# Categorical and K-theoretic Donaldson-Thomas theory of $\mathbb{C}^{3}$
(part II)

Tudor Pădurariu and Yukinobu Toda

###### Abstract.

Quasi-BPS categories appear as summands in semiorthogonal decompositions of DT
categories for Hilbert schemes of points in the three dimensional affine space
and in the categorical Hall algebra of the two dimensional affine space. In
this paper, we prove several properties of quasi-BPS categories analogous to
BPS sheaves in cohomological DT theory.

We first prove a categorical analogue of Davison’s support lemma, namely that
complexes in the quasi-BPS categories for coprime length and weight are
supported over the small diagonal in the symmetric product of the three
dimensional affine space. The categorical support lemma will be used to
determine the torsion-free generator of the torus equivariant K-theory of the
quasi-BPS category of coprime length and weight.

We next construct a bialgebra structure on the torsion free equivariant
K-theory of quasi-BPS categories for a fixed ratio of length and weight. We
define the K-theoretic BPS space as the space of primitive elements with
respect to the coproduct. We show that all localized equivariant K-theoretic
BPS spaces are one dimensional, which is a K-theoretic analogue of the
computation of (numerical) BPS invariants of the three dimensional affine
space.

##  1\. Introduction

###  1.1. Quasi-BPS categories

In [PTa, Pădc, Pădb], we studied quasi-BPS categories $\mathbb{S}(d)_{w}$ for
$d\in\mathbb{N}$ (length) and $w\in\mathbb{Z}$ (weight) in relation to
categorical Donaldson-Thomas (DT) theory and to categorical Hall algebras (of
surfaces and of quivers with potentials). They are defined to be full
subcategories of the category of matrix factorizations

(1.1)
$\displaystyle\mathbb{S}(d)_{w}:=\mathrm{MF}(\mathbb{M}(d)_{w},\mathop{\mathrm{Tr}}\nolimits
W_{d})\subset\mathrm{MF}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits W_{d}),$

where $\mathbb{M}(d)_{w}$ is a twisted noncommutative resolution first
considered by Špenko–Van den Bergh [ŠdB17], see Subsection 2.6 for more
details. Here, the stack $\mathcal{X}(d)$ and the regular function
$\mathop{\mathrm{Tr}}\nolimits W_{d}$ are given by

(1.2) $\displaystyle\mathcal{X}(d):=\operatorname{Hom}(V,V)^{\oplus 3}/GL(V),\
\mathop{\mathrm{Tr}}\nolimits W_{d}(A,B,C)=\mathop{\mathrm{Tr}}\nolimits
A[B,C],$

where $V$ is a $d$-dimensional vector space.

There is also a graded quasi-BPS category $\mathbb{S}^{\rm{gr}}(d)_{v}$, which
is equivalent (via Koszul duality) to a subcategory

$\displaystyle\mathbb{T}(d)_{v}\subset D^{b}(\mathscr{C}(d)),$

where $\mathscr{C}(d)$ is the derived moduli stack of zero-dimensional sheaves
on $\mathbb{C}^{2}$ with length $d$. We will also consider $T$-equivariant
versions of these categories, where $T=(\mathbb{C}^{*})^{2}$ is the Calabi-Yau
torus of $\mathbb{C}^{3}$. We denote by
$\mathbb{K}=K(BT)=\mathbb{Z}[q_{1}^{\pm 1},q_{2}^{\pm 1}]$ and by $\mathbb{F}$
the fractional field of $\mathbb{K}$. The ($T$-equivariant or not)
Grothendieck groups of $\mathbb{S}(d)_{w}$ and
$\mathbb{S}^{\mathrm{gr}}(d)_{w}\cong\mathbb{T}(d)_{w}$ are isomorphic.

The purpose of this paper is to prove several properties of quasi-BPS
categories analogous to BPS sheaves in cohomological DT theory, and use these
properties to compute the $T$-equivariant K-theory of quasi-BPS categories.

###  1.2. Semiorthogonal decompositions

We briefly review semiorthogonal decompositions with summands given by quasi-
BPS categories proved in [PTa, Pădc].

In our previous paper [PTa], we constructed a semiorthogonal decomposition of
the categorification $\mathcal{DT}(d)$ of the Donaldson-Thomas invariant
$\mathrm{DT}_{d}$, which is a virtual count of zero-dimensional closed
subschemes in $\mathbb{C}^{3}$. The category $\mathcal{DT}(d)$ is defined by

$\mathcal{DT}\left(d\right):=\text{MF}\left(\text{NHilb}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d}\right),$

see [PTa, Subsection 1.5]. Here $\text{NHilb}(d)$ is the noncommutative
Hilbert scheme of points

$\text{NHilb}(d):=\left(V\oplus\text{Hom}(V,V)^{\oplus
3}\right)^{\text{ss}}/GL(V).$

More precisely, in [PTa, Theorem 1.1] we showed that there is a semiorthogonal
decomposition

(1.3)
$\mathcal{DT}(d)=\left\langle\boxtimes_{i=1}^{k}\mathbb{S}(d_{i})_{v_{i}+d_{i}\left(\sum_{i>j}d_{j}-\sum_{j>i}d_{j}\right)}\mathrel{}\middle|\mathrel{}\begin{array}[]{c}0\leqslant
v_{1}/d_{1}<\cdots<v_{k}/d_{k}<1\\\
d_{1}+\cdots+d_{k}=d\end{array}\right\rangle.$

There is also a semiorthogonal decomposition of $D^{b}(\mathscr{C}(d))$ in
graded quasi-BPS categories, see [Pădc, Theorem 1.1]:

(1.4)
$D^{b}\left(\mathscr{C}(d)\right)=\left\langle\boxtimes_{i=1}^{k}\mathbb{T}(d_{i})_{v_{i}}\mathrel{}\middle|\mathrel{}\begin{array}[]{c}v_{1}/d_{1}<\cdots<v_{k}/d_{k}\\\
d_{1}+\cdots+d_{k}=d\end{array}\right\rangle.$

###  1.3. Numerical and cohomological BPS invariants

It is expected that, for any smooth Calabi-Yau threefold $X$, there are
certain deformation invariant integers called BPS invariants which determine
the Donaldson-Thomas and Gromov-Witten invariants of $X$, see [PT14, Section 2
and a half]. Denote by $\Omega_{d}$ the BPS invariants of $\mathbb{C}^{3}$ for
$d\geqslant 1$. Then

(1.5) $\Omega_{d}=-1\text{ for all }d\geqslant 1.$

The wall-crossing formula for the DT invariants of $\mathbb{C}^{3}$, see
[JS12, Section 6.3], [Tod10, Remark 5.14], says that

(1.6) $\displaystyle\sum_{d\geqslant 0}\mathrm{DT}_{d}q^{d}=\prod_{d\geqslant
1}(1-(-q)^{d})^{d\Omega_{d}}=\prod_{d\geqslant
1}\frac{1}{\left(1-(-q)^{d}\right)^{d}},$

see [PTa, Subsection 1.6] for more details.

Davison–Meinhardt [DM20] constructed a perverse sheaf $\mathcal{B}PS_{d}$ on
$\mathrm{Sym}^{d}(\mathbb{C}^{3})$, called BPS sheaf, whose Euler
characteristic recovers the BPS invariant $\Omega_{d}$. Davison [Davb, Theorem
5.1] showed that

(1.7) $\mathcal{B}PS_{d}=\Delta_{*}\mathrm{IC}_{\mathbb{C}^{3}},$

where
$\Delta\colon\mathbb{C}^{3}\hookrightarrow\mathrm{Sym}^{d}(\mathbb{C}^{3})$ is
the small diagonal. Davison [Davb, Lemma 4.1] also proved restrictions on the
support of BPS sheaves for tripled quivers with potentials and used [Davb,
Lemma 4.1] to prove purity results about stacks of representations of
preprojective algebras [Davb, Theorem A]. We refer to (1.7) as Davison’s
support lemma. The (cohomological) BPS spaces are the cohomology of the BPS
sheaf.

Properties (and computations in special cases) of BPS sheaves have
applications in the study of Hodge theory of various cohomological DT spaces
(in particular, in proving purity of the Borel-Moore homology of moduli of
objects in K$3$ categories [Davc]) and in the study of Cohomological Hall
algebras [Dava, KV].

Our point of view is that the semiorthogonal decomposition (1.3) may be
regarded as a categorification of the wall-crossing formula (1.6) and the
category $\mathbb{S}(d)_{w}$ may be regarded as a categorical analogue of the
BPS invariant (1.5) or BPS sheaf (1.7). In this paper, we make the above
heuristic more rigorous. Let $(d,w)\in\mathbb{N}\times\mathbb{Z}$ be integers
with $\gcd(d,w)=1$. We first prove a categorical analogue of (1.7), namely any
object of $\mathbb{S}(d)_{w}$ is supported over the small diagonal in
$\mathrm{Sym}^{d}(\mathbb{C}^{3})$. Further, for $n\in\mathbb{N}$, we define a
K-theoretic analogue of the cohomological BPS space:

$\mathrm{P}(nd)_{nw}\subset K_{T}\left(\mathbb{S}(nd)_{nw}\right)\cong
K_{T}\left(\mathbb{T}(nd)_{nw}\right)$

and show that
$\mathrm{P}(nd)_{nw,\mathbb{F}}:=\mathrm{P}(nd)_{nw}\otimes_{\mathbb{K}}\mathbb{F}$
is a one dimensional $\mathbb{F}$-vector space, compare with (1.5).

###  1.4. Support of matrix factorizations in quasi-BPS categories

We prove a version of the support lemma (1.7) for categories
$\mathbb{S}(d)_{w}$ with $\gcd(d,w)=1$. We consider the quotient stack
$\mathcal{X}(d)$ defined in (1.2) together with its good moduli space

$\displaystyle\pi\colon\mathcal{X}(d)\to X(d):=\operatorname{Hom}(V,V)^{\oplus
3}\sslash GL(V).$

Consider the diagram

where $\mathcal{C}oh(\mathbb{C}^{3},d)$ is the stack of sheaves with zero
dimensional support and length $d$ on $\mathbb{C}^{3}$. In Section 3, we prove
the following (see Theorem 3.1 for a more precise statement):

######  Theorem 1.1.

_(Theorem 3.1)_ For a pair $(d,w)\in\mathbb{N}\times\mathbb{Z}$ with
$\gcd(d,w)=1$, any object in $\mathbb{S}(d)_{w}$ is supported on
$\pi^{-1}(\Delta)$.

The categorical support restriction in Theorem 1.1 implies a strong constraint
on $T$-equivariant K-theory classes of objects of quasi-BPS categories. Let
$\Theta$ be the forget-the-potential map

(1.8) $\Theta\colon
K_{T}(\mathrm{MF}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits W_{d}))\to
K_{T}(\mathrm{MF}(\mathcal{X}(d),0))=\mathbb{K}[z_{1}^{\pm
1},\ldots,z_{d}^{\pm 1}]^{\mathfrak{S}_{d}},$

see [Păda, Proposition 3.6]. In Lemma 4.1, we show that if a complex
$\mathcal{F}$ is supported on $\pi^{-1}(\Delta)$, then $\Theta([\mathcal{F}])$
is divisible by the element

(1.9)
$\displaystyle(q_{1}-1)^{d-1}(q_{2}-1)^{d-1}(q_{1}q_{2}-1)^{d-1}\in\mathbb{K}.$

In [PTa], we introduced certain complexes
$\mathcal{E}_{d,w}\in\mathbb{T}(d)_{w}$ using a derived stack of pairs of
commuting matrices, both of which have spectrum of cardinality one, and have
an explicit shuffle description (2.40). In particular, Lemma 4.1 applies to
$\mathcal{E}_{d,w}$ for $\gcd(d,w)=1$. We remark that the divisibility by
(1.9) is not obvious from the shuffle description of
$\Theta([\mathcal{E}_{d,w}])$.

###  1.5. Integral generator of K-theory of quasi-BPS categories

In [PTa, Theorem 1.2], we showed that the localized (i.e. taking
$\otimes_{\mathbb{K}}\mathbb{F}$) $T$-equivariant K-theory of
$\mathbb{T}(d)_{w}$ (and thus of $\mathbb{S}(d)_{w}$) is generated by
monomials in $[\mathcal{E}_{d^{\prime},w^{\prime}}]$ for $d^{\prime}\leqslant
d$ and $\frac{w}{d}=\frac{w^{\prime}}{d^{\prime}}$. The divisibility by (1.9)
will be useful to compute the $T$-equivariant K-theory of quasi-BPS categories
without localization. For a $\mathbb{K}$-module $M$, we will use the notation
$M^{\prime}:=M/(\mathbb{K}\text{-torsion})$. Using Lemma 4.1 and Theorem 1.1,
we show that:

######  Theorem 1.2.

_(Theorem 4.4)_ Let $(d,w)\in\mathbb{N}\times\mathbb{Z}$ with $\gcd(d,w)=1$.
The $\mathbb{K}$-module $K_{T}\left(\mathbb{S}(d)_{w}\right)^{\prime}$ is free
of rank one with generator $\left[\mathcal{E}_{d,w}\right]$.

For $(d,w)\in\mathbb{N}\times\mathbb{Z}$ with $\gcd(d,w)=1$, we believe the
category $\mathbb{T}(d)_{w}$ is generated by $\mathcal{E}_{d,w}$, which in
particular implies Theorem 1.2. Further, for $\ast\in\\{\emptyset,T\\}$, we
suspect there are equivalences
$\mathbb{T}_{\ast}(1)_{0}\xrightarrow{\sim}\mathbb{T}_{\ast}(d)_{w}$, but we
do not have much evidence supporting this belief.

###  1.6. The coproduct

For the reminder of the introduction, we consider a pair
$(d,v)\in\mathbb{N}\times\mathbb{Z}$ with $\gcd(d,v)=1$. The Grothendieck
group of the category $\mathbb{T}(nd)_{nv}$ for $n>1$ contains a contribution
from partitions $a+b=n$ because the Hall product restricts to a functor

$m_{a,b}\colon\mathbb{T}(ad)_{av}\otimes\mathbb{T}(bd)_{bd}\to\mathbb{T}(nd)_{nv}$

for $a,b\geqslant 1$ with $a+b=n$, see [PTa, Lemma 4.8]. The category:

$\mathcal{D}_{d,v}:=\bigoplus_{n\geqslant 0}\mathbb{T}(nd)_{nv}$

is thus monoidal. For $n=a+b$, there is a coproduct

(1.10)
$\Delta_{a,b}\colon\mathbb{T}(nd)_{nv}\to\mathbb{T}(ad)_{av}\otimes\mathbb{T}(bd)_{bv},$

see also [Pădc, Section 5]. The construction also provides a $T$-equivariant
version. In Section 5, we prove that the Hall product is compatible with the
above coproduct:

######  Theorem 1.3.

_(Corollary 5.6)_ Let $(d,v)\in\mathbb{N}\times\mathbb{Z}$ be coprime, let
$a,b,c,e,n\in\mathbb{N}$ such that $a+b=c+e=n$, and let $S$ be the set of
tuples $(f_{1},f_{2},f_{3},f_{4})$ such that $a=f_{1}+f_{2}$, $b=f_{3}+f_{4}$,
$c=f_{1}+f_{3}$, $e=f_{2}+f_{4}$. The following diagram commutes:

${K_{T}(\mathbb{T}(ad)_{av})^{\prime}\otimes
K_{T}(\mathbb{T}(bd)_{bv})^{\prime}}$${K_{T}(\mathbb{T}(nd)_{nv})^{\prime}}$${\bigoplus_{S}\bigotimes_{i=1}^{4}K_{T}(\mathbb{T}(f_{i}d)_{f_{i}v})^{\prime}}$${K_{T}(\mathbb{T}(cd)_{cv})\otimes
K_{T}(\mathbb{T}(ed)_{ev})^{\prime},}$$\scriptstyle{m_{a,b}}$$\scriptstyle{\Delta}$$\scriptstyle{\Delta_{c,e}}$$\scriptstyle{m^{\prime}}$

where we have used the shorthand notations
$\Delta:=\bigoplus_{S}\Delta_{f_{1},f_{2}}\otimes\Delta_{f_{3},f_{4}}$ and
$m^{\prime}:=\bigoplus_{S}(m\otimes m)(1\otimes\text{sw}_{f_{2},f_{3}}\otimes
1)$.

In the proof, we first construct a version of $\Delta_{a,b}$ for the
subcategories $\mathbb{M}(nd)_{nv}\subset D^{b}(\mathcal{X}(nd))_{nv}$, see
(1.1). We then construct the functor (1.10) by applying matrix factorizations
and using the Koszul equivalence.

In Propositions 5.7 and 5.8, we show that the product and the coproduct on
$K_{T}(\mathcal{D}_{d,v})^{\prime}$ are commutative and cocommutative. We then
obtain an isomorphism

(1.11) $K_{T}(\mathcal{D}_{d,v})_{\mathbb{F}}\cong\Lambda_{\mathbb{F}},$

where $\Lambda_{\mathbb{F}}$ is the $\mathbb{F}$-algebra of symmetric
functions, see Subsection 2.12 for its definition. The elementary functions
$e_{n}\in\Lambda_{\mathbb{F}}$ are sent, up to a factor in $\mathbb{F}$, to
$\mathcal{E}_{nd,nv}$.

###  1.7. K-theoretic BPS spaces

We define the K-theoretic BPS space to be the space of primitive elements of
$K_{T}(\mathbb{T}(nd)_{nv})$ with respect to the above coproduct:

(1.12)
$\mathrm{P}(nd)_{nv}:=\mathrm{ker}\left(\bigoplus_{\begin{subarray}{c}a+b=n\\\
a,b\geqslant
1\end{subarray}}\Delta_{a,b}:K_{T}\left(\mathbb{T}(nd)_{nv}\right)\to\bigoplus_{\begin{subarray}{c}a+b=n\\\
a,b\geqslant
1\end{subarray}}K_{T}\left(\mathbb{T}(ad)_{av}\otimes\mathbb{T}(bd)_{bv}\right)\right).$

We show that the dimension over $\mathbb{F}$ of localized K-theoretic BPS
spaces for all pairs $(nd,nv)\in\mathbb{N}\times\mathbb{Z}$ is the same as the
(numerical) BPS invariants (1.5) up to a sign:

######  Proposition 1.4.

_(Corollary 5.11)_ The $\mathbb{F}$-vector space
$\mathrm{P}(nd)_{nv,\mathbb{F}}$ is one dimensional.

Using the above proposition and [PTa, Theorem 1.1], we obtain a K-theoretic
analogue of the wall-crossing formula (1.6), see Subsection 5.5 and [PTa,
Subsection 1.6] for more details.

######  Corollary 1.5.

_(Corollary 5.13)_ There is an isomorphism of $\mathbb{N}$-graded
$\mathbb{F}$-vector spaces:

(1.13) $\bigoplus_{d\geqslant
0}K_{T}(\mathcal{DT}(d))_{\mathbb{F}}\cong\bigotimes_{\begin{subarray}{c}0\leqslant
v<d\\\ \gcd(d,v)=1\end{subarray}}\left(\bigotimes_{n\geqslant
1}\mathrm{Sym}\big{(}\mathrm{P}(nd)_{nv,\mathbb{F}}\big{)}\right),$

We conjecture an integral version of Proposition 1.4, which is a version of
Theorem 1.2 for pairs $(nd,nv)$ for all $n\in\mathbb{N}$:

######  Conjecture 1.6.

The $\mathbb{K}$-module $\mathrm{P}(nd)_{nv}$ is free of rank one.

The torsion-free version of the above conjecture for $(d,v)=(1,0)$ and $n=2$
follows from the discussion in Subsection 4.3. We finally conjecture an
analogue of Theorem 1.1 for all $n\in\mathbb{N}$.

######  Conjecture 1.7.

The subspace $\mathrm{P}(nd)_{nv}\subset K_{T}(\mathbb{T}(nd)_{nv})\cong
K_{T}(\mathbb{S}(nd)_{nv})$ is supported over $\pi^{-1}(\Delta)$,
alternatively, the following composition is zero:

$\mathrm{P}(nd)_{nv}\to K_{T}(\mathbb{S}(nd)_{nv})\to
K_{T}(\mathcal{X}(nd)\setminus\pi^{-1}(\Delta)).$

###  1.8. Acknowledgements

We thank Tasuki Kinjo, Davesh Maulik, Andrei Neguţ, Raphaël Rouquier,
Christopher Ryba, Olivier Schiffmann, Špela Špenko, Eric Vasserot, and Yu Zhao
for discussions related to this work. Y. T. is supported by World Premier
International Research Center Initiative (WPI initiative), MEXT, Japan, and
Grant-in Aid for Scientific Research grant (No. 19H01779) from MEXT, Japan.

