\mathscr {X}(d)})^{\gt 0})) \in \mathbb {Z}$ . In Corollary 3.10, we show that for $\ell =0$ , the category (3.10) is $\mathbb {M}(d; \delta _d)$ .

Consider the restriction functor

(3.12) \begin{align} \mathrm {res} \colon D^b(\mathscr {X}(d)) \twoheadrightarrow D^b(\mathscr {X}(d)^{\ell \text {-ss}}). \end{align}

Lemma 3.2. The functor (3.12) restricts to the functor

(3.13) \begin{align} \mathrm {res} \colon \mathbb {M}(d; \delta _d) \to \mathbb {M}^{\ell }(d; \delta _d). \end{align}

There is an orthogonal decomposition

\begin{equation*}D^b(\mathscr {X}(d)^{\ell \text {-ss}})=\bigoplus _{w\in \mathbb {Z}}D^b(\mathscr {X}(d)^{\ell \text {-ss}})_w\end{equation*}

as in (2.5). There are analogous such decompositions for $D^b(\mathscr {X}(d)^\lambda )$ . The following lemma is a generalization of [Reference Halpern-Leistner and SamHLS20, Proposition 3.11], where a similar result was obtained when the $\mathbb {C}^{\ast }$ -rigidification of $\mathscr {X}(d)^{\ell \text {-ss}}$ is a Deligne–Mumford stack.

Lemma 3.3. Let $w\in \mathbb {Z}$ . For $\ell \in M(d)_{0, \mathbb {R}}^{W_d}$ and $\delta _d \in M(d)_{\mathbb {R}}^{W_d}$ with $\langle 1_d, \delta _d\rangle =w$ , the category $D^b(\mathscr {X}(d)^{\ell \text {-ss}})_w$ is generated by $\mathrm {res}(\mathbb {M}(d; \delta _d) )$ and objects of the form $m_{\lambda }(P)$ , where $P \in D^b(\mathscr {X}(d)^{\lambda })_w$ is generated by $\Gamma _{G^{\lambda }(d)}(\chi '') \otimes \mathcal {O}_{\mathscr {X}^{\lambda }}(d)$ such that

(3.14) \begin{align} \langle \lambda , \chi '' \rangle \lt -\tfrac {1}{2}n_{\lambda }+\langle \lambda , \delta _d \rangle , \end{align}

and $\lambda \in \{\lambda _1, \ldots , \lambda _m\}$ is an antidominant cocharacter of $T(d)$ such that $\langle \lambda , \ell \rangle =0$ . The functor $m_{\lambda }$ is the categorical Hall product for the cocharacter $\lambda$ ; see (2.7).

Proof. We explain how to modify the proof of [Reference Halpern-Leistner and SamHLS20, Proposition 3.11] to obtain the desired conclusion. The vector bundles

(3.15) \begin{equation} \Gamma _{G(d)}(\chi ) \otimes \mathcal {O}_{\mathscr {X}(d)^{\ell \text {-ss}}}, \end{equation}

for $\chi$ a dominant weight of $G(d)$ such that $\langle 1_d, \chi \rangle =w$ , generate the category $D^b(\mathscr {X}(d)^{\ell \text {-ss}})_w$ . We show that (3.15) is generated by the objects in the statement using induction on the pair $(r_{\chi }, p_{\chi })$ (with respect to the lexicographic order), where

\begin{align*} r_{\chi }:=\mathrm {min} \{r \geqslant 0 \mid \chi +\rho -\delta _d \in r\mathbf {W}(d) \} \end{align*}

and $p_{\chi }$ is the smallest possible number of $a_\beta$ equal to $-r_{\chi }$ among all ways of writing

\begin{equation*}\chi +\rho -\delta _d=\sum _{\beta \in \mathscr {A}}a_\beta \beta\end{equation*}

with $a_\beta \in [-r_{\chi }, 0]$ . If $r_{\chi } \le 1/2$ , then $\Gamma _{G(d)}(\chi ) \otimes \mathcal {O}_{\mathscr {X}(d)^{\ell \text {-ss}}}$ is an object in $\mathrm {res}(\mathbb {M}(d; \delta _d) )$ , so we may assume that $r_{\chi }\gt 1/2$ .

As in the argument of [Reference Halpern-Leistner and SamHLS20, Proposition 3.11], there is an antidominant cocharacter $\lambda$ of $ST(d)$ such that the following hold:

1. – $\langle \lambda , \chi \rangle \leqslant \langle \lambda , \mu \rangle$ for any $\mu \in -\rho +r\mathbf {W}(d)+\delta _d$ ;

2. – $\lambda ^{\perp }$ is parallel to a face in $\mathbf {W}(d)$ .

Suppose first that $\langle \lambda , \ell \rangle \neq 0$ . Then as in the proof of [Reference Halpern-Leistner and SamHLS20, Proposition 3.11], there is a complex of vector bundles consisting of $\Gamma _{G(d)}(\chi ) \otimes \mathcal {O}_{\mathscr {X}(d)}$ (which appears once) and $\Gamma _{G(d)}(\chi ') \otimes \mathcal {O}_{\mathscr {X}(d)}$ such that $(r_{\chi '}, p_{\chi '})$ is smaller than $(r_{\chi }, p_{\chi })$ and supported on the $\ell$ -unstable locus. The conclusion then follows by induction.

Suppose next that $\langle \lambda , \ell \rangle =0$ (observe that this case did not occur in [Reference Halpern-Leistner and SamHLS20, Proposition 3.11]). The object

\begin{equation*}m_{\lambda }(\Gamma _{G(d)^{\lambda }}(\chi ) \otimes \mathcal {O}_{\mathscr {X}^{\lambda }(d)})\end{equation*}

is quasi-isomorphic to a complex of vector bundles (see [Reference Pădurariu and TodaPT22a, Proposition 2.1]) consisting of $\Gamma _{G(d)}(\chi ) \otimes \mathcal {O}_{\mathscr {X}(d)}$ (which appears once) and $\Gamma _{G(d)}(\chi ') \otimes \mathcal {O}_{\mathscr {X}(d)}$ such that $(r_{\chi '}, p_{\chi '})$ is smaller than $(r_{\chi }, p_{\chi })$ ; see the last part of the proof of [Reference PădurariuPăd21, Proposition 4.1]. Since we have

\begin{align*} r_{\chi }=-\frac {\langle \lambda , \chi +\rho -\delta _d\rangle }{\langle \lambda , R^{\lambda \gt 0}(d) \rangle }\gt \tfrac {1}{2} \end{align*}

and $\lambda$ is antidominant, the inequality (3.14) holds for $\chi ''=\chi$ . The conclusion then follows using induction on $(r_{\chi }, p_{\chi })$ .

3.3 The categorical wall-crossing equivalence for symmetric quivers

In this subsection, we give some sufficient conditions for the functor (3.13) to be an equivalence. As a corollary, we obtain the categorical wall-crossing equivalence for quasi-BPS categories of symmetric quivers (and potential zero). The argument is similar to that for [Reference Halpern-Leistner and SamHLS20, Theorem 3.2], with a modification due to the existence of faces of $\mathbf {W}(d)$ parallel to hyperplanes which contain $M(d)_{0, \mathbb {R}}^{W_d}$ .