##  2\. Preliminaries

###  2.1. Notations

The spaces considered in this paper are defined over the complex field
$\mathbb{C}$ and they are quotient stacks $\mathcal{X}=A/G$, where $A$ is a dg
scheme, the derived zero locus of a section $s$ of a finite rank bundle vector
bundle $\mathcal{E}$ on a finite type separated scheme $X$ over $\mathbb{C}$,
and $G$ is a reductive group. For such a dg scheme $A$, let $\dim A:=\dim
X-\text{rank}(\mathcal{E})$, let $A^{\mathrm{cl}}:=Z(s)\subset X$ be the
(classical) zero locus, and let
$\mathcal{X}^{\mathrm{cl}}:=A^{\mathrm{cl}}/G$. We denote by
$\mathcal{O}_{\mathcal{X}}$ or $\mathcal{O}_{A}$ the structure sheaf of
$\mathcal{X}$.

For $G$ a reductive group and $A$ a dg scheme as above, denote by $A/G$ the
corresponding quotient stack and by $A/\\!\\!/G$ the quotient dg scheme with
dg-ring of regular functions $\mathcal{O}_{A}^{G}$.

For a classical stack $\mathcal{X}$ with a morphism $\mathcal{X}\to X$ to a
scheme $X$ and for a closed point $p\in X$, we denote by $\mathcal{X}_{p}$ its
formal fiber
$\mathcal{X}_{p}:=\mathcal{X}\times_{X}\operatorname{Spec}\widehat{\mathcal{O}}_{X,p}$.

For a (derived) stack $\mathcal{X}$ with an action of a torus $T$, we denote
by $D^{b}_{T}(\mathcal{X})$ the bounded derived category of $T$-equivariant
coherent sheaves on $\mathcal{X}$ and by $G_{T}(\mathcal{X})$ its Grothendieck
group. We denote by $\mathrm{Perf}_{T}(\mathcal{X})$ the subcategory of
$T$-equivariant perfect complexes and by $K_{T}(\mathcal{X})$ its Grothendieck
group. When $T$ is trivial, we drop $T$ from the notation of the above
Grothendieck groups. We introduce more notations for categories and K-theory
in Subsection 2.6.2.

Consider the two dimensional torus

(2.1)
$\displaystyle(\mathbb{C}^{\ast})^{2}\stackrel{{\scriptstyle\cong}}{{\to}}T:=\\{(t_{1},t_{2},t_{3})\in(\mathbb{C}^{\ast})^{\times
3}\mid t_{1}t_{2}t_{3}=1\\}\subset(\mathbb{C}^{\ast})^{3}.$

The isomorphism above is given by
$(t_{1},t_{2})\mapsto(t_{1},t_{2},t_{1}^{-1}t_{2}^{-1})$. We denote by
$\mathbb{K}:=K_{T}(\mathrm{pt})=\mathbb{Z}[q_{1}^{\pm 1},q_{2}^{\pm 1}]$ and
let $\mathbb{F}$ be the fraction field of $\mathbb{K}$. For $V$ a
$\mathbb{K}$-module, we use the notations
$V^{\prime}:=V/(\mathbb{K}\text{-torsion})$ and
$V_{\mathbb{F}}:=V\otimes_{\mathbb{K}}\mathbb{F}$.

For a dg-category $\mathcal{D}$, a full dg-subcategory
$\mathcal{C}\subset\mathcal{D}$ is called dense if any object in $\mathcal{D}$
is a direct summand of an object in $\mathcal{C}$.

###  2.2. Weights and partitions

#### 2.2.1.

For $d\in\mathbb{N}$, let $V$ be a $\mathbb{C}$-vector space of dimension $d$,
and let $\mathfrak{g}=\mathfrak{gl}(V):=\text{End}(V)$. When the dimension is
clear from the context, we drop $d$ from its notation. Let

$\pi\colon\mathcal{X}(d):=R(d)/G(d):=\mathfrak{gl}(V)^{\oplus 3}/GL(V)\to
X(d):=\mathfrak{gl}(V)^{\oplus 3}/\\!\\!/GL(V).$

Alternatively, $\mathcal{X}(d)$ is the stack of representations of dimension
$d$ of the quiver $Q$ with one vertex and three loops $\\{x,y,z\\}$:

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Consider the super-potential $W=x[y,z]$ of $Q$ and the regular function

(2.2) $\mathop{\mathrm{Tr}}\nolimits W=\mathop{\mathrm{Tr}}\nolimits
W_{d}:=\mathop{\mathrm{Tr}}\nolimits A[B,C]\colon\mathcal{X}(d)\to\mathbb{C},$

where $(A,B,C)\in\mathfrak{gl}(V)^{\oplus 3}$.

Fix the maximal torus $T(d)\subset GL(d)$ to be consisting of diagonal
matrices. Denote by $M=\oplus_{i=1}^{d}\mathbb{Z}\beta_{i}$ the weight space
of $T(d)$ and let $M(d)_{\mathbb{R}}:=M(d)\otimes_{\mathbb{Z}}\mathbb{R}$,
where $\beta_{1},\ldots,\beta_{d}$ is the set of simple roots. A weight
$\chi=\sum_{i=1}^{d}c_{i}\beta_{i}$ is dominant if

$\displaystyle c_{1}\leqslant\ldots\leqslant c_{d},\ $

We denote by $M^{+}\subset M$ and $M^{+}_{\mathbb{R}}\subset M_{\mathbb{R}}$
the dominant chambers. When we want to emphasize the dimension vector, we
write $M(d)$ etc. Denote by $N$ the coweight lattice of $T(d)$ and by
$N_{\mathbb{R}}:=N\otimes_{\mathbb{Z}}\mathbb{R}$. Let $\langle\,,\,\rangle$
be the natural pairing between $N_{\mathbb{R}}$ and $M_{\mathbb{R}}$.

Let $W=\mathfrak{S}_{d}$ be the Weyl group of $GL(d)$. For $\chi\in M(d)^{+}$,
let $\Gamma_{GL(d)}(\chi)$ be the irreducible representation of $GL(d)$ of
highest weight $\chi$. We drop $GL(d)$ from the notation if the dimension
vector $d$ is clear from the context. Let $w*\chi:=w(\chi+\rho)-\rho$ be the
Weyl-shifted action of $w\in W$ on $\chi\in M(d)_{\mathbb{R}}$. We denote by
$\ell(w)$ the length of $w\in W$.

Denote by $\mathcal{W}$ the multiset of $T(d)$-weights of $R(d)$. If there is
a natural action of a torus $T$ on $R(d)$, we abuse notation and write
$\mathcal{W}$ for the multiset of $(T\times T(d))$-weights of $R(d)$. For
$\lambda$ a cocharacter of $T(d)$, we denote by $N^{\lambda>0}$ the sum of
weights $\beta$ in $\mathcal{W}$ such that $\langle\lambda,\beta\rangle>0$.

#### 2.2.2.

We denote by $\rho$ half the sum of positive roots of $GL(d)$. In our
convention of the dominant chamber, it is given by

$\displaystyle\rho=\frac{1}{2}\sum_{j<i}(\beta_{i}-\beta_{j}).$

We denote by $1_{d}:=z\cdot\text{Id}$ the diagonal cocharacter of $T(d)$.
Define the weights

$\displaystyle\sigma_{d}:=\sum_{j=1}^{d}\beta_{j}\in M,\
\tau_{d}:=\frac{\sigma_{d}}{d}\in M_{\mathbb{R}}.$

#### 2.2.3.

Let $G$ be a reductive group (in this paper, $G$ will be a Levi subgroup of
$GL(d)$ for some positive integer $d$), let $T_{G}$ be a maximal torus of $G$,
let $X$ be a $G$-representation, and let

$\mathcal{X}=X/G$

be the corresponding quotient stack. Let $\mathcal{W}$ be the multiset of
$T_{G}$-weights of $X$. For $\lambda$ a cocharacter of $T_{G}$, let
$X^{\lambda}\subset X$ be the subspace generated by weights
$\beta\in\mathcal{W}$ such that $\langle\lambda,\beta\rangle=0$, let
$X^{\lambda\geqslant 0}\subset X$ be the subspace generated by weights
$\beta\in\mathcal{W}$ such that $\langle\lambda,\beta\rangle\geqslant 0$, and
let $G^{\lambda}$ and $G^{\lambda\geqslant 0}$ be the Levi and parabolic
groups associated to $\lambda$. Consider the fixed and attracting stacks

$\displaystyle\mathcal{X}^{\lambda}:=X^{\lambda}/G^{\lambda},\
\mathcal{X}^{\lambda\geqslant 0}:=X^{\lambda\geqslant 0}/G^{\lambda\geqslant
0}$

with maps

$\mathcal{X}^{\lambda}\xleftarrow{q_{\lambda}}\mathcal{X}^{\lambda\geqslant
0}\xrightarrow{p_{\lambda}}\mathcal{X}.$

Define the integer

(2.3) $n_{\lambda}:=\langle\lambda,[X^{\lambda\geqslant
0}]-[\mathfrak{g}^{\lambda\geqslant 0}]\rangle.$

#### 2.2.4.

Let $d\in\mathbb{N}$ and recall the definition of $\mathcal{X}(d)$ from
Subsection 2.2.1. For a cocharacter $\lambda:\mathbb{C}^{*}\to T(d)$, consider
the maps of fixed and attracting loci

(2.4)
$\mathcal{X}(d)^{\lambda}\xleftarrow{q_{\lambda}}\mathcal{X}(d)^{\lambda\geqslant
0}\xrightarrow{p_{\lambda}}\mathcal{X}(d).$

We say that two cocharacters $\lambda$ and $\lambda^{\prime}$ are equivalent
and write $\lambda\sim\lambda^{\prime}$ if $\lambda$ and $\lambda^{\prime}$
have the same fixed and attracting stacks as above.

We call $\underline{d}:=(d_{i})_{i=1}^{k}$ a partition of $d$ if
$d_{i}\in\mathbb{N}$ are all non-zero and $\sum_{i=1}^{k}d_{i}=d$. In Section
5, we allow partitions $\underline{d}=(d_{i})_{i=1}^{k}$ to have terms $d_{i}$
equal to zero. We similarly define partitions of
$(d,w)\in\mathbb{N}\times\mathbb{Z}$. For a cocharacter $\lambda$ of $T(d)$,
there is an associated partition $(d_{i})_{i=1}^{k}$ such that

$\mathcal{X}(d)^{\lambda\geqslant
0}\xrightarrow{q_{\lambda}}\mathcal{X}(d)^{\lambda}\cong\times_{i=1}^{k}\mathcal{X}(d_{i}).$

Define the length $\ell(\lambda):=k$.

Equivalence classes of antidominant cocharacters are in bijection with ordered
partitions $(d_{i})_{i=1}^{k}$ of $d$. For an ordered partition
$\underline{d}=(d_{i})_{i=1}^{k}$ of $d$, fix a corresponding antidominant
cocharacter $\lambda=\lambda_{\underline{d}}$ of $T(d)$ which induces the maps

$\mathcal{X}(d)^{\lambda}\cong\times_{i=1}^{k}\mathcal{X}(d_{i})\xleftarrow{q_{\lambda}}\mathcal{X}(d)^{\lambda\geqslant
0}\xrightarrow{p_{\lambda}}\mathcal{X}(d).$

We also use the notations $p_{\lambda}=p_{\underline{d}}$,
$q_{\lambda}=q_{\underline{d}}$. The categorical Hall algebra is given by the
functor $p_{\lambda*}q_{\lambda}^{*}=p_{\underline{d}*}q_{\underline{d}}^{*}$:

(2.5) $\displaystyle m=m_{\underline{d}}\colon
D^{b}(\mathcal{X}(d_{1}))\boxtimes\cdots\boxtimes D^{b}(\mathcal{X}(d_{k}))\to
D^{b}(\mathcal{X}(d)).$

We may drop the subscript $\lambda$ or $\underline{d}$ in the functors $p_{*}$
and $q^{*}$ when the cocharacter $\lambda$ or the partition $\underline{d}$ is
clear. We also use the notation $\ast$ for the Hall product.

#### 2.2.5.

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

$\bigoplus_{i=1}^{k}M(d_{i})\cong M(d),$

where the simple roots $\beta_{j}$ in $M(d_{1})$ correspond to the first
$d_{1}$ simple roots $\beta_{j}$ of $d$ etc.

#### 2.2.6.

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

$a_{0}=0<a_{1}<\cdots<a_{k-1}\leqslant a_{k}=l$

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

$\sum_{i=a_{j}+1}^{a_{j+1}}e_{i}=d_{j+1}.$

We say $\underline{e}$ is a refinement of $\underline{d}$. There is a
similarly defined order on pairs $(d,w)\in\mathbb{N}\times\mathbb{Z}$.

#### 2.2.7.

Let $A$ be a partition $(d_{i},w_{i})_{i=1}^{k}$ of $(d,w)$ and consider its
corresponding antidominant cocharacter $\lambda$. Define the weights

$\displaystyle\chi_{A}:=\sum_{i=1}^{k}w_{i}\tau_{d_{i}},\
\chi^{\prime}_{A}:=\chi_{A}+\mathfrak{g}^{\lambda>0}.$

Consider weights $\chi^{\prime}_{i}\in M(d_{i})_{\mathbb{R}}$ such that

$\chi^{\prime}_{A}=\sum_{i=1}^{k}\chi^{\prime}_{i}.$

Let $v_{i}$ be the sum of coefficients of $\chi^{\prime}_{i}$ for $1\leqslant
i\leqslant k$; alternatively, $v_{i}:=\langle
1_{d_{i}},\chi^{\prime}_{i}\rangle$. We denote the above transformation by

(2.6) $\displaystyle A\mapsto A^{\prime},\
(d_{i},w_{i})_{i=1}^{k}\mapsto(d_{i},v_{i})_{i=1}^{k}.$

Explicitly, the weights $v_{i}$ for $1\leqslant i\leqslant k$ are given by

(2.7) $\displaystyle
v_{i}=w_{i}+d_{i}\left(\sum_{j>i}d_{j}-\sum_{j<i}d_{j}\right).$

###  2.3. Polytopes

The polytope $\textbf{W}(d)$ is defined as

(2.8)
$\textbf{W}(d):=\frac{3}{2}\text{sum}[0,\beta_{i}-\beta_{j}]+\mathbb{R}\tau_{d}\subset
M(d)_{\mathbb{R}},$

where the Minkowski sum is after all $1\leqslant i,j\leqslant d$. For
$w\in\mathbb{Z}$, consider the hyperplane:

(2.9)
$\textbf{W}(d)_{w}:=\frac{3}{2}\text{sum}[0,\beta_{i}-\beta_{j}]+w\tau_{d}\subset\textbf{W}(d).$

For $r>0$ and $\lambda$ a cocharacter of $T(d)$, let $F_{r}(\lambda)$ be the
face of the polytope $2r\textbf{W}(d)$ corresponding to the cocharacter
$\lambda$, so the set of weights $\chi$ in $M(d)_{\mathbb{R}}$ such that

$\displaystyle\chi\in 2r\textbf{W}(d),\
\langle\lambda,\chi\rangle=r\langle\lambda,R(d)^{\lambda>0}\rangle.$

When $r=\frac{1}{2}$, we use the notations $F(\lambda)$. For $\chi\in
M(d)^{+}_{\mathbb{R}}$, its $r$-invariant $r(\chi)$ is the smallest real
number $r$ such that

$\chi\in 2r\textbf{W}(d).$

For a cocharacter $\lambda$ of $T(d)$, denote by

$\textbf{W}(\lambda)_{0}:=\frac{3}{2}\text{sum}[0,\beta_{i}-\beta_{j}]\subset
M(d)_{\mathbb{R}},$

where the sum is after all weights $1\leqslant i,j\leqslant d$ such that
$\langle\lambda,\beta_{i}-\beta_{j}\rangle=0$.

###  2.4. A corollary of the Borel-Weyl-Bott theorem

For future reference, we state a result from [HLS20, Section 3.2]. We continue
with the notations from Subsection 2.2.3. Let $M$ be the weight lattice of
$T_{G}$. We assume that $X$ is a symmetric $G$-representation, meaning that
for any weight $\beta$ of $X$, the weights $\beta$ and $-\beta$ appear with
the same multiplicity in $X$. Let $\chi$ be a weight in $M$. Let $\chi^{+}$ be
the dominant Weyl-shifted conjugate of $\chi$ if it exists, and zero
otherwise. For a multiset $J\subset\mathcal{W}$, let

$\sigma_{J}:=\sum_{\beta\in J}\beta.$

Let $w$ be the element of the Weyl group of minimal length such that
$w*(\chi-\sigma_{J})$ is dominant or zero. We let $\ell(J):=\ell(w)$.

######  Proposition 2.1.

Let $X$ be a symmetric $G$-representation, and let $\lambda$ be an
antidominant cocharacter of $T_{G}$. Recall the fixed and attracting stacks
and the corresponding maps

$X^{\lambda}/G^{\lambda}\xleftarrow{q_{\lambda}}X^{\lambda\geqslant
0}/G^{\lambda\geqslant 0}\xrightarrow{p_{\lambda}}X/G.$

For a weight $\chi$ in $M$, there is a quasi-isomorphism

$\left(\bigoplus_{J}\mathcal{O}_{X}\otimes\Gamma_{G}\left((\chi-\sigma_{J})^{+}\right)\left[|J|-\ell(J)\right],d\right)\xrightarrow{\sim}p_{\lambda*}q_{\lambda}^{*}\left(\mathcal{O}_{X^{\lambda}}\otimes\Gamma_{G^{\lambda}}(\chi)\right),$

where the complex on the left hand side has terms (shifted) vector bundles for
all multisets
$J\subset\\{\beta\in\mathcal{W}\mid\langle\lambda,\beta\rangle<0\\}$.

###  2.5. Matrix factorizations

In the notations from Subsection 2.2.1, we denote by

$\displaystyle\mathrm{MF}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits W_{d})$

the dg-category of matrix factorizations of the regular function
$\mathop{\mathrm{Tr}}\nolimits W_{d}$ on the smooth stack $\mathcal{X}(d)$.
Its objects consist of tuples

$\displaystyle\left(\alpha\colon F\rightleftarrows G\colon\beta\right)\text{
such that}\
\alpha\circ\beta=\beta\circ\alpha=\cdot\mathop{\mathrm{Tr}}\nolimits W_{d},\ $

where $F,G\in\operatorname{Coh}(\mathcal{X}(d))$, see [PTa, Subsection 2.6]
for details. For an object
$\mathcal{F}\in\mathrm{MF}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d})$, its internal homomorphism $R\mathcal{H}om(\mathcal{F},\mathcal{F})$
is an object of the $\mathbb{Z}/2$-graded derived category of
$\operatorname{Coh}(\mathcal{X}(d))$. The support of $\mathcal{F}$

$\displaystyle\mathrm{Supp}(\mathcal{F})\subset\mathcal{X}(d)$

is defined to be the support of $R\mathcal{H}om(\mathcal{F},\mathcal{F})$,
which is a closed substack of $\mathcal{X}(d)$. Alternatively, it is the
smallest closed substack $\mathcal{Z}\subset\mathcal{X}(d)$ such that
$\mathcal{F}|_{\mathcal{X}(d)\setminus\mathcal{Z}}\cong 0$ in
$\mathrm{MF}(\mathcal{X}(d)\setminus\mathcal{Z},\mathop{\mathrm{Tr}}\nolimits
W_{d})$.

Similarly to (2.5), for $d=d_{1}+d_{2}$ we have the categorical Hall product

$\displaystyle
m=m_{d_{1},d_{2}}\colon\mathrm{MF}(\mathcal{X}(d_{1}),\mathop{\mathrm{Tr}}\nolimits
W_{d_{1}})\boxtimes\mathrm{MF}(\mathcal{X}(d_{2}),\mathop{\mathrm{Tr}}\nolimits
W_{d_{2}})\to\mathrm{MF}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits W_{d}),$

see [Păda] for details. We sometimes write $a\ast b$ instead of $m(a,b)$.

We also consider equivariant and graded matrix factorizations for the regular
function (2.2), see [PTa, Subsection 2.6.2] for details. The group
$(\mathbb{C}^{\ast})^{3}$ acts on the linear maps corresponding to the edges
$(x,y,z)$ of the quiver $Q$ by scalar multiplication. Consider the two
dimensional subtorus

$T\cong(\mathbb{C}^{\ast})^{2}\subset(\mathbb{C}^{\ast})^{3}$

which preserves the super-potential $W=x[y,z]$, see (2.1). Then $T$ acts on
$\mathcal{X}(d)$ and preserves $\mathop{\mathrm{Tr}}\nolimits W$. We will also
consider graded matrix factorizations, where the grading is given by scaling
with weight $2$ the space $\mathfrak{gl}(V)$ for $V$ a vector space. For
example, we can choose an edge $e\in\\{x,y,z\\}$ of $Q$ and let
$\mathbb{C}^{*}$ scale with weight $2$ the linear map corresponding to $e$.
Contrary to the $T$-action, the regular function
$\mathop{\mathrm{Tr}}\nolimits W$ has weight $2$ with respect to such a
grading. The corresponding categories of matrix factorizations are denoted by

$\displaystyle\mathrm{MF}_{\ast}^{\bullet}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d})\text{ for}\ \ast\in\\{\emptyset,T\\},\
\bullet\in\\{\emptyset,\rm{gr}\\}.$

###  2.6. Quasi-BPS categories

#### 2.6.1.

For $w\in\mathbb{Z}$, we denote by $D^{b}(\mathcal{X}(d))_{w}$ the subcategory
of $D^{b}(\mathcal{X}(d))$ consisting of objects of weight $w$ with respect to
the diagonal cocharacter $1_{d}$ of $T(d)$. We have the direct sum
decomposition

$\displaystyle
D^{b}(\mathcal{X}(d))=\bigoplus_{w\in\mathbb{Z}}D^{b}(\mathcal{X}(d))_{w}.$

We define the dg subcategories

$\displaystyle\mathbb{M}(d)\subset D^{b}(\mathcal{X}(d)),\ (\mbox{resp.
}\mathbb{M}(d)_{w}\subset D^{b}(\mathcal{X}(d))_{w})$

to be generated by the vector bundles
$\mathcal{O}_{\mathcal{X}(d)}\otimes\Gamma_{GL(d)}(\chi)$, where $\chi$ is a
dominant weight of $T(d)$ such that

(2.10) $\chi+\rho\in\textbf{W}(d),\ (\mbox{resp.
}\chi+\rho\in\textbf{W}(d)_{w}).$

Note that $\mathbb{M}(d)$ decomposes into the direct sum of
$\mathbb{M}(d)_{w}$ for $w\in\mathbb{Z}$. Alternatively, the category
$\mathbb{M}(d)_{w}$ is generated by the vector bundles
$\mathcal{O}_{\mathcal{X}(d)}\otimes\Gamma$ for $\Gamma$ a
$GL(d)$-representation such that the $T(d)$-weights of $\Gamma$ are contained
in the set $\nabla_{w}$ defined by:

$\displaystyle\nabla_{w}$ $\displaystyle:=\left\\{\chi\in
M_{\mathbb{R}}\mathrel{}\middle|\mathrel{}-\frac{1}{2}n_{\lambda}\leqslant\langle\lambda,\chi\rangle\leqslant\frac{1}{2}n_{\lambda}\mbox{
for all }\lambda\colon\mathbb{C}^{\ast}\to T(d)\right\\}+w\tau_{d},$

see [HLS20, Lemma 2.9]. For a partition $A=(d_{i},w_{i})_{i=1}^{k}$ of
$(d,w)$, define

(2.11)
$\displaystyle\mathbb{M}_{A}:=\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}.$

#### 2.6.2.

Recall the regular function (2.2). We define the subcategory

$\mathbb{S}(d):=\text{MF}(\mathbb{M}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d})\subset\mathrm{MF}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits W_{d})$

to be the subcategory of matrix factorizations $\left(\alpha\colon
F\rightleftarrows G\colon\beta\right)$ with $F$ and $G$ in $\mathbb{M}(d)$. It
decomposes into the direct sum of $\mathbb{S}(d)_{w}$ for $w\in\mathbb{Z}$,
where $\mathbb{S}(d)_{w}$ is defined similarly to $\mathbb{S}(d)$ using
$\mathbb{M}(d)_{w}$.

We also consider subcategories for $\ast\in\\{\emptyset,T\\}$,
$\bullet\in\\{\emptyset,\text{gr}\\}$ defined in a similar way

$\mathbb{S}^{\bullet}_{\ast}(d):=\text{MF}^{\bullet}_{\ast}(\mathbb{M}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d})\subset\text{MF}^{\bullet}_{\ast}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d}).$

The subcategory $\mathbb{S}^{\bullet}_{\ast}(d)_{w}$ is also defined in a
similar way. For a partition $A=(d_{i},w_{i})_{i=1}^{k}$ of $(d,w)$, the
category $\mathbb{S}_{\ast,A}^{\bullet}$ is also defined similarly to (2.11).
We denote the Grothendieck group of $\mathbb{S}^{\bullet}_{\ast}(d)_{w}$ by

$\displaystyle K_{\ast}(\mathbb{S}^{\bullet}(d)_{w}),\
\ast\in\\{\emptyset,T\\},\ \bullet\in\\{\emptyset,\mathrm{gr}\\}.$

By [Todb, Corollary 3.13], there are natural isomorphisms (which hold for all
graded matrix factorizations as in Subsection 2.5):

(2.12) $\displaystyle
K(\mathbb{S}^{\mathrm{gr}}(d)_{w})\stackrel{{\scriptstyle\cong}}{{\to}}K(\mathbb{S}(d)_{w}),\
K_{T}(\mathbb{S}^{\mathrm{gr}}(d)_{w})\stackrel{{\scriptstyle\cong}}{{\to}}K_{T}(\mathbb{S}(d)_{w}).$

###  2.7. Complexes in quasi-BPS categories

Let $V$ be a $d$-dimensional complex vector space and recall that we denote by
$\mathfrak{g}=\operatorname{Hom}(V,V)$ the Lie algebra of $GL(V)$. We set

$\displaystyle\mathcal{Y}(d):=\mathfrak{g}^{\oplus 2}/GL(V),$

where $GL(V)$ acts on $\mathfrak{g}$ by conjugation. The stack
$\mathcal{Y}(d)$ is the moduli stack of representations of dimension $d$ of
the quiver with one vertex and two loops. Let $s$ be the morphism

(2.13) $\displaystyle s\colon\mathcal{Y}(d)\to\mathfrak{g},\
(X,Y)\mapsto[X,Y].$

The morphism $s$ induces a map of vector bundles
$\partial:\mathfrak{g}^{\vee}\otimes\mathcal{O}_{\mathfrak{g}^{\oplus
2}}\to\mathcal{O}_{\mathfrak{g}^{\oplus 2}}$. Let $s^{-1}(0)$ be the derived
scheme with the dg-ring of regular functions

(2.14)
$\displaystyle\mathcal{O}_{s^{-1}(0)}:=\mathcal{O}_{\mathfrak{g}^{\oplus
2}}\left[\mathfrak{g}^{\vee}\otimes\mathcal{O}_{\mathfrak{g}^{\oplus
2}}[1];d_{s}\right],$

where the differential $d_{s}$ is induced by the map $\partial$. Consider the
(derived) stack

(2.15) $\mathscr{C}(d):=s^{-1}(0)/GL(V)\hookrightarrow\mathcal{Y}(d).$

For a smooth variety $X$, we denote by $\mathscr{C}oh(X,d)$ the derived moduli
stack of zero-dimensional sheaves on $X$ with length $d$ and by
$\mathcal{C}oh(X,d)$ the classical truncation of $\mathscr{C}oh(X,d)$. Then
$\mathscr{C}(d)$ is equivalent to $\mathscr{C}oh(\mathbb{C}^{2},d)$.

For a decomposition $d=d_{1}+\cdots+d_{k}$, let
$\mathscr{C}(d_{1},\ldots,d_{k})$ be the derived moduli stack of filtrations
of coherent sheaves on $\mathbb{C}^{2}$:

(2.16) $\displaystyle 0=Q_{0}\subset Q_{1}\subset Q_{2}\subset\cdots\subset
Q_{k}$

such that each subquotient $Q_{i}/Q_{i-1}$ is a zero-dimensional sheaf on
$\mathbb{C}^{2}$ with length $d_{i}$. There exist evaluation morphisms

$\displaystyle\mathscr{C}(d_{1})\times\cdots\times\mathscr{C}(d_{k})\stackrel{{\scriptstyle
q}}{{\leftarrow}}\mathscr{C}(d_{1},\ldots,d_{k})\stackrel{{\scriptstyle
p}}{{\to}}\mathscr{C}(d),$

where $p$ is proper and $q$ is quasi-smooth. The above diagram for $k=2$
defines the categorical Hall product

(2.17) $\displaystyle m=m_{d_{1},d_{2}}=p_{\ast}q^{\ast}\colon
D^{b}(\mathscr{C}(d_{1}))\boxtimes D^{b}(\mathscr{C}(d_{2}))\to
D^{b}(\mathscr{C}(d)),$

which is a special case of the product of categorical Hall algebras for
surfaces defined by Porta–Sala [PS].

Let $T$ be the two-dimensional torus in (2.1) which acts on $\mathbb{C}^{2}$
by $(t_{1},t_{2})\cdot(x,y)=(t_{1}x,t_{2}y)$. It naturally induces an action
on $\mathscr{C}(d)$. There is also a $T$-equivariant Hall product:

(2.18) $\displaystyle m=m_{d_{1},d_{2}}=p_{\ast}q^{\ast}\colon
D^{b}_{T}(\mathscr{C}(d_{1}))\boxtimes D^{b}_{T}(\mathscr{C}(d_{2}))\to
D^{b}_{T}(\mathscr{C}(d)).$

Here, the box product is taken over $BT$. In what follows, whenever we take a
box-product in the $T$-equivariant setting, we take it over $BT$. We also use
the notation $\ast$ for the Hall product.

###  2.8. Subcategories $\mathbb{T}(d)_{v}$

Let

(2.19) $i\colon\mathscr{C}(d)\hookrightarrow\mathcal{Y}(d)$

be the natural closed immersion. Define the full triangulated subcategory

$\widetilde{\mathbb{T}}(d)_{v}\subset D^{b}(\mathcal{Y}(d))$

generated by the vector bundles
$\mathcal{O}_{\mathcal{Y}(d)}\otimes\Gamma_{GL(d)}(\chi)$ for a dominant
weight $\chi$ satisfying

$\displaystyle\chi+\rho\in\textbf{W}(d)_{v}.$

Define the full triangulated subcategory

(2.20) $\displaystyle\mathbb{T}(d)_{v}\subset D^{b}(\mathscr{C}(d))$

with objects $\mathcal{E}$ such that $i_{\ast}\mathcal{E}$ is in
$\widetilde{\mathbb{T}}(d)_{v}$. In [PTa, Lemma 4.8], we showed that the Hall
product restricts to functors

$m:\mathbb{T}(d_{1})_{v_{1}}\otimes\mathbb{T}(d_{2})_{v_{2}}\to\mathbb{T}(d)_{v}$

for $(d,v)=(d_{1},v_{1})+(d_{2},v_{2})$ and
$\frac{v_{1}}{d_{1}}=\frac{v_{2}}{d_{2}}$. Also, there is a semiorthogonal
decomposition, see [Pădb, Corollary 3.3]:

(2.23) $\displaystyle
D^{b}(\mathscr{C}(d))=\left\langle\mathbb{T}(d_{1})_{v_{1}}\boxtimes\cdots\boxtimes\mathbb{T}(d_{k})_{v_{k}}\mathrel{}\middle|\mathrel{}\begin{array}[]{c}v_{1}/d_{1}<\cdots<v_{k}/d_{k}\\\
d_{1}+\cdots+d_{k}=d\end{array}\right\rangle.$

In the above, each fully-faithful functor

$\displaystyle\mathbb{T}(d_{1})_{v_{1}}\boxtimes\cdots\boxtimes\mathbb{T}(d_{k})_{v_{k}}\hookrightarrow
D^{b}(\mathscr{C}(d))$

is given by the categorical Hall product (2.17).

Consider the grading induced by the action of $\mathbb{C}^{*}$ on
$\mathcal{X}(d)$ scaling the linear map corresponding to $Z$ with weight $2$.
The Koszul duality equivalence, also called dimensional reduction in the
literature, gives the following equivalence [Isi13, Hir17, Toda]:

(2.24) $\displaystyle\Phi\colon
D^{b}(\mathscr{C}(d))\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{MF}^{\mathrm{gr}}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits
W).$

Under this equivalence, we have that
$\Phi\colon\mathbb{T}(d)_{v}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{S}^{\mathrm{gr}}(d)_{v}$.

###  2.9. Constructions of objects in $\mathbb{T}(d)_{v}$

Here we review the construction of objects
$\mathcal{E}_{d,v}\in\mathbb{T}(d)_{v}$ (which also produces an object in
$\mathbb{T}_{T}(d)_{v}$) following [PTa, Subsection 4.3]. Let
$\mathcal{Z}\subset\mathscr{C}(1,1,\ldots,1)$ be the closed substack defined
as follows. Let $\lambda$ be the cocharacter

(2.25) $\displaystyle\lambda\colon\mathbb{C}^{\ast}\to GL(V),\
t\mapsto(t^{d},t^{d-1},\ldots,t).$

The attracting stack of $\mathcal{Y}(d)$ with respect to $\lambda$ is given by

(2.26) $\displaystyle\mathcal{Y}(d)^{\lambda\geqslant
0}:=\left(\mathfrak{g}^{\lambda\geqslant 0}\right)^{\oplus
2}\big{/}GL(V)^{\lambda\geqslant 0},$

where $GL(V)^{\lambda\geqslant 0}\subset GL(V)$ is the subgroup of upper
triangular matrices. Then the morphism (2.13) restricts to the morphism

(2.27) $\displaystyle s^{\lambda\geqslant
0}\colon\mathcal{Y}(d)^{\lambda\geqslant 0}\to\mathfrak{g}^{\lambda\geqslant
0}$

whose derived zero locus $\mathscr{C}(d)^{\lambda\geqslant 0}$ is equivalent
to $\mathscr{C}(1,\ldots,1)$. Let $X=(x_{i,j})$ and $Y=(y_{i,j})$ be elements
of $\mathfrak{g}^{\lambda\geqslant 0}$ for $1\leqslant i,j\leqslant d$, where
$x_{i,j}=y_{i,j}=0$ for $i>j$. Then the equation $s^{\lambda\geqslant
0}(X,Y)=0$ is equivalent to the equations

$\displaystyle\sum_{i\leqslant a\leqslant j}x_{i,a}y_{a,j}=\sum_{i\leqslant
a\leqslant j}y_{i,a}x_{a,j},$

for each $(i,j)$ with $i\leqslant j$. We call the above equation
$\mathbb{E}_{i,j}$. The equation $\mathbb{E}_{i,i}$ is
$x_{i,i}y_{i,i}-y_{i,i}x_{i,i}=0$, which always holds but imposes a non-
trivial derived structure on $\mathscr{C}(1,\ldots,1)$. The equation
$\mathbb{E}_{i,i+1}$ is

$\displaystyle(x_{i,i}-x_{i+1,i+1})y_{i,i+1}-(y_{i,i}-y_{i+1,i+1})x_{i,i+1}=0.$

The above equation is satisfied if the following equation $\mathbb{F}_{i,i+1}$
is satisfied:

$\displaystyle\\{x_{i,i}-x_{i+1,i+1}=0,y_{i,i}-y_{i+1,i+1}=0\\}.$

We define the closed derived substack

(2.28)
$\displaystyle\mathcal{Z}:=\mathcal{Z}(d)\subset\mathcal{Y}^{\lambda\geqslant
0}(d)$

to be the derived zero locus of the equations $\mathbb{F}_{i,i+1}$ for all $i$
and $\mathbb{E}_{i,j}$ for all $i+2\leqslant j$. We usually drop $d$ from the
notation if the dimension is clear from the context. Then $\mathcal{Z}$ is a
closed substack of $\mathscr{C}(d)^{\lambda\geqslant
0}=\mathscr{C}(1,\ldots,1)$. Note that, set theoretically, the closed substack
$\mathcal{Z}$ corresponds to filtrations (2.16) such that each $Q_{i}/Q_{i-1}$
is isomorphic to $\mathcal{O}_{x}$ for some $x\in\mathbb{C}^{2}$ independent
of $i$.

We have the diagram of attracting loci

$\displaystyle\mathscr{C}(1)^{\times
d}=\mathscr{C}(d)^{\lambda}\stackrel{{\scriptstyle
q}}{{\leftarrow}}\mathscr{C}(d)^{\lambda\geqslant 0}\stackrel{{\scriptstyle
p}}{{\to}}\mathscr{C}(d),$

where $p$ is a proper morphism. We set

(2.29) $\displaystyle
m_{i}:=\left\lceil\frac{vi}{d}\right\rceil-\left\lceil\frac{v(i-1)}{d}\right\rceil+\delta_{i}^{d}-\delta_{i}^{1}\in\mathbb{Z},$

where $\delta^{j}_{i}$ is the Kronecker delta function defined by
$\delta^{j}_{i}=1$ if $i=j$ and $\delta^{j}_{i}=0$ otherwise. For a weight
$\chi=\sum_{i=1}^{d}n_{i}\beta_{i}$ with $n_{i}\in\mathbb{Z}$, we denote by
$\mathbb{C}(\chi)$ the one dimensional $GL(V)^{\lambda\geqslant
0}$-representation given by

$\displaystyle GL(V)^{\lambda\geqslant 0}\to
GL(V)^{\lambda}=T(d)\stackrel{{\scriptstyle\chi}}{{\to}}\mathbb{C}^{\ast},$

where the first morphism is the projection.

######  Definition 2.2.

([PTa, Definition 4.2]) We define the complex $\mathcal{E}_{d,v}$ by

(2.30)
$\displaystyle\mathcal{E}_{d,v}:=p_{\ast}\left(\mathcal{O}_{\mathcal{Z}}\otimes\mathbb{C}(m_{1},\ldots,m_{d})\right)\in
D^{b}(\mathscr{C}(d))_{v}.$

The construction above is $T$-equivariant, so we also obtain an object
$\mathcal{E}_{d,v}\in D^{b}_{T}(\mathscr{C}(d))_{v}$.

In [PTa, Lemma 4.3], we showed that $\mathcal{E}_{d,v}$ is an object of
$\mathbb{T}(d)_{v}$ and $\mathbb{T}_{T}(d)_{v}$.

###  2.10. Shuffle algebras

#### 2.10.1.

Consider the $\mathbb{N}$-graded $\mathbb{K}$-module:

$\mathcal{S}h:=\bigoplus_{d\geqslant 0}\mathbb{K}\left[z_{1}^{\pm
1},\ldots,z_{d}^{\pm 1}\right]^{\mathfrak{S}_{d}}.$

We define a shuffle product on $\mathcal{S}h$ as follows. Let $\xi(x)$ be
defined by

$\displaystyle\xi(x):=\frac{(1-q_{1}^{-1}x)(1-q_{2}^{-1}x)(1-q^{-1}x^{-1})}{1-x},$

where $q:=q_{1}q_{2}$. For
$f\in\mathbb{K}\left[z_{1}^{\pm},\ldots,z_{a}^{\pm}\right]$ and
$g\in\mathbb{K}\left[z_{a+1}^{\pm},\ldots,z_{a+b}^{\pm}\right]$, we set

(2.31) $\displaystyle f\ast
g:=\frac{1}{a!b!}\mathrm{Sym}\left(fg\cdot\prod_{\begin{subarray}{c}1\leqslant
i\leqslant a,\\\ a<j\leqslant a+b\end{subarray}}\xi(z_{i}z_{j}^{-1})\right),$

where we denote by $\mathrm{Sym}(h(z_{1},\ldots,z_{d}))$ the sum of
$h(z_{\sigma(1)},\ldots,z_{\sigma(d)})$ after all permutations
$\sigma\in\mathfrak{S}_{d}$. Let $\mathcal{S}\subset\mathcal{S}h$ be the
subalgebra generated by $z_{1}^{l}$ for $l\in\mathbb{Z}$. Let
$\mathcal{S}_{\mathbb{F}}:=\mathcal{S}\otimes_{\mathbb{K}}\mathbb{F}$. It is
proved in [Nega, Theorem 4.6] that there is an isomorphism

(2.32) $\displaystyle i_{\ast}\colon\bigoplus_{d\geqslant
0}G_{T}(\mathscr{C}(d))\otimes_{\mathbb{K}}\mathbb{F}\stackrel{{\scriptstyle\cong}}{{\to}}\mathcal{S}_{\mathbb{F}}.$

The above isomorphism is induced by the algebra homomorphism which will be
defined in (2.39).