For a stability condition $\ell \in M(d)_{0, \mathbb {R}}^{W_d}$ , let

(3.16) \begin{align} \mathscr {X}(d)=\mathscr {S}_1 \sqcup \cdots \sqcup \mathscr {S}_N \sqcup \mathscr {X}(d)^{\ell \text {-ss}} \end{align}

be the Kempf–Ness stratification of $\mathscr {X}(d)$ with center $\mathscr {Z}_i \subset \mathscr {S}_i$ for $1\leqslant i\leqslant N$ . Consider the associated one-parameter subgroups

(3.17) \begin{equation} \mu _1,\ldots ,\mu _N\colon \mathbb {C}^{\ast } \to T(d); \end{equation}

see [Reference Halpern-LeistnerHL15, § 2] for a review of Kempf–Ness stratification.

Proposition 3.4. Suppose that $2\langle \mu _i, \delta _d \rangle \notin \mathbb {Z}$ for all $1\leqslant i\leqslant N$ . Then the functor (3.13) is fully faithful:

\begin{equation*}\mathrm {res} \colon \mathbb {M}(d; \delta _d) \hookrightarrow \mathbb {M}^{\ell }(d; \delta _d).\end{equation*}

Proof. For a choice $k_\bullet =(k_i)_{i=1}^N \in \mathbb {R}^N$ , there is a `window category’ $\mathbb {W}_{k_{\bullet }}^{\ell } \subset D^b(\mathscr {X}(d))$ such that the functor (3.12) restricts to the equivalence (see [Reference Halpern-LeistnerHL15, Reference Ballard, Favero and KatzarkovBFK19]):

\begin{align*} \mathrm {res} \colon \mathbb {W}_{k_{\bullet }}^{\ell } \stackrel {\sim }{\to } D^b(\mathscr {X}(d)^{\ell \text {-ss}}). \end{align*}

The subcategory $\mathbb {W}_{k_{\bullet }}^{\ell }$ consists of objects $P$ such that $P|_{\mathscr {Z}_i}$ has $\mu _i$ -weights contained in the interval $[k_i, k_i+n_{\mu _i})$ , where the width $n_{\mu _i}$ is defined by (2.24) for $\lambda =\mu _i$ . For $1\leqslant i\leqslant N$ , let

\begin{equation*}k_i:=-n_{\mu _i}/2+\langle \mu _i, \delta _d \rangle .\end{equation*}

By the assumption of $\delta _d$ , we have that $\mathbb {M}(d; \delta _d) \subset \mathbb {W}_{k_{\bullet }}^{\ell }$ . It thus follows that the functor (3.13) is fully faithful.

Proposition 3.5. For any $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ , the functor (3.13) is essentially surjective:

\begin{equation*}\mathrm {res} \colon \mathbb {M}(d; \delta _d) \twoheadrightarrow \mathbb {M}^{\ell }(d; \delta _d).\end{equation*}

Proof. We will use Lemma 3.3. We first explain that it suffices to show that

(3.18) \begin{align} \textrm {Hom}(\mathbb {M}^{\ell }(d; \delta _d), \mathrm {res}(m_{\lambda }(P)))=0, \end{align}

where $m_\lambda$ is the categorical Hall product and $\lambda$ and $P$ are given as in Lemma 3.3. If the above vanishing holds, then

\begin{align*} \textrm {Hom}(\mathrm {res}(\mathbb {M}(d; \delta _d)), \mathrm {res}(m_{\lambda }(P)))=0, \end{align*}

and hence by Lemma 3.3 there is a semiorthogonal decomposition

\begin{align*} D^b(\mathscr {X}(d)^{\ell \text {-ss}}) = \langle \mathcal {C}, \mathrm {res}(\mathbb {M}(d; \delta _d)) \rangle , \end{align*}

where $\mathcal {C}$ is the subcategory of $D^b(\mathscr {X}(d)^{\ell \text {-ss}})$ generated by objects $\mathrm {res}(m_{\lambda }(P))$ as above (or as in Lemma 3.3). Observe that $\mathbb {M}^{\ell }(d; \delta _d)$ is in the left complement of $\mathcal {C}$ in $D^b(\mathscr {X}(d)^{\ell \text {-ss}})$ . Thus the vanishing (3.18) implies that indeed (3.13) is essentially surjective.

Below we show the vanishing (3.13). The stack $\mathscr {X}^{\lambda \geqslant 0}(d)$ is the stack of filtrations

(3.19) \begin{align} 0=R_0 \subset R_1 \subset R_2 \subset \cdots \subset R_k=R \end{align}

of $Q$ -representations such that $R_i/R_{i-1}$ has dimension vector $d_i$ . By Lemma 3.1, we have that $\langle \ell , d_i\rangle =0$ for $1\leqslant i\leqslant k$ . It follows that $R$ is $\ell$ -semistable if and only if every $R_i/R_{i-1}$ is $\ell$ -semistable. Therefore there are Cartesian squares

(3.20)

where each vertical arrow is an open immersion. The stack $\mathcal {N}$ is the moduli stack of filtrations (3.19) such that each $R_i/R_{i-1}$ is $\ell$ -semistable with dimension vector $d_i$ . Using proper base change and adjunction, the vanishing (3.18) is equivalent to the vanishing of

(3.21) \begin{align} \textrm {Hom}(E, p_{\lambda \ast }^{\ell }q_{\lambda }^{\ell \ast } j^{\ast }P) =\textrm {Hom}(p_{\lambda }^{\ell \ast }E, q_{\lambda }^{\ell \ast }j^{\ast }P) \end{align}

for any $E \in \mathbb {M}^{\ell }(d; \delta _d)$ . Let $\pi$ be the following composition:

\begin{align*} \pi \colon \mathcal {N} \to \mathscr {X}(d)^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}. \end{align*}

By the local-to-global Hom spectral sequence, it is enough to show the vanishing of

(3.22) \begin{align} \pi _{\ast }\mathcal {H}om (p_{\lambda }^{\ell \ast }E, q_{\lambda }^{\ell \ast }j^{\ast }P)=0. \end{align}

Furthermore, it suffices to show the vanishing (3.22) formally locally at each point $p \in X(d)^{\ell \text {-ss}}$ . We abuse notation and also denote by $p$ the unique closed point in the fiber of $\mathscr {X}(d)^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ , and we may assume that $\lambda$ is a cocharacter of $G_p:=\mathrm {Aut}(p)$ . By Lemma 3.6 below, the top diagram of (3.20) is, formally locally over $p\in X(d)^{\ell \text {-ss}}$ , of the form

(3.23) \begin{align} A^{\lambda }/G_p^{\lambda } \leftarrow A^{\lambda \geqslant 0}/G_p^{\lambda \geqslant 0} \to A/G_p, \end{align}

where $A$ is a smooth affine scheme (of finite type over a complete local ring) with a $G_p$ -action and good moduli space map $\pi _A\colon A/G_p\to A/\!\!/ G_p$ . Then the vanishing (3.22) at $p$ holds since the $\lambda$ -weights of $p_{\lambda }^{\ell \ast }E$ are strictly larger than those of $q_{\lambda }^{\ell \ast }j^{\ast }P$ by the definition of $\mathbb {M}^{\ell }(d; \delta _d)$ and the inequality (3.14); see [Reference Halpern-LeistnerHL15, Corollary 3.17 and Amplification 3.18] and [Reference PădurariuPăd21, Proposition 4.2].