#### 2.10.2.

Let

$\displaystyle\mathcal{S}^{\prime}\subset\bigoplus_{d\geqslant
0}\mathbb{K}\left(z_{1},\ldots,z_{d}\right)^{\mathfrak{S}_{d}}$

be the $\mathbb{K}$-subalgebra generated by elements of the form

(2.33) $A^{\prime}_{k_{\bullet}}:=\mathrm{Sym}\left(\frac{z_{1}^{k_{1}}\cdots
z_{d}^{k_{d}}}{(1-q^{-1}z_{1}^{-1}z_{2})\cdots(1-q^{-1}z_{d-1}^{-1}z_{d})}\cdot\prod_{j>i}w(z_{i}z_{j}^{-1})\right)$

for various $(k_{1},\ldots,k_{d})\in\mathbb{Z}^{d}$ and $d\geqslant 1$, for
the shuffle product (2.31) where we replace $\xi(x)$ with $w(x)$ defined by

$\displaystyle w(x):=\frac{(1-q_{1}^{-1}x)(1-q_{2}^{-1}x)}{(1-x)(1-q^{-1}x)}.$

Let
$\mathcal{S}^{\prime}_{\mathbb{F}}:=\mathcal{S}^{\prime}\otimes_{\mathbb{K}}\mathbb{F}$.
Consider the morphism

(2.34) $\displaystyle\bigoplus_{d\geqslant 0}\mathbb{F}[z_{1}^{\pm
1},\ldots,z_{d}^{\pm 1}]^{\mathfrak{S}_{d}}\to\bigoplus_{d\geqslant
0}\mathbb{F}(z_{1},\ldots,z_{d})^{\mathfrak{S}_{d}}$

defined by

$\displaystyle f(z_{1},\ldots,z_{d})\mapsto
f(z_{1},\ldots,z_{d})\cdot\prod_{i\neq j}(1-q^{-1}z_{i}z_{j}^{-1})^{-1}.$

Then (2.34) induces an algebra homomorphism
$\mathcal{S}\to\mathcal{S}^{\prime}$. There is an isomorphism

(2.35)
$\mathcal{S}_{\mathbb{F}}\stackrel{{\scriptstyle\cong}}{{\to}}\mathcal{S}^{\prime}_{\mathbb{F}},$

see [PTa, Proof of Lemma 4.11]. For $(d,v)\in\mathbb{N}\times\mathbb{Z}$, we
set $A^{\prime}_{d,v}$ to be $A^{\prime}_{m_{\bullet}}$ for the choice of
$m_{\bullet}$ in (2.29). By [Negb, Equation (2.12)], we have the following
isomorphism of $\mathbb{K}$-modules:

(2.36)
$\displaystyle\mathcal{S}^{\prime}=\bigoplus_{v_{1}/d_{1}\leqslant\cdots\leqslant
v_{k}/d_{k}}\mathbb{K}\cdot A^{\prime}_{d_{1},v_{1}}\ast\cdots\ast
A^{\prime}_{d_{k},v_{k}},$

where the tuples $(d_{i},v_{i})_{i=1}^{k}$ appearing above are unordered for
subtuples $(d_{i},v_{i})_{i=a}^{b}$ with $v_{a}/d_{a}=\cdots=v_{b}/d_{b}$. We
also define

(2.37) $A_{d,v}:=\mathrm{Sym}\left(\frac{z_{1}^{m_{1}}\cdots
z_{d}^{m_{d}}}{(1-q^{-1}z_{1}^{-1}z_{2})\cdots(1-q^{-1}z_{d-1}^{-1}z_{d})}\cdot\prod_{j>i}\xi(z_{i}z_{j}^{-1})\right),$

where the exponents $m_{i}$ for $1\leqslant i\leqslant d$ are given by (2.29).

#### 2.10.3.

The $T$-equivariant Hall product (2.18) induces an associative algebra
structure

(2.38) $\displaystyle m\colon
G_{T}(\mathscr{C}(d_{1}))\otimes_{\mathbb{K}}G_{T}(\mathscr{C}(d_{2}))\to
G_{T}(\mathscr{C}(d)).$

Let $i\colon\mathscr{C}(d)\hookrightarrow\mathcal{Y}(d)$ be the closed
immersion. The pull-back via $\mathcal{Y}(d)\to BGL(d)$ gives the isomorphism

$\displaystyle\bigoplus_{d\geqslant 0}K_{T}(BGL(d))=\bigoplus_{d\geqslant
0}\mathbb{K}\left[z_{1}^{\pm 1},\ldots,z_{d}^{\pm
1}\right]^{\mathfrak{S}_{d}}\stackrel{{\scriptstyle\cong}}{{\to}}\bigoplus_{d\geqslant
0}K_{T}(\mathcal{Y}(d)).$

Therefore the push-forward by $i$ induces a morphism

(2.39) $\displaystyle i_{\ast}\colon\bigoplus_{d\geqslant
0}G_{T}(\mathscr{C}(d))\to\bigoplus_{d\geqslant 0}\mathbb{K}\left[z_{1}^{\pm
1},\ldots,z_{d}^{\pm 1}\right]^{\mathfrak{S}_{d}}.$

The product (2.38) is compatible with a shuffle algebra defined on the right
hand side of (2.39), see [PTa, Subsection 4.5]. In [PTa, Lemma 4.11], we
showed that

(2.40)
$i_{\ast}[\mathcal{E}_{d,v}]=(1-q_{1}^{-1})^{d-1}(1-q_{2}^{-1})^{d-1}A_{d,v}.$

###  2.11. Compatibility of the Hall product under the Koszul equivalence

In this subsection, we denote by $m$ the Hall product (2.17) and by
$\widetilde{m}$ the Hall product for the quiver with potential $(Q,W)$ from
Subsection 2.2.1. Using the results in [Toda, Section 2.4], Koszul duality
equivalences are compatible with the Hall products by the following
commutative diagram (see [Pădb, Proposition 3.1]):

(2.45)

where the left arrow $\widetilde{\Phi}$ is the composition of Koszul duality
equivalences (2.24) with the tensor product of

(2.46)
$\det((\mathfrak{g}^{\nu>0})^{\vee}(2))[-\dim\mathfrak{g}^{\nu>0}]=(\det
V_{1})^{-d_{2}}\otimes(\det V_{2})^{d_{1}}[d_{1}d_{2}].$

The cocharacter $\nu\colon\mathbb{C}^{\ast}\to T(d)$ is
$\nu(t)=(\overbrace{t,\ldots,t}^{d_{1}},\overbrace{1,\ldots,1}^{d_{2}})$, the
vector spaces $V_{i}$ have $\dim V_{i}=d_{i}$ for $i=1,2$, and $(1)$ is a
twist by the weight one $\mathbb{C}^{\ast}$-character, which is isomorphic to
the shift functor $[1]$ of the category of graded matrix factorizations.

###  2.12. Symmetric polynomials

Let $\Lambda$ be the $\mathbb{Z}$-algebra of symmetric polynomials [Mac79,
Chapter I, Section 2], [Sch12, Subsection 2.4]:

$\Lambda\cong\varprojlim\mathbb{Z}\left[x_{1},\ldots,x_{n}\right]^{\mathfrak{S}_{n}},$

with multiplication defined by

$f(x_{1},\ldots,x_{a})\star
g(x_{a+1},\ldots,x_{a+b}):=\sum_{\mathfrak{S}_{a+b}/\mathfrak{S}_{a}\times\mathfrak{S}_{b}}w\left(f(x_{1},\ldots,x_{a})g(x_{a+1},\ldots,x_{a+b})\right)$

and comultiplication induced by the restriction map

$\mathbb{Z}\left[x_{1},\ldots,x_{a+b}\right]^{\mathfrak{S}_{a+b}}\to\mathbb{Z}\left[x_{1},\ldots,x_{a}\right]^{\mathfrak{S}_{a}}\otimes\mathbb{Z}\left[x_{a+1},\ldots,x_{a+b}\right]^{\mathfrak{S}_{b}}.$

Alternatively, $\Lambda$ is isomorphic to the Grothendieck group of the
monoidal category

$\mathcal{R}:=\bigoplus_{n\geqslant 0}\text{Rep}(\mathfrak{S}_{n}),$

where $\text{Rep}(\mathfrak{S}_{n})$ is the abelian category of finite
dimensional $\mathfrak{S}_{n}$-representations, multiplication is given by the
induction functor

$\text{Ind}:\text{Rep}(\mathfrak{S}_{a}\times\mathfrak{S}_{b})\to\text{Rep}(\mathfrak{S}_{a+b}),$

and comultiplication is given by the restriction functor

$\text{Res}:\text{Rep}(\mathfrak{S}_{a+b})\to\text{Rep}(\mathfrak{S}_{a}\times\mathfrak{S}_{b}).$

The isomorphism $\mathcal{R}\stackrel{{\scriptstyle\cong}}{{\to}}\Lambda$ is
given by sending an irreducible $\mathfrak{S}_{n}$-representation
$W_{\lambda}$ corresponding to a partition $\lambda$ of $n$ to the Schur
function $s_{\lambda}$, see [Mac79, Chapter I, Equation (7.5)].

For $R$ a ring with a map $\mathbb{Z}\to R$, denote by
$\Lambda_{R}:=R\otimes_{\mathbb{Z}}\Lambda$ the $R$-algebra with
multiplication and comultiplication induced from those of $\Lambda$. Consider
the elementary symmetric functions

$e_{n}:=\sum_{i_{1}<\ldots<i_{n}}x_{i_{1}}\ldots x_{i_{n}}\in\Lambda$

and the power sum functions

$p_{n}:=\sum_{i}x_{i}^{n}\in\Lambda.$

We also denote by $e_{n}$ and $p_{n}$ the images of these symmetric functions
in $\Lambda_{R}$. Let $t$ be a formal variable. These functions are connected
via the identity

(2.47) $\sum_{n\geqslant 0}e_{n}t^{n}=\text{exp}\left(\sum_{n\geqslant
1}\frac{(-1)^{n+1}}{n}p_{n}t^{n}\right).$

There are isomorphisms (see [Mac79, Chapter I, Equations (2.4), (2.14)]):

(2.48)
$\Lambda_{\mathbb{Q}}\cong\mathbb{Q}[e_{1},e_{2},\ldots]\cong\mathbb{Q}[p_{1},p_{2},\ldots].$

For $n\geqslant 1$, let $P(n)$ be the free one dimensional $\mathbb{Z}$-module
with generator $p_{n}$. By (2.48), there is an isomorphism of
$\mathbb{N}$-graded $\mathbb{Q}$-vector spaces:

(2.49) $\Lambda_{\mathbb{Q}}\cong\bigotimes_{n\geqslant
1}\mathrm{Sym}\big{(}P(n)_{\mathbb{Q}}\big{)}.$

##  3\. The support of complexes in quasi-BPS categories

Recall the regular function (2.2). Consider the commutative diagram

(3.5)

where the vertical arrows are good moduli space morphisms and the horizontal
arrows are closed immersions. The left vertical arrow sends a zero-dimensional
sheaf to its support. Let
$\Delta\colon\mathbb{C}^{3}\subset\mathrm{Sym}^{d}(\mathbb{C}^{3})$ be the
small diagonal

$\displaystyle\mathbb{C}^{3}\subset\mathrm{Sym}^{d}(\mathbb{C}^{3}),\
x\mapsto(x,\ldots,x).$

We abuse notation and denote the image of $\Delta$ also by
$\Delta\cong\mathbb{C}^{3}$. We consider the pull-back
$\pi^{-1}(\Delta)\subset\mathcal{C}oh(\mathbb{C}^{3},d)$, which is a closed
substack of $\mathrm{Crit}(\mathop{\mathrm{Tr}}\nolimits
W_{d})\subset\mathcal{X}(d)$. Davison [Davb, Theorem 5.1] showed that the BPS
sheaf for the moduli stack of degree $d$ sheaves on $\mathbb{C}^{3}$ is

$\mathcal{B}PS_{d}=\Delta_{*}\mathrm{IC}_{\mathbb{C}^{3}}.$

Recall the notations involving formal completions from Subsection 2.1. The
following is the main result of this section, which will be proved in
Subsection 3.1:

######  Theorem 3.1.

Consider a pair $(d,w)\in\mathbb{N}\times\mathbb{Z}$ and let $\mathcal{F}$ be
an object in $\mathbb{S}(d)_{w}$. Assume there exists
$p\in\mathrm{Sym}^{d}(\mathbb{C}^{3})\setminus\Delta$ such that the support of
$\mathcal{F}$ intersects $\pi^{-1}(p)$. Write $p=\sum_{i=1}^{l}p^{(i)}$,
$p^{(i)}=d^{(i)}x^{(i)}$, $d^{(i)}\in\mathbb{Z}_{>0}$, $x^{(i)}\neq
x^{(i^{\prime})}$ for $i\neq i^{\prime}$ and $l\geqslant 2$. Then there exist
non-zero objects
$\mathcal{F}_{i}\in\mathrm{MF}(\mathcal{X}_{p^{(i)}}(d^{(i)}),\mathop{\mathrm{Tr}}\nolimits
W_{d^{(i)}})_{w^{(i)}}$ for $1\leqslant i\leqslant l$ with

$\frac{w^{(1)}}{d^{(1)}}=\cdots=\frac{w^{(l)}}{d^{(l)}}$

and $w^{(1)}+\cdots+w^{(l)}=w$. In particular, if $\gcd(d,w)=1$, then any
object in $\mathbb{S}(d)_{w}$ is supported on $\pi^{-1}(\Delta)$. The same
result holds for the categories $\mathbb{S}^{\bullet}_{\ast}(d)_{w}$
introduced in Subsection 2.6.2.

In Subsection 4.1, we discuss divisibility properties of complexes supported
on $\pi^{-1}(\Delta)$, see Lemma 4.1. In Subsection 4.2, we use Theorem 3.1
and Lemma 4.1 to show that $K_{T}(\mathbb{S}(d)_{w})^{\prime}$ is a free
$\mathbb{K}$-module with generator $[\mathcal{E}_{d,w}]$ when $\gcd(d,w)=1$.

###  3.1. Proof of the main result

###### Proof of Theorem 3.1.

Suppose that an object $\mathcal{F}\in\mathbb{S}(d)_{w}$ has support not
contained in $\pi^{-1}(\Delta)$. Then there is
$p\in\mathrm{Sym}^{d}(\mathbb{C}^{3})\setminus\Delta$ such that
$\mathcal{F}|_{\mathcal{X}_{p}(d)}\neq 0$ in
$\mathrm{MF}(\mathcal{X}_{p}(d),\mathop{\mathrm{Tr}}\nolimits W_{d})$, where
$\mathcal{X}_{p}(d)$ is the formal fiber of $p$ along the good moduli space
morphism $\mathcal{X}(d)\to X(d)$. Since $p\notin\Delta$, it is written as

$\displaystyle p=\sum_{i=1}^{l}p^{(i)},\ p^{(i)}=d^{(i)}x^{(i)},\
d^{(i)}\in\mathbb{Z}_{>0},\ x^{(i)}\neq x^{(i^{\prime})}\mbox{ for }i\neq
i^{\prime},\ l\geqslant 2.$

The unique closed point in $\pi^{-1}(p)$ corresponds to the semisimple
$Q$-representation

$\displaystyle R=\bigoplus_{i=1}^{l}V^{(i)}\otimes R^{(i)},$

where $R^{(i)}$ is the one-dimensional $Q$-representation corresponding to
$\mathcal{O}_{x^{(i)}}$ and $V^{(i)}$ is a $d^{(i)}$-dimensional vector space
such that $d^{(1)}+\cdots+d^{(l)}=d$. Below we write a basis of $V^{(i)}$ as
$\beta_{1}^{(i)},\ldots,\beta_{d^{(i)}}^{(i)}$ and set

$\displaystyle\\{\beta_{1},\ldots,\beta_{d}\\}=\\{\beta_{1}^{(1)},\ldots,\beta_{d^{(1)}}^{(1)},\beta_{1}^{(2)},\ldots,\beta_{d^{(2)}}^{(2)},\ldots\\}.$

The étale slice theorem implies that

(3.6)
$\displaystyle\mathcal{X}_{p}(d)\cong\widehat{\operatorname{Ext}}_{Q}^{1}(R,R)/G_{p},$

where $G_{p}=\mathrm{Aut}(R)=\prod_{i=1}^{l}GL(V^{(i)})$ and
$\widehat{\operatorname{Ext}}^{1}_{Q}(R,R)$ is the formal fiber of the origin
along the morphism
$\operatorname{Ext}^{1}_{Q}(R,R)\to\operatorname{Ext}_{Q}^{1}(R,R)/\\!\\!/G_{p}$.
By Lemma 3.2, the Ext-group $\operatorname{Ext}^{1}_{Q}(R,R)$ is computed as
follows:

(3.7)
$\displaystyle\operatorname{Ext}_{Q}^{1}(R,R)=\bigoplus_{i=1}^{l}\mathrm{End}(V^{(i)},V^{(i)})^{\oplus
3}\oplus\bigoplus_{i\neq j}\operatorname{Hom}(V^{(i)},V^{(j)})^{\oplus 2}.$

Note that the maximal torus $T(d)\subset GL(V)$ is contained in $G_{p}$.

We define

$\displaystyle\mathbb{S}_{p}(d)_{w}\subset\mathrm{MF}(\mathcal{X}_{p}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d})$

to be the full subcategory generated by matrix factorizations whose entries
are of the form $\Gamma_{G_{p}}(\chi)\otimes\mathcal{O}$, where $\chi$ is a
$G_{p}$-dominant $T(d)$-weight satisfying

(3.8) $\displaystyle\chi+\rho_{p}\in\textbf{W}_{p}(d)_{w}.$

Here, $\rho_{p}$ is half the sum of positive roots of $G_{p}$ and
$\textbf{W}_{p}(d)_{w}$ is defined as in (2.9) for the $G_{p}$-representation
$\operatorname{Ext}_{Q}^{1}(R,R)$:

$\displaystyle\textbf{W}_{p}(d)_{w}:=\frac{1}{2}\mathrm{sum}[0,\beta]+w\tau_{d},$

where the Minkowski sum is after all weights $\beta$ in
$\operatorname{Ext}_{Q}^{1}(R,R)$. By Lemma 3.5, we have
$\mathcal{F}|_{\mathcal{X}_{p}(d)}\in\mathbb{S}_{p}(d)_{w}$, in particular
$\mathbb{S}_{p}(d)_{w}\neq 0$.

Below, in order to simplify the notation, we treat the case $l=2$. Since any
zero-dimensional sheaf $Q$ on $\mathbb{C}^{3}$ supported on $p$ decomposes
into $Q^{(1)}\oplus Q^{(2)}$, where $Q^{(i)}$ is supported on $x^{(i)}$, we
have

(3.9)
$\displaystyle\widehat{\mathcal{C}oh}_{p}(\mathbb{C}^{3},d)=\mathrm{Crit}(\mathop{\mathrm{Tr}}\nolimits
W_{d}|_{\mathcal{X}_{p}(d)})=\widehat{\mathcal{C}oh}_{p^{(1)}}(\mathbb{C}^{3},d^{(1)})\times\widehat{\mathcal{C}oh}_{p^{(2)}}(\mathbb{C}^{3},d^{(2)}).$

Here $\widehat{\mathcal{C}oh}_{p}(\mathbb{C}^{3},d)$ is the formal fiber of
the left vertical arrow in (3.5) at $p$. Indeed, by Lemma 3.3, we can show
that, by replacing the isomorphism (3.6) if necessary, the regular function
$\mathop{\mathrm{Tr}}\nolimits W_{d}$ restricted to $\mathcal{X}_{p}(d)$ is
written as

(3.10) $\displaystyle\mathop{\mathrm{Tr}}\nolimits
W_{d}|_{\mathcal{X}_{p}(d)}=\mathop{\mathrm{Tr}}\nolimits
W_{d^{(1)}}\boxplus\mathop{\mathrm{Tr}}\nolimits W_{d^{(2)}}\boxplus q.$

Here, $\mathop{\mathrm{Tr}}\nolimits W_{d^{(i)}}$ is the regular function
(2.2) on $\mathcal{X}(d^{(i)})$ restricted to $\mathcal{X}_{p^{(i)}}(d^{(i)})$
and $q$ is a non-degenerate $G_{p}$-invariant quadratic form on $U\oplus
U^{\vee}$ given by $q(u,v)=\langle u,v\rangle$, where $U$ is the following
self-dual $G_{p}$-representation

$\displaystyle
U:=\operatorname{Hom}(V^{(1)},V^{(2)})\oplus\operatorname{Hom}(V^{(2)},V^{(1)}).$

The decomposition (3.10) in particular implies (3.9).