We have used the following lemma.

Proof. Consider the stack $\Theta =\mathbb {A}^1/\mathbb {C}^{\ast }$ . Since $\mathcal {N}$ is the moduli stack of filtrations of $\ell$ -semistable objects (3.19), the top diagram of (3.20) is a component of the diagram

(3.24) \begin{align} \mathrm {Map}(B\mathbb {C}^{\ast }, \mathscr {X}(d)^{\ell \text {-ss}}) \leftarrow \mathrm {Map}(\Theta , \mathscr {X}(d)^{\ell \text {-ss}}) \to \mathscr {X}(d)^{\ell \text {-ss}}, \end{align}

where the horizontal arrows are evaluation maps for mapping stacks from $B\mathbb {C}^{\ast }$ or $\Theta$ ; see [Reference Halpern-LeistnerHL18]. Specifically, $\mathcal {N}$ is an open and closed substack of $\mathrm {Map}(\Theta , \mathscr {X}(d)^{\ell \text {-ss}})$ , and the top diagram in (3.20) is the restriction of (3.24) to $\mathcal {N}$ . By the Luna étale slice theorem, the pull-back of $\mathscr {X}(d)^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ via $X(d)_p^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ is of the form $A/G_p$ , where $A$ is a smooth affine scheme of finite type over $X(d)_p^{\ell \text {-ss}}$ with an action of $G_p$ and good moduli space map $\pi _A\colon A/G_p\to A/\!\!/ G_p$ . Since the mapping stacks from $B\mathbb {C}^{\ast }$ or $\Theta$ commute with pull-backs of maps to good moduli spaces (see [Reference Halpern-LeistnerHL18, Corollary 1.30.1]), the pull-back of the top diagram in (3.20) via $X(d)_p^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ consists of compatible connected components of the stacks:

(3.25) \begin{align} \mathrm {Map}(B\mathbb {C}^{\ast }, A/G_p) \leftarrow \mathrm {Map}(\Theta , A/G_p) \to A/G_p. \end{align}

Such connected components are of the form (3.23) for some cocharacter $\lambda$ of $G_p$ (see [Reference Halpern-LeistnerHL18, Theorem 1.37]), and thus the conclusion follows.

Recall the cocharacters $\{\lambda _1,\ldots ,\lambda _m\}$ from (3.1) and the cocharacters $\{\mu _1,\ldots ,\mu _N\}$ from (3.17).

Propositions 3.4 and 3.5 imply the following result.

Theorem 3.8. Suppose that the pair $(\ell , \delta _d)\in M(d)_{0,\mathbb {R}}^{W_d}\times M(d)^{W_d}_{\mathbb {R}}$ satisfies $\delta _d \in U_{\ell }$ . Then the functor (3.13) is an equivalence

\begin{equation*}\mathrm {res} \colon \mathbb {M}(d; \delta _d) \stackrel {\sim }{\to } \mathbb {M}^{\ell }(d; \delta _d).\end{equation*}

We mention two corollaries of Theorem 3.8.

Corollary 3.9. For weights $\ell , \ell ' \in M(d)_{0, \mathbb {R}}^{W_d}$ and $\delta _d \in U_{\ell } \cap U_{\ell '}$ , there is an equivalence

\begin{align*} \mathbb {M}^{\ell }(d; \delta _d) \simeq \mathbb {M}^{\ell '}(d; \delta _d). \end{align*}

3.4 The categorical wall-crossing equivalence for symmetric quivers with potential

Let $(Q, W)$ be a symmetric quiver with potential. Similarly to (3.10), we define a category

(3.26) \begin{align} \mathbb {S}^{\ell }(d; \delta _d) \subset \mathrm {MF}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W). \end{align}

First, for an object $A\in \mathrm {MF}(B\mathbb {C}^*, 0)$ , write $A=\bigoplus _{w\in \mathbb {Z}} A_w$ with $A_w\in \mathrm {MF}(B\mathbb {C}^*, 0)_w$ . Consider the set of weights

\begin{equation*}\mathrm {wt}(A):=\{w\mid A_w\neq 0\}\subset \mathbb {Z}.\end{equation*}

Then (3.26) is the subcategory of $\mathrm {MF}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W)$ containing objects $P$ such that for any map $\nu \colon B\mathbb {C}^{\ast } \to \mathscr {X}(d)^{\ell \text {-ss}}$ with $\nu ^{\ast } \mathop {\textrm {Tr}} W=0$ , the set $\mathrm {wt} (\nu ^{\ast }P)$ satisfies the condition (3.11). Note that if $\nu ^{\ast } \mathop {\textrm {Tr}} W \neq 0$ , then $\mathrm {MF}(B\mathbb {C}^{\ast }, \nu ^{\ast } \mathop {\textrm {Tr}} W)=0$ so that the condition on weights in $\nu ^{\ast }P$ is vacuous in this case.

In the graded case (of a tripled quiver), the subcategory

\begin{align*} \mathbb {S}^{\mathrm {gr}, \ell }(d; \delta _d) \subset \mathrm {MF}^{\mathrm {gr}}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W) \end{align*}

is defined to be the pull-back of $\mathbb {S}^{\ell }(d; \delta _d)$ by the forget-the-grading functor

(3.27) \begin{align} \mathrm {forg} \colon \mathrm {MF}^{\textrm {gr}}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W) \to \mathrm {MF}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W). \end{align}

Let $\bullet \in \{\emptyset , \mathrm {gr}\}$ . The following results are proved as Theorem 3.8, Corollary 3.9, and Corollary 3.10.

Theorem 3.12. Suppose that the pair $(\ell , \delta _d)\in M(d)_{0,\mathbb {R}}^{W_d}\times M(d)^{W_d}_{\mathbb {R}}$ satisfies $\delta _d \in U_{\ell }$ . Then the restriction functor induces an equivalence

\begin{equation*}\mathrm {res} \colon \mathbb {S}^{\bullet }(d; \delta _d) \stackrel {\sim }{\to } \mathbb {S}^{\bullet , \ell }(d; \delta _d).\end{equation*}

Corollary 3.13. For weights $\ell , \ell ' \in M(d)_{0, \mathbb {R}}^{W_d}$ , and $\delta _d \in U_{\ell } \cap U_{\ell '}$ , there is an equivalence

\begin{align*} \mathbb {S}^{\bullet , \ell }(d; \delta _d) \simeq \mathbb {S}^{\bullet , \ell '}(d; \delta _d). \end{align*}

3.5 The categorical wall-crossing for preprojective algebras

In this subsection we will use the notation from §§ 2.2.6 and 2.2.7. Consider a quiver $Q^{\circ }=(I, E^{\circ })$ . Let $Q^{\circ , d}=(I, E^{\circ , d})$ be its doubled quiver. For $\ell \in M(d)_{0, \mathbb {R}}^{W_d}$ , the moment map $\mu \colon T^*R^\circ (d)\to \mathfrak {g}(d)$ induces a map $\mu ^{\ell \text {-ss}}\colon (T^*R^\circ (d))^{\ell \text {-ss}}\to \mathfrak {g}(d)$ . Let

\begin{equation*}\mathscr {P}(d)^{\ell \text {-ss}}:=(\mu ^{\ell \text {-ss}} )^{-1}(0) /G(d) \subset \mathscr {P}(d)\end{equation*}

be the derived open substack of $\ell$ -semistable representations of the preprojective algebra of $Q^\circ$ . Consider the restriction functor

(3.28) \begin{align} \mathrm {res} \colon D^b(\mathscr {P}(d)) \twoheadrightarrow D^b(\mathscr {P}(d)^{\ell \text {-ss}}). \end{align}

The closed immersion (2.10) restricts to the closed immersion

(3.29) \begin{align} j \colon \mathscr {P}(d)^{\ell \text {-ss}} \hookrightarrow \mathscr {Y}(d)^{\ell \text {-ss}}. \end{align}

For $\delta _d \in M(d)_{\mathbb {R}}^{W_d}$ , define the subcategory

(3.30) \begin{align} \mathbb {T}^{\ell }(d; \delta _d) \subset D^b(\mathscr {P}(d)^{\ell \text {-ss}}) \end{align}

with objects $\mathcal {E}$ such that for any map $\nu \colon B\mathbb {C}^{\ast } \to \mathscr {P}(d)^{\ell \text {-ss}}$ , we have

(3.31) \begin{align} \mathrm {wt}(\nu ^{\ast }j^{\ast }j_{\ast }\mathcal {E}) \subset [{-}\tfrac {1}{2}n_{\nu }, \tfrac {1}{2}n_{\nu } ] +\mathrm {wt}(\nu ^{\ast }\delta _d). \end{align}

Here, we let $n_{\nu }:=\mathrm {wt}(\det (\nu ^{\ast }\mathbb {L}_{\mathscr {X}(d)}|_{\mathscr {P}(d)})^{\nu \gt 0})$ , where $\mathscr {X}(d)$ is the moduli stack of representations (2.14) of the tripled quiver $Q$ of $Q^{\circ }$ .

Proposition 3.16. The Koszul equivalence (2.15) descends to an equivalence

(3.32) \begin{align} \Theta \colon D^b(\mathscr {P}(d)^{\ell \text {-ss}}) \stackrel {\sim }{\to } \mathrm {MF}^{\textrm {gr}}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W), \end{align}

which restricts to an equivalence

(3.33) \begin{align} \Theta \colon \mathbb {T}^{\ell }(d; \delta _d) \stackrel {\sim }{\to } \mathbb {S}^{\mathrm {gr}, \ell }(d; \delta _d). \end{align}

Proof. Let $\eta \colon \mathscr {X}(d) \to \mathscr {Y}(d)$ be the projection. Then we have

(3.34) \begin{align} \mathrm {Crit}(\mathop {\textrm {Tr}} W) \cap \mathscr {X}(d)^{\ell \text {-ss}} \subset \eta ^{-1}(\mathscr {Y}(d)^{\ell \text {-ss}}) \subset \mathscr {X}(d)^{\ell \text {-ss}}, \end{align}

where the first inclusion is proved in [Reference Halpern-LeistnerHL20, Lemma 4.3.22] and the second inclusion is immediate from the definition of $\ell$ -stability. We obtain equivalences

\begin{align*} D^b(\mathscr {P}(d)^{\ell \text {-ss}}) \stackrel {\sim }{\to } \mathrm {MF}^{\textrm {gr}}(\eta ^{-1}(\mathscr {Y}(d)^{\ell \text {-ss}}), \mathop {\textrm {Tr}} W) \stackrel {\sim }{\leftarrow } \mathrm {MF}^{\textrm {gr}}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W), \end{align*}

where the first equivalence is the Koszul equivalence in Theorem 2.5 and the second equivalence follows from (3.34) together with the fact that matrix factorizations are supported on the critical locus; see (2.3). Therefore we obtain the equivalence (3.32).

For an object $\mathcal {E} \in D^b(\mathscr {P}(d)^{\ell \text {-ss}})$ , the object $P=\Theta (\mathcal {E})$ is in $\mathbb {S}^{\textrm {gr}, \ell }(d; \delta _d)$ if and only if for any map $\nu \colon B\mathbb {C}^{\ast } \to \mathscr {X}(d)^{\ell \text {-ss}}$ with $\nu ^{\ast }\mathop {\textrm {Tr}} W=0$ , the set $\mathrm {wt}(\nu ^{\ast }\mathrm {forg}(P))$ satisfies the weight condition (3.11). As $P$ is supported on $\mathrm {Crit}(\mathop {\textrm {Tr}} W)$ , we may assume that the image of $\nu$ is contained in $\mathrm {Crit}(\mathop {\textrm {Tr}} W) \cap \mathscr {X}(d)^{\ell \text {-ss}}$ . By the $\mathbb {C}^{\ast }$ -equivariance of $P$ for the fiberwise weight $2$ -action on $\eta \colon \mathscr {X}(d)\to \mathscr {Y}(d)$ and upper semicontinuity, we have that

\begin{equation*}\mathrm {wt}(\nu ^{\ast }\mathrm {forg}(P) )\subset \mathrm {wt}(\nu '^{\ast }\mathrm {forg}(P) ),\end{equation*}

where $\nu '$ is the composition

\begin{align*} \nu ' \colon B\mathbb {C}^{\ast } \stackrel {\nu }{\to } \mathscr {X}(d) \stackrel {\eta }{\to } \mathscr {Y}(d) \stackrel {0}{\hookrightarrow } \mathscr {X}(d). \end{align*}

The image of $\nu '$ lies in $\mathscr {P}(d)^{\ell \text {-ss}} \hookrightarrow \mathscr {Y}(d)^{\ell \text {-ss}}\stackrel {0}{\hookrightarrow } \mathscr {X}(d)^{\ell \text {-ss}}$ . Therefore we may assume that the image of $\nu$ is contained in $\mathscr {P}(d)^{\ell \text {-ss}}$ . Since the object $P$ is represented by

\begin{align*} P=\Theta (\mathcal {E})= (\mathcal {E} \otimes _{\mathcal {O}_{\mathscr {P}(d)}}\mathcal {O}_{\mathscr {X}(d)})|_{\mathscr {X}(d)^{\ell \text {-ss}}} =(j_{\ast }\mathcal {E} \otimes _{\mathcal {O}_{\mathscr {Y}(d)}} \mathcal {O}_{\mathscr {X}(d)})|_{\mathscr {X}(d)^{\ell \text {-ss}}}, \end{align*}

it follows that $\mathrm {wt}(\nu ^{\ast }\mathrm {forg}(P))$ satisfies the condition (3.11) if and only if $\mathrm {wt}(\nu ^{\ast }(j_{\ast }\mathcal {E}) )$ satisfies the condition (3.31). Therefore $\Theta (\mathcal {E})$ is in $\mathbb {S}^{\textrm {gr}, \ell }(d; \delta _d)$ if and only if $\mathcal {E}$ is in $\mathbb {T}^{\ell }(d; \delta _d)$ .

By combining Proposition 3.16 with Theorem 3.12, Corollary 3.13, and Corollary 3.14, we obtain the following.