We have the following diagram

where $\mathcal{U}$ is the vector bundle on
$\mathcal{X}_{p^{(1)}}(d^{(1)})\times\mathcal{X}_{p^{(2)}}(d^{(2)})$
determined by the $G_{p}$-representation $U$, $i$ is the closed immersion
$x\mapsto(x,0)$, and $j$ is the natural morphism induced by the formal
completion which induces the isomorphism on critical loci of
$\mathop{\mathrm{Tr}}\nolimits W_{d}$. We have the following functors

(3.11)
$\displaystyle\Psi:=j^{\ast}i_{\ast}p^{\ast}\colon\mathrm{MF}(\mathcal{X}_{p^{(1)}}(d^{(1)}),\mathop{\mathrm{Tr}}\nolimits
W_{d^{(1)}})$
$\displaystyle\boxtimes\mathrm{MF}(\mathcal{X}_{p^{(2)}}(d^{(2)}),\mathop{\mathrm{Tr}}\nolimits
W_{d^{(2)}})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{MF}(\mathcal{U}\oplus\mathcal{U}^{\vee},\mathop{\mathrm{Tr}}\nolimits
W_{d})\stackrel{{\scriptstyle
j^{\ast}}}{{\hookrightarrow}}\mathrm{MF}(\mathcal{X}_{p}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d}).$

Here the first arrow is an equivalence by Knörrer periodicity (see [Hir17,
Theorem 4.2]), and the second arrow is fully-faithful with dense image (see
[Todc, Lemma 6.4]). By Lemma 3.4, the above functor restricts to the fully-
faithful functor:

(3.12) $\displaystyle\bigoplus_{\begin{subarray}{c}w^{(1)}+w^{(2)}=w\\\
w^{(1)}/d^{(1)}=w^{(2)}/d^{(2)}\end{subarray}}\mathbb{S}_{p^{(1)}}(d^{(1)})_{w^{(1)}}\boxtimes\mathbb{S}_{p^{(2)}}(d^{(2)})_{w^{(2)}}\to\mathbb{S}_{p}(d)_{w}.$

Below we show that the above functor (3.12) has dense image, and thus the
conclusion follows. By the semiorthogonal decomposition (2.23) together with
(2.45), we have

(3.13) $\displaystyle\mathrm{MF}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d})$
$\displaystyle=\left\langle\ast_{i=1}^{k}\mathbb{S}(d_{i})_{v_{i}+d_{i}(\sum_{i>j}d_{j}-\sum_{j>i}d_{j})}\mathrel{}\middle|\mathrel{}\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}},d_{1}+\cdots+d_{k}=d\right\rangle.$

The weights $w_{i}:=v_{i}+d_{i}(\sum_{i>j}d_{j}-\sum_{j>i}d_{j})$ are computed
as in Equation (2.7):

(3.14)
$\displaystyle\sum_{i=1}^{k}w_{i}\tau_{d_{i}}=\sum_{i=1}^{k}v_{i}\tau_{d_{i}}-\mathfrak{g}^{\lambda>0},$

where $\mathfrak{g}$ is the Lie algebra of $GL(d)$ and $\lambda$ is the
antidominant cocharacter corresponding to the decomposition
$d=d_{1}+\cdots+d_{k}$:

(3.15)
$\displaystyle\lambda(t)=(\overbrace{t^{k},\ldots,t^{k}}^{d_{1}},\overbrace{t^{k-1},\ldots,t^{k-1}}^{d_{2}},\ldots,\overbrace{t,\ldots,t}^{d_{k}}).$

For $x\in\mathbb{C}^{3}$, let
$q=d[x]:=(x,\ldots,x)\in\mathrm{Sym}^{d}(\mathbb{C}^{3})$. Since $G_{q}=GL(V)$
and $\mathcal{X}_{q}(d)\cong\mathfrak{g}^{\oplus 3}_{d[0]}/GL(V)$, the
argument showing (3.13) applies verbatim to show the semiorthogonal
decomposition

(3.16)
$\displaystyle\mathrm{MF}(\mathcal{X}_{q}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d})$
$\displaystyle=\left\langle\ast_{i=1}^{k}\mathbb{S}_{q}(d_{i})_{v_{i}+d_{i}(\sum_{i>j}d_{j}-\sum_{j>i}d_{j})}\mathrel{}\middle|\mathrel{}\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}},d_{1}+\cdots+d_{k}=d\right\rangle.$

From (3.16), the left hand side of (3.11) admits a semiorthogonal
decomposition S whose summands are of the form

(3.17)
$\displaystyle\ast_{i=1}^{a}\mathbb{S}_{p_{i}^{(1)}}(d_{i}^{(1)})_{v_{i}^{(1)}+d_{i}^{(1)}(\sum_{i>j}d_{j}^{(1)}-\sum_{j>i}d_{j}^{(1)})}\boxtimes\ast_{i=1}^{b}\mathbb{S}_{p_{i}^{(2)}}(d_{i}^{(2)})_{v_{i}^{(2)}+d_{i}^{(2)}(\sum_{i>j}d_{j}^{(2)}-\sum_{j>i}d_{j}^{(2)})}.$

Here, the left hand side in (3.12) comes to the rightmost part of the
semiorthogonal decomposition S, $p_{i}^{(j)}=d_{i}^{(j)}x^{(j)}$, the integers
$d_{i}^{(j)}$ satisfy

$\displaystyle d_{1}^{(1)}+\cdots+d_{a}^{(1)}=d^{(1)},\
d_{1}^{(2)}+\cdots+d_{b}^{(2)}=d^{(2)},$

and we have the inequalities

$\displaystyle\frac{v_{1}^{(1)}}{d_{1}^{(1)}}<\cdots<\frac{v_{a}^{(1)}}{d_{a}^{(1)}},\
\frac{v_{1}^{(2)}}{d_{1}^{(2)}}<\cdots<\frac{v_{b}^{(2)}}{d_{b}^{(2)}}.$

We write

$\displaystyle\left\\{\frac{v_{1}^{(1)}}{d_{1}^{(1)}},\ldots,\frac{v_{a}^{(1)}}{d_{a}^{(1)}},\ldots,\frac{v_{1}^{(2)}}{d_{1}^{(2)}},\ldots,\frac{v_{b}^{(2)}}{d_{b}^{(2)}}\right\\}=\\{\mu_{1}<\cdots<\mu_{k}\\},$

where $k$ is the number of distinct elements in the left hand side. For
$1\leqslant i\leqslant k$, we replace $(d_{i}^{(j)},v_{i}^{(j)})$ by

$\displaystyle(d_{i}^{(j)},v_{i}^{(j)})\mapsto\begin{cases}(d_{l}^{(j)},v_{l}^{(j)}),&\mbox{
if }\mu_{i}=v_{l}^{(j)}/d_{l}^{(j)}\mbox{ for some }l,\\\ (0,0),&\mbox{
otherwise}.\end{cases}$

The subcategory (3.17) of (3.12) is unchanged under the above replacement.
Therefore we may assume that $a=b=k$ and, by setting
$(d_{i},v_{i})=(d_{i}^{(1)},v_{i}^{(1)})+(d_{i}^{(2)},v_{i}^{(2)})$, we have

$\displaystyle\frac{v_{i}}{d_{i}}=\frac{v_{i}^{(j)}}{d_{i}^{(j)}},\
\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}}.$

Here the first identity holds whenever $(d_{i}^{(j)},v_{i}^{(j)})\neq(0,0)$.

Let us take decompositions $V^{(j)}=\oplus_{i=1}^{k}V_{i}^{(j)}$ with $\dim
V_{i}^{(j)}=d_{i}^{(j)}$ and a cocharacter as in (3.15)

(3.18) $\displaystyle\lambda\colon\mathbb{C}^{\ast}\to GL(V^{(1)})\times
GL(V^{(2)})$

which acts on $V_{i}^{(j)}$ with weight $k+1-i$. The diagrams of attracting
loci with respect to $\lambda$ give the commutative diagram, see [Todc,
Proposition 2.5]:

Here, the vertical arrows are categorical Hall products determined by
$\lambda$, $p_{i}:=p_{i}^{(1)}+p_{i}^{(2)}$, the bottom horizontal arrow is
(3.11), and the top horizontal arrow is the composition of the Knörrer
periodicity equivalences with the tensor product of (a shift of)

$\displaystyle\det\left((U^{\lambda>0})^{\vee}\right)=\bigotimes_{i=1}^{k}\left((\det
V_{i}^{(1)})^{\sum_{i>j}d_{j}^{(2)}-\sum_{j>i}d_{j}^{(2)}}\otimes(\det
V_{i}^{(2)})^{\sum_{i>j}d_{j}^{(1)}-\sum_{j>i}d_{j}^{(1)}}\right).$

It follows that the functor (3.11) sends (3.17) to

(3.19)
$\displaystyle\ast_{i=1}^{k}\mathbb{S}_{p_{i}}(d_{i})_{v_{i}+d_{i}(\sum_{i>j}d_{j}-\sum_{j>i}d_{j})}\subset\mathrm{MF}(\mathcal{X}_{p}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d}).$

Let us take a decomposition

$w=v_{1}+\cdots+v_{k}=w_{1}+\cdots+w_{k},$

where $w_{i}$ is given in (3.14). Then by Lemma 3.6, the subcategory (3.19)
for $k\geqslant 2$ is right orthogonal to $\mathbb{S}_{p}(d)_{w}$. Together
with the semiorthogonal decomposition $\mathrm{S}$ with summands (3.17), we
conclude that the functor (3.12) has dense image. Indeed, for an object
$A\in\mathbb{S}_{p}(d)_{w}$, there is a distinguished triangle

(3.20) $\displaystyle A_{1}\xrightarrow{\alpha}A\xrightarrow{\beta}A_{2}$

where $A_{1}$ is a direct summand of an object $\Psi(B_{1})$ for some $B_{1}$
in the left hand side of (3.12) and $A_{2}$ is a direct summand of an object
$\Psi(B_{2})$ for some $B_{2}$ in (3.17) for $a=b\geqslant 2$. Then
$\Psi(B^{\prime})$ is an object of (3.19) for $k\geqslant 2$, hence $\beta$ is
a zero map by Lemma 3.6. So $\alpha$ is an isomorphism, hence the functor
(3.12) has dense image. ∎

We have postponed several lemmas, which are given below:

######  Lemma 3.2.

The $\operatorname{Ext}$-group $\operatorname{Ext}_{Q}^{1}(R,R)$ is computed
as (3.7).

###### Proof.

By the Euler pairing computation, we have $\chi_{Q}(R^{(i)},R^{(j)})=-2$.
Since $\operatorname{Hom}(R^{(i)},R^{(j)})=\mathbb{C}\delta_{ij}$, we have

$\displaystyle\operatorname{Ext}_{Q}^{1}(R^{(i)},R^{(j)})=\begin{cases}\mathbb{C}^{2},&i\neq
j,\\\ \mathbb{C}^{3},&i=j.\end{cases}$

Therefore (3.7) holds. ∎

######  Lemma 3.3.

By replacing the isomorphism (3.6) if necessary, the identity (3.10) holds.

###### Proof.

We may assume $x^{(1)}=(0,0,1)$ and $x^{(2)}=(0,0,0)$. We write an element of
$\operatorname{Ext}_{Q}^{1}(R,R)$ as

(3.21)
$\displaystyle(X^{(1)},Y^{(1)},Z^{(1)},X^{(2)},Y^{(2)},Z^{(2)},A^{(12)},A^{(21)},B^{(12)},B^{(21)}),$

where $X^{(i)}\in\mathrm{End}(V^{(i)})$ and
$A^{(ij)}\in\operatorname{Hom}(V^{(i)},V^{(j)})$. We have the morphism of
algebraic stacks

$\displaystyle\nu\colon\widehat{\operatorname{Ext}}_{Q}^{1}(R,R)/G_{p}\to\mathcal{X}_{p}(d)$

which sends (3.21) to

$\displaystyle\left(\begin{pmatrix}X^{(1)}&A^{(12)}\\\ A^{(21)}&X^{(2)}\\\
\end{pmatrix},\ \begin{pmatrix}Y^{(1)}&B^{(12)}\\\ B^{(21)}&Y^{(2)}\\\
\end{pmatrix},\ \begin{pmatrix}Z^{(1)}+I&0\\\ 0&Z^{(2)}\\\
\end{pmatrix}\right)\in\mathfrak{g}^{\oplus 3}.$

Indeed, the above correspondence is $GL(V)$-equivariant using the embedding
$G_{p}\subset GL(V)$, so it determines a morphism $\nu$. Note that $\nu(0)$
corresponds to the polystable $Q$-representation $R=(V^{(1)}\otimes
R^{(1)})\oplus(V^{(2)}\otimes R^{(2)})$, where $R^{(i)}$ corresponds to
$\mathcal{O}_{x^{(i)}}$.

We now explain that the morphism $\nu$ is étale at $\nu(0)$. The tangent
complex of $\mathcal{X}(d)$ at $\nu(0)$ is

$\displaystyle\mathbb{T}_{\mathcal{X}(d)}|_{\nu(0)}=\left(\mathrm{End}(V)\to\mathrm{End}(V)^{\oplus
3}\right),\ \alpha\mapsto(0,0,[\alpha,u]),\ u=\begin{pmatrix}I&0\\\ 0&0\\\
\end{pmatrix}.$

The kernel of the above map is
$\mathrm{End}(V^{(1)})\oplus\mathrm{End}(V^{(2)})$, and the cokernel is
$\operatorname{Ext}_{Q}^{1}(R,R)$, so the morphism $\nu$ induces a quasi-
isomorphism on tangent complexes at $\nu(0)$.

A straightforward computation shows that

$\displaystyle\nu^{\ast}\mathop{\mathrm{Tr}}\nolimits W_{d}=$
$\displaystyle\mathrm{Tr}(Z^{(1)}[X^{(1)},Y^{(1)}])+\mathrm{Tr}(Z^{(2)}[X^{(2)},Y^{(2)}])$
$\displaystyle+\mathrm{Tr}(A^{(12)}(B^{(21)}+B^{(21)}Z^{(1)}-Z^{(2)}B^{(21)}))$
$\displaystyle+\mathrm{Tr}(B^{(12)}(Z^{(2)}A^{(21)}-A^{(21)}Z^{(1)}-A^{(21)})).$

By the following $G_{p}$-equivariant variable change

$\displaystyle A^{(21)}\mapsto Z^{(2)}A^{(21)}-A^{(21)}Z^{(1)}-A^{(21)},\
B^{(21)}\mapsto B^{(21)}+B^{(21)}Z^{(1)}-Z^{(2)}B^{(21)},$

we obtain the identity (3.10). ∎

######  Lemma 3.4.

The equivalence (3.11) restricts to the functor (3.12).

###### Proof.

(cf. the proof of [KT21, Theorem 2.7]) For a cocharacter
$\lambda\colon\mathbb{C}^{\ast}\to T(d)$, let

$\displaystyle
n_{\lambda,p}:=\left\langle\lambda,\mathbb{L}^{\lambda>0}_{\mathcal{X}_{p}(d)}\big{|}_{0}\right\rangle,\
n_{\lambda,p}^{\prime}:=\left\langle\lambda,\mathbb{L}^{\lambda>0}_{\mathcal{X}_{p^{(1)}}(d^{(1)})\times\mathcal{X}_{p^{(2)}}(d^{(2)})}\big{|}_{0}\right\rangle.$

Define the sets of weights

$\displaystyle\nabla_{p,w}$ $\displaystyle:=\left\\{\chi\in
M_{\mathbb{R}}\mathrel{}\middle|\mathrel{}-\frac{1}{2}n_{\lambda,p}\leqslant\langle\lambda,\chi\rangle\leqslant\frac{1}{2}n_{\lambda,p}\mbox{
for all }\lambda\right\\}+w\tau_{d},$ $\displaystyle\nabla^{\prime}_{p,w}$
$\displaystyle:=\left\\{\chi\in
M_{\mathbb{R}}\mathrel{}\middle|\mathrel{}-\frac{1}{2}n^{\prime}_{\lambda,p}\leqslant\langle\lambda,\chi\rangle\leqslant\frac{1}{2}n^{\prime}_{\lambda,p}\mbox{
for all }\lambda\right\\}+w\tau_{d}.$

Then an object in the right (resp. left) hand side of (3.11) lies in the right
(resp. left) hand side of (3.12) if and only if its $T(d)$-weights are
contained in $\nabla_{p,w}$ (resp. $\nabla_{p,w}^{\prime}$), see [HLS20, Lemma
2.9]. Let $\mathcal{E}$ be an object in the left hand side of (3.12). Using a
Koszul resolution, the complex $\Psi(\mathcal{E})$ is generated by
$p^{\ast}(\mathcal{E})\otimes\wedge^{\bullet}U$. Let $\chi$ be a $T(d)$-weight
of $p^{\ast}(\mathcal{E})\otimes\wedge^{\bullet}U$. We then have that

$\displaystyle-\frac{1}{2}n_{\lambda,p}^{\prime}+\langle\lambda,U^{\lambda<0}\rangle\leqslant\langle\lambda,\chi\rangle\leqslant\frac{1}{2}n_{\lambda,p}^{\prime}+\langle\lambda,U^{\lambda>0}\rangle.$

Since $U$ is self-dual, we have
$\langle\lambda,U^{\lambda<0}\rangle=-\langle\lambda,U^{\lambda>0}\rangle$.
Moreover, from (3.7), it is easy to see that

$\displaystyle\frac{1}{2}n_{\lambda,p}^{\prime}+\langle\lambda,U^{\lambda>0}\rangle=\frac{1}{2}n_{\lambda,p}.$

Therefore $\Psi(\mathcal{E})$ is an object of the right hand side of (3.12). ∎

######  Lemma 3.5.

If a $T(d)$-weight $\chi$ satisfies (3.8), then

(3.22) $\displaystyle\chi+\rho\in\mathbf{W}(d)_{w}.$

Conversely if a $GL(V)$-dominant $T(d)$-weight $\chi$ satisfies (3.22), then
it satisfies (3.8).

###### Proof.

If a $T(d)$-weight $\chi$ satisfies (3.8), it is written as

(3.23)
$\displaystyle\chi+\rho_{p}=\sum_{i,j,a}c_{ij}^{(a)}(\beta_{i}^{(a)}-\beta_{j}^{(a)})+\sum_{i,j,a\neq
b}c_{ij}^{(ab)}(\beta_{i}^{(a)}-\beta_{j}^{(b)})$

where $0\leqslant c_{ij}^{(a)}\leqslant 3/2$ and $0\leqslant
c_{ij}^{(ab)}\leqslant 1$. Since we have

(3.24)
$\displaystyle\rho-\rho_{p}=\frac{1}{2}\sum_{i,j,a<b}(\beta_{i}^{(b)}-\beta_{j}^{(a)}),$

the weight $\chi+\rho$ satisfies (3.22). Conversely, if $\chi$ is a
$GL(V)$-dominant $T(d)$-weight satisfying (3.22), from [PTa, Proposition 3.5]
it is written as

$\displaystyle\chi+\rho=\sum_{i>j}c_{ij}(\beta_{i}-\beta_{j})$

for $0\leqslant c_{ij}\leqslant 3/2$. Therefore from (3.24), the weight
$\chi+\rho_{p}$ is written as (3.23). ∎

######  Lemma 3.6.

The subcategory (3.19) for $k\geqslant 2$ is right orthogonal to
$\mathbb{S}_{p}(d)_{w}$ for $w=w_{1}+\cdots+w_{k}$, were $w_{i}$ is given in
(3.14).

###### Proof.

Recall that $w_{i}=v_{i}+d_{i}(\sum_{i>j}d_{j}-\sum_{j>i}d_{j})$. Choose
weights $\psi_{i}\in\textbf{W}_{p_{i}}(d_{i})_{0}$ for $1\leqslant i\leqslant
k$. From the proof of semiorthogonality in [Pădc, Proposition 4.3], it is
enough to show that the $r$-invariant of the weight

(3.25)
$\displaystyle\sum_{i=1}^{k}\psi_{i}+\sum_{i=1}^{k}w_{i}\tau_{d_{i}}-\sum_{i=1}^{k}\rho_{p_{i}}+\rho_{p}$

with respect to the polytope $\textbf{W}_{p}(d)$ is bigger than $1/2$. Here,
the polytope $\textbf{W}_{p}(d)$ is defined as follows:

$\displaystyle\textbf{W}_{p}(d):=\frac{1}{2}\mathrm{sum}[0,\beta]+\mathbb{R}\tau_{d},$

where the sum is after all weights in $\operatorname{Ext}_{Q}^{1}(R,R)$.

Suppose the contrary, i.e. the weight (3.25) lies in $\textbf{W}_{p}(d)_{w}$.
Let $\lambda$ be the cocharacter as in (3.18), and write it as
$\lambda=(\lambda^{(1)},\lambda^{(2)})$, where $\lambda^{(j)}$ is the
cocharacter of $GL(V^{(j)})$. We set $\mathfrak{g}=\mathrm{End}(V)$ and
$\mathfrak{g}^{(j)}=\mathrm{End}(V^{(j)})$. The weight (3.25) is written as

$\displaystyle\sum_{i=1}^{k}\psi_{i}+\sum_{i=1}^{k}v_{i}\tau_{d_{i}}-\mathfrak{g}^{\lambda>0}-\frac{1}{2}(\mathfrak{g}^{(1)})^{\lambda^{(1)}>0}-\frac{1}{2}(\mathfrak{g}^{(2)})^{\lambda^{(2)}>0}.$

By the assumption, the above weight is an element of $\textbf{W}_{p}(d)_{w}$.
Then using an argument as in the proof of Lemma 3.5, we have that

(3.26)
$\displaystyle\sum_{i=1}^{k}\psi_{i}+\sum_{i=1}^{k}v_{i}\tau_{d_{i}}-\frac{3}{2}\mathfrak{g}^{\lambda>0}\in\textbf{W}(d)_{w}.$

Note that we have

$\displaystyle\left\langle\lambda,\sum_{i=1}^{k}\psi_{i}+\sum_{i=1}^{k}v_{i}\tau_{d_{i}}-\frac{3}{2}\mathfrak{g}^{\lambda>0}-w\tau_{d}\right\rangle=\sum_{i=1}^{k}(k+1-i)d_{i}\left(\frac{v_{i}}{d_{i}}-\frac{w}{d}\right)-\left\langle\lambda,\frac{3}{2}\mathfrak{g}^{\lambda>0}\right\rangle.$

For $1\leqslant i\leqslant k$, define

$\tilde{v}_{i}:=d_{i}\left(\frac{v_{i}}{d_{i}}-\frac{w}{d}\right).$

Then $\tilde{v}_{1}+\cdots+\tilde{v}_{k}=0$ and
$\tilde{v}_{1}+\cdots+\tilde{v}_{l}<0$ for $1\leqslant l<k$. Therefore

$\displaystyle\sum_{i=1}^{k}(k+1-i)d_{i}\left(\frac{v_{i}}{d_{i}}-\frac{w}{d}\right)=\sum_{l=1}^{k}\left(\sum_{i=1}^{l}\tilde{v}_{i}\right)<0.$

It follows that

(3.27)
$\displaystyle\left\langle\lambda,\sum_{i=1}^{k}\psi_{i}+\sum_{i=1}^{k}v_{i}\tau_{d_{i}}-\frac{3}{2}\mathfrak{g}^{\lambda>0}-w\tau_{d}\right\rangle<-\left\langle\lambda,\frac{3}{2}\mathfrak{g}^{\lambda>0}\right\rangle.$

On the other hand, we claim that for any weight $\chi\in\textbf{W}(d)_{0}$, we
have that

(3.28)
$\langle\lambda,\chi\rangle\geqslant-\left\langle\lambda,\frac{3}{2}\mathfrak{g}^{\lambda>0}\right\rangle.$

We thus obtain a contradiction with (3.27) and (3.26). To prove (3.28), write
$\chi=\sum_{i,j}c_{ij}(\beta_{i}-\beta_{j})$ with $0\leqslant c_{ij}\leqslant
3/2$. Then

$\displaystyle\langle\lambda,\chi\rangle\geqslant\sum_{\langle\lambda,\beta_{i}-\beta_{j}\rangle<0}c_{ij}\langle\lambda,\beta_{i}-\beta_{j}\rangle\geqslant\frac{3}{2}\sum_{\langle\lambda,\beta_{i}-\beta_{j}\rangle<0}\langle\lambda,\beta_{i}-\beta_{j}\rangle=-\left\langle\lambda,\frac{3}{2}\mathfrak{g}^{\lambda>0}\right\rangle.$

∎

##  4\. Integral generator of equivariant K-theory of quasi-BPS categories

Recall the graded quasi-BPS category $\mathbb{S}^{\rm{gr}}(d)_{v}$, which is
equivalent via Koszul duality to

$\mathbb{T}(d)_{v}\subset D^{b}(\mathscr{C}(d)).$

In [PTa], we proved that the monomials
$[\mathcal{E}_{d_{1},v_{1}}]\ast\cdots\ast[\mathcal{E}_{d_{k},v_{k}}]$ for
$v_{i}/d_{i}=v/d$ give a basis of the $\mathbb{F}$-vector space
$K_{T}(\mathbb{T}(d)_{v})\otimes_{\mathbb{K}}\mathbb{F}$. In this section, we
consider the torsion free equivariant K-theory defined by

$\displaystyle
K_{T}(\mathbb{T}(d)_{v})^{\prime}:=K_{T}(\mathbb{T}(d)_{v})/(\mathbb{K}\text{-torsion}).$

It is conjectured by Schiffmann–Vasserot [SV13, Conjecture 7.13] that
$K_{T}(\mathscr{C}(d))$ is torsion free as a $\mathbb{K}$-module, so
conjecturally $K_{T}(\mathbb{T}(d)_{v})^{\prime}$ is isomorphic to
$K_{T}(\mathbb{T}(d)_{v})$. We compute $K_{T}(\mathbb{T}(d)_{v})^{\prime}$
when $\gcd(d,v)=1$, and when $(d,v)=(2,0)$. Theorem 3.1 plays a key role in
the case of $\gcd(d,v)=1$.