Theorem 3.17. Suppose that the pair $(\ell , \delta _d)\in M(d)_{0,\mathbb {R}}^{W_d}\times M(d)^{W_d}_{\mathbb {R}}$ satisfies $\delta _d \in U_{\ell }$ . Then the restriction functor (3.28) induces an equivalence

\begin{equation*}\mathrm {res} \colon \mathbb {T}(d; \delta _d) \stackrel {\sim }{\to } \mathbb {T}^{\ell }(d; \delta _d).\end{equation*}

Corollary 3.18. For $\ell , \ell ' \in M(d)_{0, \mathbb {R}}^{W_d}$ and $\delta _d \in U_{\ell } \cap U_{\ell '}$ , there is an equivalence

\begin{align*} \mathbb {T}^{\ell }(d; \delta _d) \simeq \mathbb {T}^{\ell '}(d; \delta _d). \end{align*}

3.6 Quasi-BPS categories under Knörrer periodicity

In this subsection, we apply Corollaries 3.10 and 3.14 to obtain an equivalence of quasi-BPS categories under Knörrer periodicity, which is a particular case of the Koszul equivalence. Other than the use of Corollaries 3.10 and 3.14, the current subsection is independent of the other results and constructions discussed in § 3. We will use the results of this subsection in § 4 and in [Reference Pădurariu and TodaPT23b].

We will use the notation from § 2.2. For a symmetric quiver $Q$ and $d \in \mathbb {N}^I$ , let $U$ be a $G(d)$ -representation. Consider the closed immersion

\begin{align*} j \colon \mathscr {X}(d):=R(d)/G(d) \hookrightarrow \mathscr {Y}=(R(d) \oplus U)/G(d) \end{align*}

into $(R(d)\oplus \{0\})/G(d)$ . We consider the quotient stack $\mathscr {X}^\gimel$ and regular function $f^\gimel$ :

\begin{align*} \mathscr {X}^\gimel =(R(d) \oplus U \oplus U^{\vee })/G(d) \stackrel {f}{\to } \mathbb {C}, \quad f(x, u, u')=\langle u, u'\rangle . \end{align*}

We consider a Cartesian diagram

where $\eta$ is the natural projection. Let $\mathbb {C}^{\ast }$ act with weight $0$ on $R(d)\oplus U$ and with weight $2$ on $U^\vee$ . In this case, the Koszul equivalence in Theorem 2.5 is given by (see [Reference TodaTod19, Remark 2.3.5])

(3.35) \begin{align} \Theta =s_{\ast }v^{\ast } \colon D^b(\mathscr {X}(d)) \stackrel {\sim }{\to } \mathrm {MF}^{\textrm {gr}}(\mathscr {X}^\gimel , f). \end{align}

Such an equivalence is also called Knörrer periodicity [Reference OrlovOrl06, Reference HiranoHir17]. For a weight $\delta ^\gimel _d \in M(d)_{\mathbb {R}}^{W_d}$ , define the subcategory

(3.36) \begin{align} \mathbb {S}^{\textrm {gr}}(d; \delta ^\gimel _d) \subset \mathrm {MF}^{\textrm {gr}}(\mathscr {X}^\gimel , f) \end{align}

in a way similar to (2.28). By Lemma 2.10, it consists of matrix factorizations whose factors are direct sums of vector bundles $\mathcal {O}_{\mathscr {X}^\gimel }\otimes \Gamma$ , where $\Gamma$ is a $G(d)$ -representations such that for any weight $\chi '$ of $\Gamma$ and any cocharacter $\lambda$ of $T(d)$ , we have

(3.37) \begin{align} \langle \lambda , \chi '-\delta ^\gimel _d \rangle \in [{-}\tfrac {1}{2}n^\gimel _{\lambda }, \tfrac {1}{2}n^\gimel _{\lambda } ]. \end{align}

Here, we define $n^\gimel _{\lambda }$ by

(3.38) \begin{align} n^\gimel _{\lambda }=\langle \lambda , \mathbb {L}_{\mathscr {X}^\gimel }^{\lambda \gt 0} \rangle = n_{\lambda }+\langle \lambda , U^{\lambda \gt 0} \rangle + \langle \lambda , (U^{\vee })^{\lambda \gt 0} \rangle , \end{align}

where we recall the definition of $n_{\lambda }$ from (2.24). The following is the main result we prove in this subsection.

Proposition 3.20. Let $\delta ^\gimel _d=\delta _d-\frac {1}{2}\det U\in M(d)^{W_d}_{\mathbb {R}}$ . The equivalence (3.35) restricts to the equivalence

(3.39) \begin{align} \Theta \colon \mathbb {M}(d; \delta _d) \stackrel {\sim }{\to } \mathbb {S}^{\textrm {gr}}(d; \delta ^\gimel _d). \end{align}

Proof. We first note that, by Lemma 2.6, an object $\mathcal {E} \in D^b(\mathscr {X}(d))$ satisfies $\Theta (\mathcal {E}) \in \mathbf {S}^{\textrm {gr}}(d; \delta ^\gimel _d)$ if and only if $j_{\ast }\mathcal {E}$ is generated by vector bundles $\mathcal {O}_{\mathscr {Y}} \otimes \Gamma '$ , where any weight $\chi '$ of $\Gamma '$ satisfies (3.37).

By Lemma 2.10, the category $\mathbb {M}(d; \delta _d)$ is generated by vector bundles $\mathcal {O}_{\mathscr {X}(d)}\otimes \Gamma$ such that any weight $\chi$ of $\Gamma$ satisfies (2.25) for any $\lambda$ . Consider the Koszul resolution

(3.40) \begin{align} j_{\ast }(\Gamma \otimes \mathcal {O}_{\mathscr {X}(d)}) =\Gamma \otimes \mathrm {Sym}_{\mathscr {Y}} (\mathcal {U}^{\vee }[1]), \end{align}

where $\mathcal {U} \to \mathscr {Y}$ is the vector bundle associated with the $G(d)$ -representation $U$ . Therefore, the category $j_{\ast }\mathbb {M}(d; \delta )$ is generated by vector bundles $\mathcal {O}_{\mathscr {Y}} \otimes \Gamma '$ such that any weight $\chi '$ of $\Gamma '$ satisfies

(3.41) \begin{align} \langle \lambda , \chi '-\delta _d \rangle \in \left [{-}\frac {n_{\lambda }}{2}+ \langle \lambda , (U^{\vee })^{\lambda \lt 0}), \frac {n_{\lambda }}{2}+ \langle \lambda , (U^{\vee })^{\lambda \gt 0}) \right ] \end{align}

for any $\lambda$ . By (3.38), we have

(3.42) \begin{align} \frac {n^\gimel _{\lambda }}{2} =\frac {n_{\lambda }}{2}+\langle \lambda , (U^{\vee })^{\lambda \gt 0}\rangle +\frac {1}{2}\langle \lambda , U \rangle =\frac {n_{\lambda }}{2}- \langle \lambda , (U^{\vee })^{\lambda \lt 0}\rangle -\frac {1}{2}\langle \lambda , U \rangle . \end{align}

Therefore (3.41) implies (3.37) for $\delta ^\gimel _d=\delta _d-\frac {1}{2} \det U$ , and hence the functor (3.35) sends $\mathbb {M}(d; \delta )$ to $\mathbb {S}^{\textrm {gr}}(d; \delta ^\gimel _d)$ , which shows the fully faithfulness of (3.39).