###  4.1. Sheaves supported over the small diagonal

Let

$\displaystyle\Omega_{\mathscr{C}(d)}[-1]:=\operatorname{Spec}\mathrm{Sym}\left(\mathbb{T}_{\mathscr{C}(d)}[1]\right)$

be the $(-1)$-shifted cotangent of $\mathscr{C}(d)$. An object $\mathcal{E}\in
D^{b}(\mathscr{C}(d))$ has singular support [AG15]:

$\displaystyle\mathrm{Supp}^{\rm{sg}}(\mathcal{E})\subset(\Omega_{\mathscr{C}(d)}[-1])^{\rm{cl}}.$

We identify the right hand side with
$\mathrm{Crit}(\mathop{\mathrm{Tr}}\nolimits W_{d})\subset\mathcal{X}(d)$,
which is isomorphic to the moduli stack of zero-dimensional coherent sheaves
on $\mathbb{C}^{3}$ of length $d$. Under the Koszul duality equivalence
(2.24), we have by [Toda, Proposition 2.3.9] that

$\displaystyle\mathrm{Supp}^{\rm{sg}}(\mathcal{E})=\mathrm{Supp}(\Phi(\mathcal{E})).$

In the following lemma, we show that the K-theory class of an object in
$D^{b}_{T}(\mathscr{C}(d))$ with singular support contained in
$\pi^{-1}(\Delta)$ has a certain divisibility property. The proof is inspired
by the proof of wheel conditions in [Zha, Theorem 2.9, Corollary 2.10], [Nega,
Proposition 2.11]:

######  Lemma 4.1.

For any $\mathcal{E}\in D^{b}_{T}(\mathscr{C}(d))$ whose singular support is
contained in $\pi^{-1}(\Delta)$, the element $i_{\ast}[\mathcal{E}]\in
K_{T}(\mathcal{Y}(d))=\mathbb{K}[z_{1}^{\pm},\ldots,z_{d}^{\pm
1}]^{\mathfrak{S}_{d}}$ is divisible by

(4.1)
$\displaystyle(q_{1}-1)^{d-1}(q_{2}-1)^{d-1}(q_{1}q_{2}-1)^{d-1}\in\mathbb{K}.$

###### Proof.

By [PTa, Lemma 4.9], it is enough to show that, for
$\mathcal{F}\in\mathrm{MF}_{T}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits
W)$ supported on $\pi^{-1}(\Delta)$, its image under the forget-the-potential
map (1.8):

(4.2)
$\displaystyle\Theta([\mathcal{F}]):=[\mathcal{F}^{0}]-[\mathcal{F}^{1}]\in
K_{T}(\mathrm{MF}(\mathcal{X}(d),0))=\mathbb{K}[z_{1}^{\pm
1},\ldots,z_{d}^{\pm d}]^{\mathfrak{S}_{d}}$

is divisible by $(q_{1}-1)^{d-1}(q_{2}-1)^{d-1}(q_{1}q_{2}-1)^{d-1}$. We
consider the morphism

$\displaystyle h\colon\mathbb{C}^{d-1}\setminus\\{0\\}$
$\displaystyle\to\mathrm{Crit}(\mathop{\mathrm{Tr}}\nolimits
W_{d})\subset\mathfrak{g}^{\oplus 3}$ $\displaystyle(t_{1},\ldots,t_{d-1})$
$\displaystyle\mapsto\left(0,0,(t_{1},\ldots,t_{d-1},0)\right),$

where $(t_{1},\ldots,t_{d-1},0)$ is the diagonal matrix. Let
$T=(\mathbb{C}^{\ast})^{2}$ act on $\mathbb{C}^{d-1}$ with weight
$q_{1}^{-1}q_{2}^{-1}$ and let the maximal torus $T(d)\subset GL(d)$ act on
$\mathbb{C}^{d-1}$ trivially. Then the above morphism $h$ is $T\times
T(d)$-equivariant, so it induces a morphism

$\displaystyle h\colon(\mathbb{C}^{d-1}\setminus\\{0\\})/(T\times
T(d))\to\mathfrak{g}^{\oplus 3}/(T\times T(d))\to\mathfrak{g}^{\oplus
3}/(T\times GL(d)).$

By the localization sequence

$\displaystyle K_{T\times T(d)}(\\{0\\})\to K_{T\times
T(d)}(\mathbb{C}^{d-1})\to K_{T\times
T(d)}(\mathbb{C}^{d-1}\setminus\\{0\\})\to 0,$

we have an isomorphism

$\displaystyle K_{T\times
T(d)}(\mathbb{C}^{d-1}\setminus\\{0\\})\cong\mathbb{K}[z^{\pm
1}_{1},\ldots,z^{\pm 1}_{d}]/(1-q_{1}^{-1}q_{2}^{-1})^{d-1}.$

Note that we have

$\displaystyle\pi\circ
h(t_{1},\ldots,t_{d-1})=\\{(0,0,t_{1}),\ldots,(0,0,t_{d-1}),(0,0,0)\\}\in\mathrm{Sym}^{d}(\mathbb{C}^{3}),$

in particular the image of $h$ does not intersect
$\pi^{-1}(\Delta)\subset\mathrm{Crit}(\mathop{\mathrm{Tr}}\nolimits W_{d})$.
We thus have that

$h^{\ast}(\Theta([\mathcal{F}]))=0\text{ in }K_{T\times
T(d)}(\mathbb{C}^{d-1}\setminus\\{0\\}).$

Therefore the element (4.2) is divisible by $(q_{1}q_{2}-1)^{d-1}$. By
replacing $h$ with

$\displaystyle(t_{1},\ldots,t_{d-1})$
$\displaystyle\mapsto(0,(t_{1},\ldots,t_{d-1},0),0),$
$\displaystyle(t_{1},\ldots,t_{d-1})$
$\displaystyle\mapsto((t_{1},\ldots,t_{d-1},0),0,0),$

the element (4.2) is also divisible by $(q_{1}-1)^{d-1}$, $(q_{2}-1)^{d-1}$,
respectively. ∎

By combining the above lemma with Theorem 3.1, we obtain the following:

######  Corollary 4.2.

For any object $\mathcal{E}\in\mathbb{T}_{T}(d)_{v}$ with $\gcd(d,v)=1$, the
element

$\displaystyle i_{\ast}[\mathcal{E}]\in
K_{T}(\mathcal{Y}(d))=\mathbb{K}[z_{1}^{\pm 1},\ldots,z_{d}^{\pm
1}]^{\mathfrak{S}_{d}}$

is divisible by (4.1). In particular, the element
$i_{\ast}[\mathcal{E}_{d,v}]$ in (2.40) for $\gcd(d,v)=1$ is divisible by
(4.1).

######  Remark 4.3.

It is not clear from the expression (2.40) that $i_{\ast}[\mathcal{E}_{d,v}]$
is divisible by (4.1) when $\gcd(d,v)=1$. Further, the condition $\gcd(d,v)=1$
is necessary for the above divisibility. Indeed, for $d=2$, a direct
computation shows that

$\displaystyle i_{\ast}[\mathcal{E}_{2,0}]$
$\displaystyle=(1-q_{1}^{-1})(1-q_{2}^{-1})(1-q_{1}^{-1}-q_{2}^{-1}-q_{1}^{-1}q_{2}^{-1}+z_{1}^{-1}z_{2}+z_{1}z_{2}^{-1}),$
$\displaystyle i_{\ast}[\mathcal{E}_{2,1}]$
$\displaystyle=(1-q_{1}^{-1})(1-q_{2}^{-1})(1-q_{1}^{-1}q_{2}^{-1})(z_{1}+z_{2}).$

The element $i_{\ast}[\mathcal{E}_{2,0}]$ is not divisible by (4.1). In
particular, the singular support of $\mathcal{E}_{2,0}$ is not included in
$\pi^{-1}(\Delta)$.

###  4.2. Integral generator for the coprime case

Recall that $K_{T}(\mathbb{T}(d)_{v})$ is expected to be torsion free as a
$\mathbb{K}$-module. We also expect that $K_{T}(\mathbb{T}(d)_{v})$ is freely
generated by $[\mathcal{E}_{d,v}]$ if $\gcd(d,v)=1$. The following
proposition, which is an application of Corollary 4.2, gives evidence towards
this expectation.

######  Theorem 4.4.

Consider a pair $(d,v)\in\mathbb{N}\times\mathbb{Z}$ with $\gcd(d,v)=1$. There
is an isomorhism of $\mathbb{K}$-modules:

$\displaystyle
K_{T}(\mathbb{T}(d)_{v})^{\prime}\cong\mathbb{K}[\mathcal{E}_{d,v}].$

###### Proof.

By the isomorphism (2.32), the $\mathbb{K}$-module
$K_{T}(\mathbb{T}(d)_{v})^{\prime}$ is isomorphic to the image of

$\displaystyle i_{\ast}\colon K_{T}(\mathbb{T}(d)_{v})\to\mathbb{K}[z_{1}^{\pm
1},\ldots,z_{d}^{\pm}]^{\mathfrak{S}_{d}}.$

It is enough to show that the image of the above morphism is generated by
$i_{\ast}[\mathcal{E}_{d,v}]$ as a $\mathbb{K}$-module. By Corollary 4.2, we
have

$\displaystyle\mathrm{Im}(i_{\ast})\subset(q_{1}-1)^{d-1}(q_{2}-1)^{d-1}(q_{1}q_{2}-1)^{d-1}\mathbb{K}[z_{1}^{\pm
1},\ldots,z_{d}^{\pm 1}]^{\mathfrak{S}_{d}}.$

Moreover, $\mathrm{Im}(i_{\ast})\otimes_{\mathbb{K}}\mathbb{F}$ is generated
by $i_{\ast}[\mathcal{E}_{d,v}]$ over $\mathbb{F}$ by [PTa, Theorem 4.12]. It
is thus enough to show that $i_{\ast}[\mathcal{E}_{d,v}]$ is written as

$\displaystyle
i_{\ast}[\mathcal{E}_{d,v}]=(q_{1}-1)^{d-1}(q_{2}-1)^{d-1}(q_{1}q_{2}-1)^{d-1}\cdot
E,$

where $E$ is not divisible by a non-unit element in $\mathbb{K}$.

By (2.40), we have that

(4.3) $\displaystyle\mathrm{Sym}\left(\frac{z_{1}^{m_{1}}\ldots
z_{d}^{m_{d}}}{(1-q^{-1}z_{1}^{-1}z_{2})\cdots(1-q^{-1}z_{d-1}^{-1}z_{d})}\cdot\prod_{j>i}\xi(z_{i}z_{j}^{-1})\right)=(q_{1}q_{2}-1)^{d-1}\cdot
E.$

By setting

(4.4)
$f_{d}(z_{1},\ldots,z_{d}):=\prod_{i=1}^{d-1}(z_{i+1}-q_{1}^{-1}z_{i})(z_{i+1}-q_{2}^{-1}z_{i})\cdot\\\
\prod_{j>i+1}(z_{j}-q_{1}^{-1}z_{i})(z_{j}-q_{2}^{-1}z_{i})(z_{j}-qz_{i}),$

we can write (4.3) as

(4.5) $\displaystyle(-q)^{\frac{1}{2}(d-1)(d-2)}(z_{1}\cdots
z_{d})^{2-d}\cdot\mathrm{Sym}\left(\frac{z_{1}^{m_{1}}\cdots
z_{d-1}^{m_{d-1}}z_{d}^{m_{d}-1}f_{d}(z_{1},\ldots,z_{d})}{\prod_{j>i}(z_{j}-z_{i})}\right).$

Plug in $z_{i}=q_{1}^{i}$ for $1\leqslant i\leqslant d$ in the formula (4.5).
The only non-zero term in the sum above corresponds to the identity
permutation. The factors of this term not in $\mathbb{K}^{*}$ divide

(4.6) $q_{1}^{a+1}-1,\,q_{1}^{a}-q_{2}^{-1},\text{ or }q_{1}^{a}-q_{2}\text{
for some }a\geqslant 1.$

Next plug in $z_{i}=q_{2}^{i}$ for $1\leqslant i\leqslant d$ in the formula
(4.5). The only non-zero term in the sum of (4.5) corresponds to the identity
permutation. The factors of this term not in $\mathbb{K}^{*}$ divide

(4.7) $q_{2}^{a+1}-1,\,q_{2}^{a}-q_{1}^{-1},\text{ or }q_{2}^{a}-q_{1}\text{
for some }a\geqslant 1.$

The only factors which divide terms in both sets (4.6) and (4.7) are
$q_{1}-q_{2}$ and $q_{1}q_{2}-1$. The factor $q_{1}q_{2}-1$ appears with
multiplicity $d-1$ as it corresponds to $z_{i+1}-q_{2}^{-1}z_{i}$ for
$1\leqslant i\leqslant d-1$.

It suffices to show that $q_{1}-q_{2}$ does not divide (4.5). We will be using
computations from [Negb, Section 2]. Note that $q_{1},q_{2},q$ from our paper
correspond to $q_{1}^{-1},q_{2}^{-1},q^{-1}$ in loc. cit. Further, the weight
$v\in\mathbb{Z}$ corresponds to $k\in\mathbb{Z}$ in loc. cit. By [Negb,
Equation (2.35)], the equality $P_{d,k}=E_{d,k}$ for $\gcd(d,k)=1$ in loc.
cit., see [Negb, Equations (2.6) and (2.35)], and the isomorphism between
shuffle algebras
$\mathcal{S}_{\mathbb{F}}\xrightarrow{\sim}\mathcal{S}^{\prime}_{\mathbb{F}}$,
we can write

$(q_{1}q_{2}-1)^{d-1}\cdot E=y+t\text{ in
}\mathcal{S}\otimes_{\mathbb{K}}\mathbb{K}\left[\frac{1}{1-q_{2}^{-1}},\frac{1}{1-q^{-1}}\right]$

for $t$ a $\mathbb{K}$-torsion element and

(4.8)
$y:=\frac{(1-q_{2}^{-1})(1-q)^{v}}{(1-q_{2}^{-1})^{v}(1-q^{-1})}\cdot\mathrm{Sym}\left(\frac{z_{1}^{m_{1}}\ldots
z_{d}^{m_{d}}}{(1-q_{2}z_{1}^{-1}z_{2})\cdots(1-q_{2}z_{d-1}^{-1}z_{d})}\cdot\prod_{j>i}\xi(z_{i}z_{j}^{-1})\right).$

It suffices to show that $q_{1}-q_{2}$ does not divide $y$. By setting

(4.9)
$g_{d}(z_{1},\ldots,z_{d}):=\prod_{i=1}^{d-1}(z_{i+1}-q_{1}^{-1}z_{i})(z_{i+1}-qz_{i})\cdot\\\
\prod_{j>i+1}(z_{j}-q_{1}^{-1}z_{i})(z_{j}-q_{2}^{-1}z_{i})(z_{j}-qz_{i}),$

we can write $y$ as

(4.10) $\displaystyle(-q_{2})^{-\frac{1}{2}(d-1)(d-2)}(z_{1}\cdots
z_{d})^{2-d}\cdot\mathrm{Sym}\left(\frac{z_{1}^{m_{1}}\ldots
z_{d-1}^{m_{d-1}}z_{d}^{m_{d}-1}g_{d}(z_{1},\ldots,z_{d})}{\prod_{j>i}(z_{j}-z_{i})}\right).$

It suffices to show that $q_{1}-q_{2}$ does not divide (4.10). Let
$z_{i}=q^{-i}$ for $1\leqslant i\leqslant d$. The only non-zero term in the
sum of (4.10) corresponds to the identity permutation. The factors of this
term not in $\mathbb{K}^{*}$ divide

$q^{-a-1}-1,q^{-a}-q_{1}^{-1},q^{-a}-q_{1}\text{ for some }a\geqslant 1.$

None of these polynomials is divisible by $q_{1}-q_{2}$, and the conclusion
thus follows. ∎

###  4.3. Integral generator for $K_{T}(\mathbb{T}(2)_{0})^{\prime}$

The computation of $K_{T}(\mathbb{T}(d)_{v})^{\prime}$ is more subtle when
$\gcd(d,v)>1$, since the monomials
$[\mathcal{E}_{d_{1},v_{1}}]\ast\cdots\ast[\mathcal{E}_{d_{k},v_{k}}]$ for
$v_{i}/d_{i}=v/d$ do not generate it over $\mathbb{K}$, see Remark 4.8. We
need to find other objects giving $\mathbb{K}$-basis of
$K_{T}(\mathbb{T}(d)_{v})^{\prime}$. Here we give a computation for
$(d,v)=(2,0)$.

Let $V$ be a two-dimensional vector space and let
$\mathfrak{sl}\subset\mathfrak{g}=\mathrm{End}(V)$ be its traceless part. Note
that

$\displaystyle[\mathfrak{sl}]=1+z_{1}^{-1}z_{2}+z_{1}z_{2}^{-1}\in
K(BGL(2))=\mathbb{Z}[z_{1}^{\pm 1},z_{2}^{\pm 1}]^{\mathfrak{S}_{2}}.$

The structure sheaf of the classical truncation $\mathscr{C}(2)^{\rm{cl}}$
fits into the exact sequence

(4.11) $\displaystyle
0\to\mathcal{O}_{\mathcal{Y}(2)}(q_{1}^{-1}q_{2}^{-2})\oplus\mathcal{O}_{\mathcal{Y}(2)}(q_{1}^{-2}q_{2}^{-1})\stackrel{{\scriptstyle
A}}{{\to}}\mathfrak{sl}\otimes\mathcal{O}_{\mathcal{Y}(2)}(q_{1}^{-1}q_{2}^{-1})\stackrel{{\scriptstyle
B}}{{\to}}\mathcal{O}_{\mathcal{Y}(2)}\to\mathcal{O}_{\mathscr{C}(2)^{\rm{cl}}}\to
0.$

Here over $(X,Y)\in\mathfrak{g}^{\oplus 2}$, the maps $A$, $B$ are given by

$\displaystyle A|_{(X,Y)}=(2X-\mathop{\mathrm{Tr}}\nolimits X\cdot
I,2Y-\mathop{\mathrm{Tr}}\nolimits Y\cdot I),\
B|_{(X,Y)}(Z)=\mathop{\mathrm{Tr}}\nolimits(Z[X,Y]).$

We set $M_{1}:=\mathcal{O}_{\mathscr{C}(2)^{\rm{cl}}}(q_{1}q_{2})$. We also
define a coherent sheaf $M_{2}$ on $\mathscr{C}(2)^{\rm{cl}}$ by the exact
sequence

(4.12) $\displaystyle
0\to\mathcal{O}_{\mathcal{Y}(2)}(q_{1}^{-1}q_{2}^{-1})\stackrel{{\scriptstyle
B^{\vee}}}{{\to}}\mathfrak{sl}\otimes\mathcal{O}_{\mathcal{Y}(2)}\stackrel{{\scriptstyle
A^{\vee}}}{{\to}}\mathcal{O}_{\mathcal{Y}(2)}(q_{1})\oplus\mathcal{O}_{\mathcal{Y}(2)}(q_{2})\to
M_{2}\to 0.$

The sequences (4.11), (4.12) are the Eagon-Northcott complex and the
Buchsbaum-Rim complex associated with
$A^{\vee}\colon\mathfrak{sl}\otimes\mathcal{O}_{\mathcal{Y}(2)}\to\mathcal{O}_{\mathcal{Y}(2)}(q_{1})\oplus\mathcal{O}_{\mathcal{Y}(2)}(q_{2})$,
respectively. In particular they are exact, see [Eis95, Theorem A2.10]. From
the exact sequences (4.11), (4.12), we have $M_{1},M_{2}\in\mathbb{T}(2)_{0}$.