To show essential surjectivity of (3.39), let $\mathcal {E} \in D^b(\mathscr {X}(d))$ be such that $j_{\ast }\mathcal {E}$ is generated by the vector bundles $\mathcal {O}_{\mathscr {Y}} \otimes \Gamma '$ , where any weight $\chi '$ of $\Gamma '$ satisfies (3.37). We will show that $\mathcal {E}\in \mathbb {M}^{\ell =0}(d; \delta _d)$ and thus that $\mathcal {E}\in \mathbb {M}(d; \delta _d)$ by Corollary 3.10.

Let $\nu \colon B\mathbb {C}^{\ast } \to \mathscr {X}(d)$ be a map, which corresponds to a point $x \in R(d)$ and a cocharacter $\lambda \colon \mathbb {C}^{\ast } \to T(d)$ which fixes $x$ . By the condition (3.37) for weights of $\Gamma '$ , we have

\begin{align*} \mathrm {wt}^{\mathrm {max}}(\nu ^{\ast }j^{\ast }j_{\ast }\mathcal {E}) \leqslant \tfrac {1}{2} n_{\lambda }' +\langle \lambda , \delta _d' \rangle ; \end{align*}

see (3.9) for the definition of $\mathrm {wt}^{\mathrm {max}}$ . On the other hand, by the Koszul resolution (3.40), we have

\begin{align*} \mathrm {wt}^{\mathrm {max}}(\nu ^{\ast }j^{\ast }j_{\ast }\mathcal {E}) = \mathrm {wt}^{\mathrm {max}}(\nu ^{\ast }\mathcal {E}) -\langle \lambda , (U^{\vee })^{\lambda \gt 0} \rangle . \end{align*}

Therefore we have

\begin{align*} \mathrm {wt}^{\mathrm {max}}(\nu ^{\ast }\mathcal {E}) \leqslant \frac {n_{\lambda }'}{2}+\langle \lambda , \delta _d' \rangle -\langle \lambda , (U^{\vee })^{\lambda \gt 0} \rangle =\frac {n_{\lambda }}{2}+\langle \lambda , \delta _d\rangle , \end{align*}

where the last equality follows from (3.42). The lower bound

\begin{equation*}\mathrm {wt}^{\mathrm {min}}(\nu ^{\ast }\mathcal {E}) \geqslant -\frac {n_{\lambda }}{2}+\langle \lambda , \delta _d\rangle \end{equation*}

is proved similarly. We then have that

\begin{equation*}\mathrm {wt}(\nu ^{\ast }\mathcal {E})\subset \left [{-}\frac {n_{\lambda }}{2}+\langle \lambda , \delta _d\rangle , \frac {n_{\lambda }}{2}+\langle \lambda , \delta _d\rangle \right ].\end{equation*}

Thus $\mathcal {E} \in \mathbb {M}(d; \delta _d)^{\ell =0}$ , and then $\mathcal {E} \in \mathbb {M}(d; \delta _d)$ by Corollary 3.10.

Let $W$ be a potential of $Q$ . With an abuse of notation, we denote by $\mathop {\textrm {Tr}} W \colon \mathscr {X}^\gimel \to \mathbb {C}$ the pull-back of $\mathop {\textrm {Tr}} W \colon \mathscr {X}(d) \to \mathbb {C}$ by the natural projection $\mathscr {X}^\gimel \to \mathscr {X}(d)$ . There is an equivalence similar to (3.35), also called Knörrer periodicity (see [Reference HiranoHir17, Theorem 4.2] and [Reference OrlovOrl06]):

(3.43) \begin{align} \Theta =s_{\ast }v^{\ast } \colon \mathrm {MF}(\mathscr {X}(d), \mathop {\textrm {Tr}} W) \stackrel {\sim }{\to } \mathrm {MF}(\mathscr {X}^\gimel , \mathop {\textrm {Tr}} W+f). \end{align}

The subcategory

\begin{align*} \mathbb {S}(d; \delta _d) \subset \mathrm {MF}(\mathscr {X}^\gimel , \mathop {\textrm {Tr}} W+f) \end{align*}

is defined similarly to (3.36). The following proposition is proved as Proposition 3.20, using Corollary 3.14 instead of Corollary 3.10.

Proposition 3.21. Let $\delta ^\gimel _d=\delta _d-\frac {1}{2}\det U\in M(d)^{W_d}_{\mathbb {R}}$ . The equivalence (3.43) restricts to the equivalence

\begin{align*} \Theta \colon \mathbb {S}(d; \delta _d) \stackrel {\sim }{\to } \mathbb {S}(d; \delta ^\gimel _d). \end{align*}

4.1 Semiorthogonal decompositions

The following is the main result in this section, which provides a semiorthogonal decomposition of $D^b(\mathscr {X}^f(d)^{\text {ss}})$ in products of quasi-BPS categories of $Q$ . Recall the definitions of a good weight from Definition 2.12, the category $\mathbb {M}(d;\delta )$ from Definition (2.9), and the Weyl-invariant real weights $\tau _d\mbox{ and } \sigma _d$ from § 2.2.2. Recall also the convention about the product of categories of matrix factorizations from § 2.1.

Theorem 4.1. Let $Q$ be a symmetric quiver such that the number of loops at each vertex $i\in I$ has the same parity. Let $d\in \mathbb {N}^I$ , let $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ , and let $\mu \in \mathbb {R}$ be such that $\delta _d+\mu \sigma _d$ is a good weight. For a partition $(d_i)_{i=1}^k$ of $d$ , let $\lambda$ be an associated antidominant cocharacter and define the weights $\delta _{d_i}\in M(d_i)_{\mathbb {R}}^{W_{d_i}}$ and $\theta _i \in \frac {1}{2} M(d_i)^{W_{d_i}}$ for $1\leqslant i\leqslant k$ by

(4.1) \begin{equation} \sum _{i=1}^k \delta _{d_i}=\delta _d, \quad \sum _{i=1}^k \theta _i=-\frac {1}{2}R(d)^{\lambda \gt 0}+\frac {1}{2}\mathfrak {g}(d)^{\lambda \gt 0}. \end{equation}

There is a semiorthogonal decomposition

(4.2) \begin{equation} D^b(\mathscr {X}^f(d)^{\text {ss}})=\bigg \langle \bigotimes _{i=1}^k \mathbb {M}(d_i; \theta _i+\delta _{d_i}+v_i\tau _{d_i}): \mu \leqslant \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}\lt 1+\mu \bigg \rangle , \end{equation}

where the right-hand side in (4.2) is over all partitions $(d_i)_{i=1}^k$ of $d$ and real numbers $(v_i)_{i=1}^k\in \mathbb {R}^k$ such that the sum of coefficients of $\theta _{i}+\delta _{d_i}+v_i\tau _{d_i}$ is an integer for all $1\leqslant i\leqslant k$ . The order on the summands is as in § 4.6 .

The functor from a summand on the right-hand side to $D^b(\mathscr {X}^f(d)^{\text {ss}})$ is the composition of the Hall product with the pull-back along the forget-the-framing map $\mathscr {X}^f(d)^{\mathrm {ss}}\to \mathscr {X}(d)$ . Further, the decomposition (4.2) is $X(d)$ -linear for the map $\pi _{f,d}\colon \mathscr {X}^f(d)^{\mathrm {ss}}\to \mathscr {X}(d)\xrightarrow {\pi _{X,d}} X(d)$ .

The same argument also applies to obtain the following similar semiorthogonal decomposition using Theorem 4.15 for unframed moduli stacks. Note that this decomposition is different from the one discussed in [Reference PădurariuPăd21, Theorem 1.1], which we could not use to obtain a decomposition of $D^b(\mathscr {X}^f(d)^{\mathrm {ss}} )$ as in Theorem 4.1.

Theorem 4.2. Let $Q$ be a symmetric quiver such that the number of loops at each vertex $i\in I$ has the same parity. Let $d\in \mathbb {N}^I$ and let $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ . For a partition $(d_i)_{i=1}^k$ of $d$ , define the weights $\delta _{d_i}\in M(d_i)_{\mathbb {R}}^{W_{d_i}}$ and $\theta _i \in \frac {1}{2} M(d_i)^{W_{d_i}}$ for $1\leqslant i\leqslant k$ as in (4.1). There is a semiorthogonal decomposition

(4.3) \begin{equation} D^b(\mathscr {X}(d))=\bigg \langle \bigotimes _{i=1}^k \mathbb {M}(d_i; \theta _i+\delta _{d_i}+v_i\tau _{d_i}) : \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k} \bigg \rangle , \end{equation}

where the right-hand side in (4.3) is over all partitions $(d_i)_{i=1}^k$ of $d$ and real numbers $(v_i)_{i=1}^k\in \mathbb {R}^k$ such that the sum of coefficients of $\theta _{i}+\delta _{d_i}+v_i\tau _{d_i}$ is an integer for all $1\leqslant i\leqslant k$ . The order on the summands is as in § 4.6 .

The functor from a summand on the right-hand side to $D^b(\mathscr {X}^f(d)^{\text {ss}} )$ is given by the Hall product. The decomposition (4.3) is $X(d)$ -linear.

The plan of proof for both of Theorems 4.1 and 4.2 is as follows: we first prove them in a particular case, that of very symmetric quivers; then we use Knörrer periodicity to obtain the more general statements stated above.

Using [Reference Pădurariu and TodaPT22a, Proposition 2.3] and [Reference PădurariuPăd22, Proposition 2.1], one obtains versions for quivers with potentials of Theorems 4.1 and 4.2; see § 4.12. Note that both theorems apply to tripled quivers and thus to tripled quivers with potential.

4.2 Very symmetric quivers

We introduce a class of quivers for which we can prove Theorems 4.1 and 4.4 using the arguments employed for the quiver with one vertex and three loops in [Reference Pădurariu and TodaPT22a, Reference PădurariuPăd23].

The first step in proving Theorem 4.1 is the following.

Until § 4.8, we assume that the quiver $Q$ is very symmetric.

The proof of Theorem 4.5 follows closely the proof of [Reference Pădurariu and TodaPT22a, Theorem 3.2]. As in loc. cit., the claim follows from the following semiorthogonal decomposition for subcategories of $D^b(\mathscr {X}(d))$ using `window categories’. Recall the definition of the categories $\mathbb {D}(d; \delta )$ from (2.26).

Theorem 4.6. Let $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ and let $\mu \in \mathbb {R}$ be such that $\delta _d+\mu \sigma _d$ is a good weight. Then there is a semiorthogonal decomposition

(4.4) \begin{equation} \mathbb {D}(d; \delta _d+\mu \sigma _d) =\bigg \langle \bigotimes _{i=1}^k \mathbb {M}(d_i; \theta _i+\delta _{d_i}+v_i\tau _{d_i}) \bigg \rangle , \end{equation}

where the right-hand side is as in Theorem 4.1 .

Proof of Theorem 4.5 assuming Theorem 4.6. We briefly explain why Theorem 4.6 implies Theorem 4.5; for full details see [Reference Pădurariu and TodaPT22a, proof of Theorem 3.2]. Consider the morphisms

(4.5) \begin{equation} \mathscr {X}^f(d)^{\text {ss}} \stackrel {j}{\hookrightarrow } \mathscr {X}^f(d)\stackrel {\pi }{\twoheadrightarrow } \mathscr {X}(d), \end{equation}

where $j$ is an open immersion and $\pi$ is the natural projection. Let

\begin{align*} \mathbb {E}(d; \delta _d+\mu \sigma _d) \subset D^b(\mathscr {X}^f(d)) \end{align*}

be the subcategory generated by the complexes $\pi ^*(D)$ for $D\in \mathbb {D}(d; \delta _d+\mu \sigma _d)$ . If $\delta _d+\mu \sigma _d$ is a good weight, then there is an equivalence of categories

\begin{equation*}j^*\colon \mathbb {E}(d; \delta _d+\mu \sigma _d)\xrightarrow {\sim } D^b(\mathscr {X}^f(d)^{\text {ss}});\end{equation*}

see [Reference Pădurariu and TodaPT22a, proof of Proposition 3.16]. The equivalence follows from the theory of `window categories’ of Halpern-Leistner [Reference Halpern-LeistnerHL15] and of Ballard, Favero, and Katzarkov [Reference Ballard, Favero and KatzarkovBFK19], and from the description of window categories via explicit generators (due to Halpern-Leistner and Sam [Reference Halpern-Leistner and SamHLS20]) for the self-dual representation of $G(d)$ :

\begin{align*} R^f(d) \oplus V(d)^{\vee }=R(d)\oplus V(d)\oplus V(d)^{\vee }. \end{align*}

Furthermore, there is a semiorthogonal decomposition

\begin{align*} D^b(\mathscr {X}^f(d))= \langle \pi ^{\ast } D^b(\mathscr {X}(d))_w : w \in \mathbb {Z} \rangle \end{align*}

and equivalences $\pi ^{\ast } \colon D^b(\mathscr {X}(d))_w \stackrel {\sim }{\to } \pi ^{\ast }D^b(\mathscr {X}(d))_w$ for each $w\in \mathbb {Z}$ . Therefore, by Theorem 4.6, there is a semiorthogonal decomposition

\begin{align*} \mathbb {E}(d; \delta _d+\mu \sigma _d)= \bigg \langle \bigotimes _{i=1}^k \mathbb {M}(d_i; \theta _i+\delta _{d_i}+v_i\tau _{d_i})\bigg \rangle , \end{align*}

where the right-hand side is as in Theorem 4.5. Therefore the theorem holds.

4.3 Decompositions of weights

The proof of Theorem 4.6 follows closely the proof of [Reference Pădurariu and TodaPT22a, Proposition 3.12].

The proof of Theorem 4.6 uses the decomposition of categorical Hall algebras for quivers with potential in quasi-BPS categories from [Reference PădurariuPăd21]. The summands in this semiorthogonal decomposition are indexed by decompositions of weights of $T(d)$ , which we now briefly review.