######  Proposition 4.5.

As a $\mathbb{K}$-module, we have

(4.13) $\displaystyle
K_{T}(\mathbb{T}(2)_{0})^{\prime}=\mathbb{K}[M_{1}]\oplus\mathbb{K}[M_{2}].$

###### Proof.

It is enough to show that the image of

(4.14) $\displaystyle i_{\ast}\colon
K_{T}(\mathbb{T}(2)_{0})\to\mathbb{K}[z_{1}^{\pm 1},z_{2}^{\pm
1}]^{\mathfrak{S}_{2}}$

is generated by $i_{\ast}M_{1}$, $i_{\ast}M_{2}$ and that
$i_{\ast}M_{1},i_{\ast}M_{2}$ are linearly independent over $\mathbb{F}$.
Below we omit $i_{\ast}$ from the notation $i_{\ast}M_{j}$. From the exact
sequences (4.11), (4.12), we have

(4.15) $\displaystyle[M_{1}]=q_{1}q_{2}+q_{1}^{-1}+q_{2}^{-1}-\mathfrak{sl},\
[M_{2}]=q_{1}^{-1}q_{2}^{-1}+q_{1}+q_{2}-\mathfrak{sl}.$

A direct computation shows that

(4.16) $\displaystyle[\mathcal{E}_{1,0}\ast\mathcal{E}_{1,0}]$
$\displaystyle=1+q_{1}^{-1}+q_{2}^{-1}+q_{1}^{-1}q_{2}^{-2}+q_{1}^{-2}q_{2}^{-1}+q_{1}^{-2}q_{2}^{-2}-2\mathfrak{sl}q_{1}^{-1}q_{2}^{-1}$
$\displaystyle=q_{1}^{-1}q_{2}^{-1}([M_{1}]+[M_{2}]),$
$\displaystyle[\mathcal{E}_{2,0}]$
$\displaystyle=(1-q_{1}^{-1})(1-q_{2}^{-1})(\mathfrak{sl}-q_{1}^{-1}-q_{2}^{-1}-q_{1}^{-1}q_{2}^{-1})$
$\displaystyle=(q_{1}^{-1}+q_{2}^{-1})[M_{1}]-(q_{1}^{-1}q_{2}^{-1}+1)[M_{2}].$

Therefore we have

(4.17)
$\displaystyle\mathbb{K}[\mathcal{E}_{2,0}]\oplus\mathbb{K}[\mathcal{E}_{1,0}\ast\mathcal{E}_{1,0}]\subset\mathbb{K}[M_{1}]\oplus\mathbb{K}[M_{2}].$

Since we have

(4.18)
$\displaystyle\det\begin{pmatrix}q_{1}^{-1}q_{2}^{-1}&q_{1}^{-1}q_{2}^{-1}\\\
q_{1}^{-1}+q_{2}^{-1}&-(q_{1}^{-1}q_{2}^{-1}+1)\\\
\end{pmatrix}=-q_{1}^{-1}q_{2}^{-1}(1+q_{1}^{-1})(1+q_{2}^{-1})\in\mathbb{K}\setminus\\{0\\},$

the embedding (4.17) is an isomorphism after taking
$\otimes_{\mathbb{K}}\mathbb{F}$. In particular, $[M_{1}]$, $[M_{2}]$ are
linearly independent over $\mathbb{F}$.

By the above argument, we have

$\displaystyle
K_{T}(\mathbb{T}(2)_{0})\otimes_{\mathbb{K}}{\mathbb{F}}=\mathbb{F}[\mathcal{E}_{2,0}]\oplus\mathbb{F}[\mathcal{E}_{1,0}\ast\mathcal{E}_{1,0}]=\mathbb{F}[M_{1}]\oplus\mathbb{F}[M_{2}],$

where the first isomorphism is proved in [PTa, Theorem 4.12]. It follows that
any element in the image of (4.14) is written as $a_{1}[M_{1}]+a_{2}[M_{2}]$
for $a_{1},a_{2}\in\mathbb{F}$. As it lies in $\mathbb{K}[z_{1}^{\pm
1},z_{2}^{\pm 1}]$, from (4.15) we have

$\displaystyle
a_{1}(q_{1}q_{2}+q_{1}^{-1}+q_{2}^{-1})+a_{2}(q_{1}^{-1}q_{2}^{-1}+q_{1}+q_{2})\in\mathbb{K},\
a_{1}+a_{2}\in\mathbb{K}.$

By solving the above equation, there exist $b,c\in\mathbb{K}$ such that

(4.19) $\displaystyle a_{1}=\frac{b}{\Delta},\ a_{2}=-\frac{b}{\Delta}+c$

where $\Delta$ is given by

$\displaystyle\Delta$
$\displaystyle:=q_{1}^{-1}+q_{2}^{-1}+q_{1}q_{2}-q_{1}-q_{2}-q_{1}^{-1}q_{2}^{-1}$
$\displaystyle=q_{1}^{-1}q_{2}^{-1}(q_{1}-1)(q_{2}-1)(q_{1}q_{2}-1).$

It is enough to show that $b$ is divisible by $\Delta$. If $a_{1},a_{2}$ are
given by (4.19), then $a_{1}[M_{1}]+a_{2}[M_{2}]=b+c[M_{2}]$. As we assumed
that $a_{1}[M_{1}]+a_{2}[M_{2}]$ lies in the image of (4.14), by the wheel
condition [Nega, Proposition 2.11] we have

(4.20)
$\displaystyle(b+c[M_{2}])\big{|}_{z_{i}=q_{1}^{-1}z_{j}=q_{1}^{-1}q_{2}^{-1}z_{k}}=(b+c[M_{2}])\big{|}_{z_{i}=q_{2}^{-1}z_{j}=q_{1}^{-1}q_{2}^{-1}z_{k}}=0$

unless $i=j=k$, see Remark 4.6. By setting $i=k=1$ and $j=2$, we obtain
$b|_{q_{1}q_{2}=1}=0$. Similarly we also obtain $b|_{q_{1}=1}=b|_{q_{2}=1}=0$,
so $b$ is divisible by $\Delta$. ∎

######  Remark 4.6.

In [Nega, Proposition 2.11], it is assumed that $i$, $j$, and $k$ are pairwise
distinct for the identity (4.20), but the proof in loc. cit. works unless
$i=j=k$. Indeed, using the notation in loc. cit., let $\phi_{e}=(x_{ij})$ for
$x_{bc}\neq 0$, $x_{ij}=0$ for $(i,j)\neq(b,c)$, $\phi_{e}^{\ast}=(y_{ij})$
for $y_{ab}\neq 0$, $y_{ij}=0$ for $(i,j)\neq(a,b)$. Then
$\phi_{e}^{\ast}\phi_{e}$ is concentrated on $(a,c)$ with value
$y_{ab}x_{bc}\neq 0$, $\phi_{e}\phi_{e}^{\ast}$ is concentrated on $(b,b)$
(either zero or non-zero). Therefore if $[\phi_{e},\phi_{e}^{\ast}]=0$ holds,
then we must have $a=b=c$. Unless $a=b=c$, we have $\mu_{n}^{-1}(0)\cap
V_{e}=\emptyset$ in the notation of the proof of [Nega, Proposition 2.11]. The
rest of the argument is verbatim.

######  Remark 4.7.

Note that

$\mathscr{C}(2)^{\rm{cl}}\cong[R/GL(2)]\times\mathbb{C}^{2},$

where $R$ is the determinantal variety of $(3\times 2)$-matrices, and
$M_{1}\oplus M_{2}$ gives a noncommutative resolution of
$R\times\mathbb{C}^{2}$ by [BLdB10, Theorem A].

######  Remark 4.8.

Since (4.18) is not invertible in $\mathbb{K}$, the inclusion (4.17) is not an
isomorphism if we do not take $\otimes_{\mathbb{K}}\mathbb{F}$.

######  Remark 4.9.

For $(d,v)\in\mathbb{N}\times\mathbb{Z}$ with $\gcd(d,v)=1$ and $n\geqslant
1$, let $P_{nd,nv}$ be defined as in [Neg14, Equation (1.2)]:

(4.21)
$P_{nd,nv}:=\frac{(q_{1}^{-1}-1)^{nd}(q_{2}^{-1}-1)^{nd}}{(q_{1}^{-n}-1)(q_{2}^{-n}-1)}\cdot\\\
\text{Sym}\left(\frac{\prod_{i=1}^{nd}z_{i}^{\big{\lfloor}\frac{iv}{d}\big{\rfloor}-\big{\lfloor}\frac{(i-1)v}{d}\big{\rfloor}}}{\prod_{i=1}^{nd-1}\left(1-q^{-1}z_{i+1}z_{i}^{-1}\right)}\sum_{s=0}^{n-1}q^{-s}\frac{z_{d(n-1)+1}\ldots
z_{d(n-s)+1}}{z_{d(n-1)}\ldots
z_{d(n-s)}}\prod_{i<j}\xi\left(\frac{z_{i}}{z_{j}}\right)\right).$

Then we have

(4.22) $\displaystyle
P_{2,0}=q_{1}^{-2}q_{2}^{-2}(q_{1}-1)(q_{2}-1)(q_{1}q_{2}-1)=q_{1}^{-1}q_{2}^{-1}(M_{1}-M_{2}).$

Together with (4.16), we have

(4.23) $\displaystyle\mathbb{K}[P_{1,0}\ast
P_{1,0}]\oplus\mathbb{K}[P_{2,0}]\subsetneq\mathbb{K}[M_{1}]\oplus\mathbb{K}[M_{2}]$

with cokernel $\mathbb{Z}/2$. The inclusion (4.23) is an isomorphism after
$\otimes_{\mathbb{Z}}\mathbb{Q}$.

More generally, we expect that

$\displaystyle
K_{T}(\mathbb{S}(nd)_{nv})_{\mathbb{Q}}^{\prime}=\bigoplus_{n_{1}+\cdots+n_{k}=n}\mathbb{K}_{\mathbb{Q}}[P_{n_{1}d,n_{1}v}\ast\cdots\ast
P_{n_{k}d,n_{k}v}].$

More details will be discussed in [PTb].

##  5\. The coproduct on quasi-BPS categories and K-theoretic BPS spaces

Recall that $Q$ is the quiver with one vertex and three edges $x,y,z$. Recall
the regular function (2.2) induced from the super-potential $W:=x[y,z]$. We
will denote by $\widetilde{m}$ the Hall product (2.5) of $(Q,W)$ and by $m$
the Hall product (2.17).

In Subsection 5.1, we define the coproduct $\widetilde{\Delta}$ for the
categories $\mathbb{S}^{\bullet}_{\ast}(d)_{w}$. In Subsection 5.2, we prove
the compatibility of the product and coproduct for the quasi-BPS categories
$\mathbb{S}^{\bullet}_{T}(d)_{w}$, see Theorem 5.2. In Subsection 5.3, we use
the Koszul equivalence to define the coproduct $\Delta$ for the categories
$\mathbb{T}_{\ast}(d)_{v}$ and to check the compatibility between the product
and the coproduct for the quasi-BPS categories $\mathbb{T}_{T}(d)_{v}.$

Recall the definition of $\mathcal{D}_{d,v}$ for
$(d,v)\in\mathbb{N}\times\mathbb{Z}$ for $\gcd(d,v)=1$. In Subsection 5.4, we
show that $K(\mathcal{D}_{d,v})_{\mathbb{F}}$ is isomorphic to the
$\mathbb{F}$-algebra of symmetric polynomials, see Proposition 5.11. Further,
we consider the space of primitive elements

$\mathrm{P}(nd,nv)\subset K_{T}(\mathbb{T}(nd)_{nv})$

which we regard as an analogue of cohomological BPS spaces in K-theory, see
Proposition 5.11 and the equality (1.5).

Note that the definition of Hall multiplication involves attracting stacks of
antidominant cocharacters. The definition of the coproduct is through
attracting stacks of dominant cocharacters. In this section, we will use
dominant cocharacters.

###  5.1. Preliminaries

Let $d\in\mathbb{N}$. Recall the stack of representation of $Q$ of dimension
$d$:

$\mathcal{X}(d):=R(d)/G(d):=\mathfrak{g}^{\oplus 3}/GL(d).$

In this section, we allow partitions which have terms equal to zero. For
$w\in\mathbb{Z}$, let $H_{d,w}$ be the set of partitions
$(d_{i},w_{i})_{i=1}^{k}$ of $(d,w)$ with $d_{i}\geqslant 1$ for
$i\in\\{1,\ldots,k\\}$ such that for
$v_{i}:=w_{i}-d_{i}\left(\sum_{i>j}d_{j}-\sum_{j>i}d_{j}\right)$ for
$i\in\\{1,\ldots,k\\}$, we have

(5.1) $\frac{v_{1}}{d_{1}}=\ldots=\frac{v_{k}}{d_{k}}.$

For a partition $A=(d_{i},w_{i})_{i=1}^{k}$ with terms possibly equal to zero,
let $I\subset\\{1,\ldots,k\\}$ be the subset of $i$ with $d_{i}\geqslant 1$,
and define the partition of $(d,w)$ with non-zero terms
$\overline{A}:=(d_{i},w_{i})_{i\in I}$.

#### 5.1.1.

For $\lambda$ and $\mu$ two cocharacters, let $A_{\lambda}$ be the set of
$(T(d)\times T)$-weights $\beta$ of $R(d)$ such that
$\langle\lambda,\beta\rangle>0$, let $I^{\mu}_{\lambda}\subset A_{\lambda}$ be
the set of weights such that $\langle\mu,\beta\rangle<0$, and let
$A^{\mu}_{\lambda}\subset A_{\lambda}$ be the set of weights such that
$\langle\mu,\beta\rangle=0$. Let $J^{\mu}_{\lambda}$ be the set of weights
$\beta$ of $\mathfrak{g}$ such that $\langle\lambda,\beta\rangle>0$ and
$\langle\mu,\beta\rangle<0$. Define the weights

$\displaystyle N^{\mu}_{\lambda}:=\sum_{I^{\mu}_{\lambda}}\beta,\
\mathfrak{g}^{\mu}_{\lambda}:=\sum_{J^{\mu}_{\lambda}}\beta.$

#### 5.1.2.

Let $\lambda$ and $\mu$ be dominant cocharacters of $T(d)$ with associated
partitions $(d_{i})_{i=1}^{k}$ and $(e_{i})_{i=1}^{s}$, respectively. Let
$W\cong\mathfrak{S}_{d}$ be the Weyl group of $GL(d)$, let
$W^{\lambda}\cong\times_{i=1}^{k}\mathfrak{S}_{d_{i}}$ be the Weyl group of
$GL(d)^{\lambda}$, and let $W^{\mu}$ be the Weyl group of $GL(d)^{\mu}$.
Define the set of cosets

$S^{\mu}_{\lambda}:=W^{\mu}\backslash W/W^{\lambda}.$

A coset $C\in S^{\mu}_{\lambda}$ corresponds to partitions $(f_{ij})$ for
$1\leqslant i\leqslant k$ and $1\leqslant j\leqslant s$ such that

$\displaystyle\sum_{j=1}^{s}f_{ij}=d_{i}\mbox{ for }1\leqslant i\leqslant k,\
\sum_{i=1}^{k}f_{ij}=e_{j}\mbox{ for }1\leqslant j\leqslant s.$

Let $\nu$ be a dominant cocharacter corresponding to the partition

(5.2)
$(f_{11},\cdots,f_{1s},f_{21},\cdots,f_{2s},\cdots,f_{k1},\cdots,f_{ks})$

and let $\kappa$ be a dominant cocharacter corresponding to the partition

(5.3)
$(f_{11},\cdots,f_{k1},f_{12},\cdots,f_{k2},\cdots,f_{1s},\cdots,f_{ks}).$

Consider the permutation $w$ of $\mathfrak{S}_{d}$ of minimal length which
permutes blocks of consecutive integers

(5.4)
$w=w_{C}\colon(f_{11},\cdots,f_{1s},f_{21},\cdots,f_{2s},\cdots,f_{k1},\cdots,f_{ks})\mapsto\\\
(f_{11},\cdots,f_{k1},f_{12},\cdots,f_{k2},\cdots,f_{1s},\cdots,f_{ks}).$

Consider the partition $D$ with terms $(f_{ij},u_{ij})$, where $f_{ij}$ are
ordered as on (5.2), and assume that $\overline{D}$ is in $H_{d,w}$. Then
there also exists a partition $E$ with terms $(f_{ij},\widetilde{u}_{ij})$
where $f_{ij}$ are ordered as on (5.3) such that $\overline{E}$ is in
$H_{d,w}$. Consider the order $1\leqslant l\leqslant ks$ of the pairs $(i,j)$
as on the first line on (5.4). Define the functor

$\displaystyle\text{sw}_{C}:\boxtimes_{l=1}^{ks}\mathbb{M}(f_{l})_{u_{l}}$
$\displaystyle\to\boxtimes_{l=1}^{ks}\mathbb{M}(f_{w(l)})_{\widetilde{u}_{w(l)}},$
$\displaystyle\boxtimes_{l=1}^{ks}\mathcal{A}_{l}$
$\displaystyle\mapsto\boxtimes_{l=1}^{ks}\mathcal{A}_{w(l)}.$

which permutes the factors as in (5.4). Define

$\displaystyle\widetilde{\text{sw}}_{C}\colon\boxtimes_{i=1}^{k}\boxtimes_{j=1}^{s}\mathbb{M}(f_{ij})_{u_{ij}}$
$\displaystyle\to\boxtimes_{j=1}^{s}\boxtimes_{i=1}^{k}\mathbb{M}(f_{ij})_{\widetilde{u}_{ij}},$
$\displaystyle\mathcal{A}$
$\displaystyle\mapsto\mathrm{sw}_{C}\left(\mathcal{A}\otimes\mathcal{O}(-N^{w^{-1}\mu}_{\lambda}+\mathfrak{g}^{w^{-1}\mu}_{\lambda})[|I^{w^{-1}\mu}_{\lambda}|-|J^{w^{-1}\mu}_{\lambda}|]\right),$

where
$\mathcal{O}(-N^{w^{-1}\mu}_{\lambda}+\mathfrak{g}^{w^{-1}\mu}_{\lambda})$ is
a one dimensional representation of $G^{\nu}\cong G^{\kappa}$. Consider the
maps

$\mathcal{X}(d)^{\kappa}\xleftarrow{q_{\kappa^{-1}\mu}}\left(\mathcal{X}(d)^{\mu}\right)^{\kappa^{-1}\geqslant
0}\xrightarrow{p_{\kappa^{-1}\mu}}\mathcal{X}(d)^{\mu}.$

Finally, define

$\widehat{m}_{CB}=p_{\kappa^{-1}\mu*}q_{\kappa^{-1}\mu}^{*}\widetilde{\text{sw}}_{\nu}\colon\mathbb{M}_{C}\to\mathbb{M}_{B}.$

There are analogous such functors for categories of (equivariant and/ or
graded) matrix factorizations

$\widehat{m}_{CB}\colon\mathbb{S}^{\bullet}_{\ast,C}\to\mathbb{S}^{\bullet}_{\ast,B}.$

#### 5.1.3.

We introduce some more maps and functors needed in the rest of this section.
Let $\lambda$ and $\mu$ be dominant cocharacters and let $\nu$ be a dominant
cocharacter corresponding to a partition in $S^{\mu}_{\lambda}$. The map
$p_{\lambda^{-1}}$ factors as
$p_{\lambda^{-1}}=\pi_{\lambda^{-1}}\iota_{\lambda^{-1}}$, where

$\displaystyle\pi_{\lambda^{-1}}$
$\displaystyle:R(d)/G(d)^{\lambda^{-1}\geqslant 0}\to R(d)/G(d),$
$\displaystyle\iota_{\lambda^{-1}}$ $\displaystyle:R(d)^{\lambda^{-1}\geqslant
0}/G(d)^{{\lambda^{-1}}\geqslant 0}\to R(d)/G(d)^{{\lambda^{-1}}\geqslant 0}.$

There are similarly defined maps

$\displaystyle\pi_{\kappa^{-1}\mu}$
$\displaystyle:R(d)^{\mu}/\left(G(d)^{\mu}\right)^{{\kappa^{-1}}\geqslant
0}\to R(d)^{\mu}/G(d)^{\mu},$ $\displaystyle\iota_{\kappa^{-1}\mu}$
$\displaystyle:\left(R(d)^{\mu}\right)^{\kappa^{-1}\geqslant
0}/\left(G(d)^{\mu}\right)^{\kappa^{-1}\geqslant 0}\to
R(d)^{\mu}/\left(G(d)^{\mu}\right)^{\kappa^{-1}\geqslant 0}.$

#### 5.1.4.

Let $\mu$ be a dominant cocharacter of $T(d)$, let $b\in\mathbb{Z}$, and let
$D^{b}\left(\mathcal{X}(d)^{\mu\geqslant 0}\right)_{\leqslant b}$ be the
subcategory of $D^{b}\left(\mathcal{X}(d)^{\mu\geqslant 0}\right)$ generated
by complexes $q_{\mu}^{*}\mathcal{A}$ for $\mathcal{A}\in
D^{b}\left(\mathcal{X}(d)^{\mu}\right)_{i}$ and $i\leqslant b$. There is a
semiorthogonal decomposition

$D^{b}\left(\mathcal{X}(d)^{\mu\geqslant 0}\right)_{\leqslant
b}=\Big{\langle}D^{b}\left(\mathcal{X}(d)^{\mu\geqslant 0}\right)_{\leqslant
b-1},D^{b}\left(\mathcal{X}(d)^{\mu\geqslant 0}\right)_{b}\Big{\rangle}$

and there are equivalences
$q_{\mu}^{*}:D^{b}\left(\mathcal{X}(d)^{\mu}\right)_{b}\xrightarrow{\sim}D^{b}\left(\mathcal{X}(d)^{\mu\geqslant
0}\right)_{b}$. We define the functor

$\beta_{b}:D^{b}\left(\mathcal{X}(d)^{\mu\geqslant 0}\right)_{\leqslant b}\to
D^{b}\left(\mathcal{X}(d)^{\mu\geqslant 0}\right)_{b}$

to be the projection with respect to the above semiorthogonal decomposition.