Before stating the results, we introduce some notation. For a summand $d_a$ of a partition of $d$ , denote by $M(d_a)\subset M(d)$ the subspace as in the decomposition from § 2.2.3. Assume that $\ell$ is a partition of a dimension $d_a\in \mathbb {N}^I$ or, alternatively, that $\ell$ is an edge of the tree $\mathcal {T}$ introduced in § 2.2. Let $\lambda _{\ell }$ be the corresponding antidominant cocharacter of $T(d_a)$ . Recall the set

\begin{equation*}\mathscr {A}=\{(\beta ^a_i-\beta ^b_j)^{\times A}\mid a,b\in I,\: 1\leqslant i\leqslant i\leqslant d^a,\: 1\leqslant j\leqslant d^b\}\end{equation*}

from (2.20). Let

\begin{equation*}\mathscr {A}_{\ell }\subset M(d_a)\cap \mathscr {A}\end{equation*}

be the multiset of weights in $M(d_a)\cap \mathscr {A}$ such that $\langle \lambda _{\ell }, \beta \rangle \gt 0$ . Define $N_{\ell }$ by

\begin{align*} N_\ell :=\sum _{\beta \in \mathscr {A}_\ell }\beta . \end{align*}

Proposition 4.8. Let $\chi$ be a dominant weight in $M(d)_{\mathbb {R}}$ , let $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ , and let $w=\langle 1_{d}, \chi -\delta _d\rangle \in \mathbb {R}$ . Then the following exist:

1. (1) a path of partitions $T$ (see § 2.2) with decomposition $(d_i)_{i=1}^k$ at the end vertex;

2. (2) coefficients $r_\ell$ for $\ell \in T$ such that $r_{\ell }\gt 1/2$ if $\ell$ corresponds to a partition with length greater than $1$ and $r_{\ell }=0$ otherwise and such that, furthermore, if $\ell , \ell '\in T$ are vertices corresponding to partitions with length greater than 1 and with a path from $\ell$ to $\ell '$ , then $r_{\ell }\gt r_{\ell '}\gt {1}/{2}$ ;

3. (3) dominant weights $\psi _i\in \mathbf {W}(d_i)$ for $1\leqslant i\leqslant k$ such that

(4.6) \begin{equation} \chi +\rho -\delta _d=-\sum _{\ell \in T}r_\ell N_\ell +\sum _{i=1}^k\psi _i+w\tau _d. \end{equation}

Proof. The above is proved in [Reference PădurariuPăd22, § 3.1.2]; see also [Reference PădurariuPăd23, § 3.2.8].

We briefly explain the process of obtaining the decomposition (4.6). Choose $r$ such that $\chi +\rho -\delta _d-w\tau _d$ is on the boundary of $2r\mathbf {W}(d)$ (that is, let $r$ be the $r$ -invariant of $\chi +\rho -w\tau _d$ ). The first partition $\ell _1$ in $T$ corresponds to the face of $2r\mathbf {W}(d)$ which contains $\chi +\rho -\delta _d-w\tau _d$ in its interior. Assume that $\ell _1$ corresponds to a partition $(e_i)_{i=1}^s$ . Then there exist weights $\chi '_i\in M(e_i)_0$ for $1\leqslant i\leqslant s$ such that

\begin{equation*}\chi +\rho -\delta _d-w\tau _d+r_{\ell _1}N_{\ell _1}=\sum _{i=1}^s \chi '_i.\end{equation*}

By the choice of $r$ and $\ell _1$ , the weights $\chi '_i$ are inside the polytopes $2r\mathbf {W}(e_i)$ . One repeats the process above to further decompose the weights $\chi '_i$ until the decomposition (4.6) is obtained.

Let $w\in \mathbb {Z}$ and assume that $\langle 1_d, \delta _d\rangle =w$ . Let $M(d)^+_w$ be the subset of $M(d)$ of integral dominant weights $\chi$ with $\langle 1_d, \chi \rangle =w$ . We denote by $L^d_{\delta _d}$ the set of all paths of partitions $T$ with coefficients $r_\ell$ for $\ell \in T$ satisfying (2) from the statement of Proposition 4.9 for a dominant integral weight $\chi \in M(d)^+_w$ . By Proposition 4.8, there is a map

\begin{equation*}\Upsilon \colon M(d)^+_w\to L^d_{\delta _d}, \,\, \Upsilon (\chi )=(T, r_\ell ).\end{equation*}

The set $L^d_{\delta _d}$ was used in [Reference PădurariuPăd21] to index summands in semiorthogonal decompositions of $D^b(\mathscr {X}(d))_w$ . We will show in the next subsection that $L^d_{\delta _d}$ has an explicit description.

4.4 Partitions associated to dominant weights

We continue with the notation from § 4.3. Using the following proposition, the weight $-\sum _{\ell \in T}r_\ell N_\ell$ from (4.6) is a linear combinations of $\tau _{d_i}$ for $1\leqslant i\leqslant k$ .

Proof. This follows from a direct computation; for example when $k=2$ , one computes directly that

\begin{equation*}R(d)^{\lambda \gt 0}=A \sum _{a, b \in I}\sum _{d_1^a\lt i \leqslant d^a, 1 \leqslant j\leqslant d_1^b}(\beta ^b_j-\beta ^a_i)=A(\underline {d}_2\sigma _{d_1}-\underline {d}_1\sigma _{d_2}).\end{equation*}

Consider the decomposition (4.6) and let $\lambda$ be the antidominant cocharacter corresponding to $(d_i)_{i=1}^k$ . We define $v_i \in \mathbb {R}$ for $1\leqslant i \leqslant k$ by

(4.7) \begin{align} \sum _{i=1}^k v_i \tau _{d_i}=-\sum _{\ell \in T}\Big(r_{\ell }-\frac {1}{2}\Big)N_{\ell }+w\tau _d=-\sum _{\ell \in T} r_{\ell } N_{\ell } +\frac {1}{2}R(d)^{\lambda \gt 0}+w\tau _d. \end{align}

Here the right-hand side is a linear combination of $\tau _{d_i}$ by Proposition 4.9, so $v_i \in \mathbb {R}$ is well-defined. We define the weights $\theta _i \in M(d_i)^{W_{d_i}}_{\mathbb {R}}$ by

(4.8) \begin{align} \sum _{i=1}^k \theta _i=-\frac {1}{2}R(d)^{\lambda \gt 0}+\frac {1}{2}\mathfrak {g}(d)^{\lambda \gt 0}. \end{align}

Then we rewrite (4.4) as

(4.9) \begin{align} \chi =\sum _{i=1}^k \theta _i+\sum _{i=1}^k v_i\tau _{d_i}+\sum _{i=1}^k(\psi _i-\rho _i+\delta _{d_i}), \end{align}

where $\rho _i$ is half the sum of the positive roots of $\mathfrak {g}(d_i)$ . The next proposition follows as in [Reference Pădurariu and TodaPT22a, Proposition 3.5].

Proposition 4.11. Let $\chi$ be a dominant weight in $M(d)$ and consider the weights $(v_i)_{i=1}^k$ from (4.7). Then

(4.10) \begin{equation} \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}. \end{equation}

Let $T^d_w$ be the set of tuples $A=(d_i, v_i)_{i=1}^k$ of $(d,w)$ such that $(d_i)_{i=1}^k$ is a partition of $d$ and the real numbers $(v_i)_{i=1}^k\in \mathbb {R}^k$ are as follows:

1. – $\sum _{i=1}^k v_i=w$ ;

2. – the inequality (4.10) holds;

3. – for each $1\leqslant i\leqslant k$ , the sum of coefficients of $\theta _i+v_i\tau _{d_i}+\delta _{d_i}$ is an integer.