#### 5.1.5.

Let $B=(d_{i},w_{i})_{i=1}^{k}$ be a partition of $(d,w)$ in $H_{d,w}$. Let
$\mu$ be a dominant cocharacter for the partition $(d_{i})_{i=1}^{k}$ and let
$b:=\frac{n_{\mu}}{2}=\langle\mu,\mathfrak{g}^{\mu>0}\rangle$, see (2.3) for
the definition of $n_{\lambda}$.

######  Lemma 5.1.

There is a functor

(5.5)
$\displaystyle\widetilde{\Delta}_{B}:=\left(q_{\mu}^{*}\right)^{-1}\beta_{b}\,p_{\mu}^{*}:\mathbb{M}(d)_{w}\to\mathbb{M}_{B}:=\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}.$

###### Proof.

It suffices to show that the image of $\widetilde{\Delta}_{B}$ is in
$\mathbb{M}_{B}$. Let $\chi$ be a dominant weight of $T(d)$ such that
$\chi+\rho\in\textbf{W}(d)_{w}$. If $\chi+\rho$ is not on the face $F(\mu)$ of
the polytope $\textbf{W}(d)$, then

$\beta_{b}p_{\mu}^{*}\left(\mathcal{O}_{\mathcal{X}(d)}\otimes\Gamma_{GL(d)}(\chi)\right)=0.$

If $\chi+\rho$ is on the face $F(\mu)$, then by [Pădc, Corollary 3.4] we can
write

(5.6)
$\chi+\rho-w\tau_{d}=\frac{1}{2}N^{\mu>0}+\sum_{i=1}^{k}\left(\psi_{i}+\rho_{i}\right),$

where $\psi_{i}\in M(d_{i})$ and $\psi_{i}+\rho_{i}\in\textbf{W}(d_{i})_{0}$.
In particular, we have that

$\chi=\sum_{i=1}^{k}\psi_{i}+\mathfrak{g}^{\mu>0}+w\tau_{d}.$

Write $\chi=\sum_{i=1}^{k}\chi_{i}$. We have

$\displaystyle\langle 1_{d_{i}},\chi_{i}\rangle=\frac{d_{i}w}{d}+\langle
1_{d_{i}},\mathfrak{g}^{\mu>0}\rangle=v_{i}+\langle
1_{d_{i}},\mathfrak{g}^{\mu>0}\rangle=w_{i}.$

Further, we have that $\chi_{i}=\psi_{i}+w_{i}\tau_{d_{i}}$, so
$\chi_{i}+\rho_{i}\in\textbf{W}(d_{i})_{w_{i}}$. Therefore we have

$\left(q_{\mu}^{*}\right)^{-1}\beta_{b}\,p_{\mu}^{*}\left(\mathcal{O}_{\mathcal{X}(d)}\otimes\Gamma_{GL(d)}(\chi)\right)=\boxtimes_{i=1}^{k}\left(\mathcal{O}_{\mathcal{X}(d_{i})}\otimes\Gamma_{GL(d_{i})}(\chi_{i})\right)\in\mathbb{M}_{B}.$

∎

Let $\chi$ be a dominant weight with $\chi+\rho\in\textbf{W}(d)$. Note that

(5.7)
$\widetilde{\Delta}_{B}\left(\mathcal{O}_{\mathcal{X}(d)}\otimes\Gamma_{GL(d)}(\chi)\right)=\begin{cases}\mathcal{O}_{\mathcal{X}(d)^{\mu}}\otimes\Gamma_{GL(d)^{\mu}}(\chi),\text{
if }\chi\in F(\mu),\\\ 0,\text{ otherwise}.\end{cases}$

The functor (5.5) induces a functor

(5.8)
$\widetilde{\Delta}_{B}:=\left(q_{\mu}^{*}\right)^{-1}\beta_{b}\,p_{\mu}^{*}:\mathbb{S}^{\bullet}_{\ast}(d)_{w}\to\mathbb{S}^{\bullet}_{\ast,B}.$

Let $A=(d_{i},w_{i})_{i=1}^{k}$ and $C=(f_{i},u_{i})_{i=1}^{l}$ be partitions
in $H_{d,w}$ such that $C$ is a refinement of $A$, see Subsection 2.2.6. One
can analogously define functors

$\displaystyle\widetilde{\Delta}_{AC}\colon\mathbb{M}_{A}\to\mathbb{M}_{C},\
\widetilde{\Delta}_{AC}\colon\mathbb{S}^{\bullet}_{\ast,A}\to\mathbb{S}^{\bullet}_{\ast,C}$

for categories $\mathbb{S}^{\bullet}_{\ast}$ as in Subsection 2.6.2.

###  5.2. Compatibility between the product and the coproduct

In this section, we show that $\widetilde{m}$ and $\widetilde{\Delta}$ are
compatible. Recall the forget-the-potential map (1.8):

$\Theta\colon
K_{T}\left(\mathrm{MF}\left(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits
W\right)\right)\to K_{T}\left(\mathrm{MF}(\mathcal{X}(d),0)\right).$

Recall that $K_{T}(\mathbb{S}_{A})_{\mathbb{F}}\hookrightarrow
K_{T}(\mathbb{M}_{A})_{\mathbb{F}}$, see [PTa, Theorem 4.12 and Equation
(4.36)]. Let

$K_{T}(\mathbb{S}_{A})^{\prime}:=K_{T}\left(\mathbb{S}_{A}\right)/(\mathbb{K}\text{-torsion})\stackrel{{\scriptstyle\cong}}{{\to}}\text{image}\left(\Theta\colon
K_{T}(\mathbb{S}_{A})\to K_{T}(\mathbb{M}_{A})\right).$

######  Theorem 5.2.

Consider a pair $(d,w)\in\mathbb{N}\times\mathbb{Z}$. Let $\lambda$ and $\mu$
be dominant cocharacters with associated partitions
$A=(d_{i},w_{i})_{i=1}^{k}$ and $B=(e_{i},v_{i})_{i=1}^{s}$ in $H_{d,w}$. Let
$S^{B}_{A}$ be the set of partitions $C=(f_{i},u_{i})_{i=1}^{l}$ with $l=ks$
with $\overline{C}$ in $H_{d,w}$ and such that

$\displaystyle f_{(i-1)s+1}+f_{(i-1)s+2}+\ldots+f_{is}$
$\displaystyle=d_{i}\text{ for }1\leqslant i\leqslant k,$ $\displaystyle
f_{j}+f_{s+j}+\ldots+f_{(k-1)s+j}$ $\displaystyle=e_{j}\text{ for }1\leqslant
j\leqslant s.$

Then the following diagram commutes:

${K_{T}(\mathbb{S}_{A})^{\prime}}$${K_{T}(\mathbb{S}(d)_{w})^{\prime}}$${\bigoplus_{C\in
S^{B}_{A}}K_{T}(\mathbb{S}_{C})^{\prime}}$${K_{T}(\mathbb{S}_{B})^{\prime}.}$$\scriptstyle{\widetilde{m}_{A}}$$\scriptstyle{\bigoplus\widetilde{\Delta}_{AC}}$$\scriptstyle{\widetilde{\Delta}_{B}}$$\scriptstyle{\bigoplus\widehat{m}_{CB}}$

Theorem 5.2 is a $T$-equivariant version of [Pădc, Theorem 5.2] and the same
proof works to show this statement. However, we present an alternative proof
which first proves a categorical statement about complexes in
$D^{b}(\mathcal{X}(d)^{\mu})$ which is stronger than the results in loc. cit.
and is of independent interest for computations in categorical Hall algebras.

Note that the compatibility of the product and coproduct for localized
K-theory, either in the above setting or in the setting of Corollary 5.6,
follows by a direct computation and Propositions 5.7 and 5.8.

######  Proposition 5.3.

Let $A,B$ be as in Theorem 5.2. For $1\leqslant i\leqslant k$, let $\chi_{i}$
be a dominant weight of $T(d_{i})$ for $1\leqslant i\leqslant k$ such that
$\chi_{i}+\rho_{i}\in\mathbf{W}(d_{i})$. Let $\chi:=\sum_{i=1}^{k}\chi_{i}$.
For any $C\in S^{B}_{A}$, there are natural maps:

(5.9)
$\widehat{m}_{CB}\widetilde{\Delta}_{AC}\left(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi)\right)\to\widetilde{\Delta}_{B}\widetilde{m}_{A}\left(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi)\right)$

such that there is an isomorphism:

(5.10) $\bigoplus_{C\in
S^{B}_{A}}\widehat{m}_{CB}\widetilde{\Delta}_{AC}\left(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi)\right)\xrightarrow{\sim}\widetilde{\Delta}_{B}\widetilde{m}_{A}\left(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi)\right).$

###### Proof of Theorem 5.2.

By Proposition 5.3, the following diagram commutes:

${K_{T}(\mathbb{M}_{A})}$${K_{T}(\mathbb{M}(d)_{w})}$${\bigoplus_{C\in
S^{B}_{A}}K_{T}(\mathbb{M}_{C})}$${K_{T}(\mathbb{M}_{B}).}$$\scriptstyle{\widetilde{m}_{A}}$$\scriptstyle{\bigoplus\widetilde{\Delta}_{AC}}$$\scriptstyle{\widetilde{\Delta}_{B}}$$\scriptstyle{\bigoplus\widehat{m}_{CB}}$

The maps $\widetilde{m}$ and $\widetilde{\Delta}$ are compatible with the
forget-the-potential map (1.8) by [Păda, Proposition 3.6] and [Pădc,
Proposition 5.1], respectively. The conclusion thus follows. ∎

###### Proof of Proposition 5.3.

The argument follows closely the proof of [Pădd, Theorem 5.2]. We give an
overview of the proof. We use a Koszul resolution to compute
$\widetilde{m}_{A}\left(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi)\right)$
in terms of $\mathcal{O}\otimes\Gamma_{GL(d)}(\theta)$, where
$\theta=(\chi-\sigma_{I})^{+}$ for $\sigma_{I}$ a partial sum of weights
pairing positively with $\lambda$. Let
$\mathcal{O}\otimes\Gamma_{GL(d)}(\theta)$ be a vector bundle appearing in the
Koszul resolution with non-zero $\widetilde{\Delta}_{B}$. Then the weight
$\theta$ is on a face of $\textbf{W}(d)$, see (5.7). We use Proposition 5.4 to
characterize the highest weights $\theta$ on a face of $\textbf{W}(d)$ in
terms of partitions $C\in S^{B}_{A}$. The proof then follows from a direct
comparison with the right hand side of (5.10). Let $\sigma_{I}$ be a sum such
that $\chi-\sigma_{I}$ is on a face of $\textbf{W}(d)$ corresponding to $C\in
S^{B}_{A}$ with associated permutation $w$, see (5.4). The swap morphism
appears because conjugating $\chi-\sigma_{I}$ to
$\theta=(\chi-\sigma_{I})^{+}$ first requires to act by $w$.

The multiplication $\widetilde{m}_{A}$ is defined as
$\widetilde{m}_{A}=p_{\lambda^{-1}*}q_{\lambda^{-1}}^{*}$. Let
$\chi:=\sum_{i=1}^{k}\chi_{i}$. Consider the Koszul resolution:

$\displaystyle\widetilde{m}_{A}\left(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi)\right)$
$\displaystyle=\pi_{\lambda^{-1}*}\iota_{\lambda^{-1}*}q_{\lambda^{-1}}^{*}\left(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi)\right)$
$\displaystyle\cong\pi_{\lambda^{-1}*}\left(\bigoplus_{I\subset
A_{\lambda}}\mathcal{O}_{R(d)}\otimes\Gamma_{GL(d)^{\lambda\leqslant
0}}(\chi-\sigma_{I})[|I|],d\right),$

see Proposition 2.1 for the notation. The differential $d$ is induced by
multiplication with generators $(e_{i})_{i=1}^{t}$ of the polynomial ring
$\mathbb{C}\left[R(d)^{\lambda<0}\right]\cong\mathbb{C}[e_{1},\ldots,e_{t}]$.
Fix $C\in S^{B}_{A}$, consider the associated dominant cocharacters $\nu$ and
$\kappa$, and let $w=w_{C}\in\mathfrak{S}_{d}$ as in (5.4). Let
$\varphi:=w^{-1}\mu$. There are natural maps of complexes

(5.11) $\pi_{\lambda^{-1}*}\left(\bigoplus_{J\subset
A^{\varphi}_{\lambda}}\mathcal{O}_{R(d)}\otimes\Gamma_{GL(d)^{\lambda\leqslant
0}}(\chi)\otimes\mathcal{O}\left(-N^{\varphi}_{\lambda}-\sigma_{J}\right)[\lvert
I^{\varphi}_{\lambda}\rvert+\lvert J\rvert],d\right)\to\\\
\pi_{\lambda^{-1}*}\left(\bigoplus_{I\subset
A_{\lambda}}\mathcal{O}_{R(d)}\otimes\Gamma_{GL(d)^{\lambda\leqslant
0}}(\chi)\otimes\mathcal{O}(-\sigma_{I})[|I|],d\right)$

induced by the inclusion of sets

$I_{C}:=\\{N^{\varphi}_{\lambda}+\sigma_{J}\mid J\subset
A^{\varphi}_{\lambda}\\}\subset\\{\sigma_{I}\mid I\subset A_{\lambda}\\}.$

The differential $d$ of the complex on the first line is induced by
multiplication with generators of the polynomial ring
$\mathbb{C}\left[\left(R(d)^{\varphi}\right)^{\lambda<0}\right]$. For
$C,C^{\prime}$ different elements in $S^{B}_{A}$, we have that $I_{C}\cap
I_{C^{\prime}}=\emptyset$. If $\sigma\in\\{\sigma_{I}\mid I\subset
A_{\lambda}\\}\setminus\left(\bigcup_{C\in S^{B}_{A}}I_{C}\right)$, then
$\mathcal{O}_{R(d)}\otimes\Gamma_{GL(d)^{\lambda\leqslant
0}}(\chi)\otimes\mathcal{O}(-\sigma_{I})$ has $s\mu$-weights strictly less
than $\frac{n_{\mu}}{2}=\langle\mu,\mathfrak{g}^{\mu>0}\rangle$ for all
$s\in\mathfrak{S}_{d}$, see Proposition 5.4. It follows that

(5.12)
$\widetilde{\Delta}_{\mu}\pi_{\lambda^{-1}*}\left(\mathcal{O}_{R(d)}\otimes\Gamma_{GL(d)^{\lambda\leqslant
0}}(\chi)\otimes\mathcal{O}(-\sigma_{I})\right)=0$

for such sums $\sigma$, see (5.7). It suffices to show that, for $C\in
S^{B}_{A}$, we have natural isomorphisms of complexes:

(5.13) $\widetilde{\Delta}_{B}\pi_{\lambda^{-1}*}\left(\bigoplus_{J\subset
A^{\varphi}_{\lambda}}\mathcal{O}_{R(d)}\otimes\Gamma_{GL(d)^{\lambda\leqslant
0}}(\chi)\otimes\mathcal{O}\left(-N^{\varphi}_{\lambda}-\sigma_{J}\right)[|I^{\varphi}_{\lambda}|+|J|],d\right)\\\
\cong\widehat{m}_{CB}\widetilde{\Delta}_{AC}\left(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi)\right).$

Using (5.11), we obtain the natural maps (5.9), and further we obtain the
isomorphism (5.10) using the vanishing (5.12).

For $J\subset A^{\varphi}_{\lambda}$, we have that

$w*\left(\chi-N^{\varphi}_{\lambda}-\sigma_{J}\right)+\rho\in
F(\mu)\subset\textbf{W}(d),$

see Proposition 5.4. Let $\widetilde{w}_{J}\in\mathfrak{S}_{d}$ be the element
of minimal length such that
$\widetilde{w}_{J}*\left(\chi-N^{\varphi}_{\lambda}-\sigma_{J}\right)$ is
dominant or zero. Observe that $\widetilde{w}_{\emptyset}=w$. However, for
general $J$, we have that $\widetilde{w}_{J}=s_{J}\circ w$ for a permutation
$s_{J}$ in $\times_{i=1}^{s}\mathfrak{S}_{e_{i}}\cong W^{\mu}$ and
$\ell(\widetilde{w}_{J})=\ell(w)+\ell(s_{J})$. By the Borel-Bott-Weyl theorem,
there is a natural isomorphism

$\displaystyle\pi_{\lambda^{-1}*}\left(\mathcal{O}_{R(d)}\otimes\Gamma_{GL(d)^{\lambda\leqslant
0}}\left(\chi-N^{\varphi}_{\lambda}-\sigma_{J}\right)\right)\cong$
$\displaystyle\pi_{\lambda^{-1}*}\left(\mathcal{O}_{R(d)}\otimes\Gamma_{GL(d)^{\lambda\leqslant
0}}\left(w*(\chi-N^{\varphi}_{\lambda}-\sigma_{J})\right)[-|J^{\varphi}_{\lambda}|]\right)$

and further there are natural isomorphisms

(5.14)
$\displaystyle\widetilde{\Delta}_{B}\pi_{\lambda^{-1}*}\left(\bigoplus_{J\subset
A^{\varphi}_{\lambda}}\mathcal{O}_{R(d)}\otimes\Gamma_{GL(d)^{\lambda\leqslant
0}}\left(\chi-N^{\varphi}_{\lambda}-\sigma_{J}\right)[|I^{\varphi}_{\lambda}|+|J|],d\right)$
$\displaystyle\cong\widetilde{\Delta}_{B}\pi_{\lambda^{-1}*}\left(\bigoplus_{J\subset
A^{\varphi}_{\lambda}}\mathcal{O}_{R(d)}\otimes\Gamma_{GL(d)^{\lambda\leqslant
0}}\left(w*(\chi-N^{\varphi}_{\lambda}-\sigma_{J})\right)[|I^{\varphi}_{\lambda}|-|J^{\varphi}_{\lambda}|+|J|],d\right)$
$\displaystyle\cong\pi_{\kappa^{-1}\mu*}\left(\bigoplus_{J\subset
A^{\varphi}_{\lambda}}\mathcal{O}_{R(d)^{\mu}}\otimes\Gamma_{(GL(d)^{\mu})^{\kappa\leqslant
0}}\left(w*(\chi-N^{\varphi}_{\lambda}-\sigma_{J})\right)[|I^{\varphi}_{\lambda}|-|J^{\varphi}_{\lambda}|+|J|],d\right).$

We have that $\mathcal{X}(d)^{\nu}=\mathcal{X}(d)^{\kappa}$ and
$G(d)^{\nu}=G(d)^{\kappa}$. For a weight $\theta$ of $T(d)$, denote by

$\Psi\left(\mathcal{O}_{\mathcal{X}(d)^{\kappa}}\otimes\Gamma_{G(d)^{\kappa}}(\theta)\right):=\mathcal{O}_{\mathcal{X}(d)^{\kappa}}\otimes\Gamma_{G(d)^{\kappa}}(w*\theta).$

For a subset $J\subset A^{\varphi}_{\lambda}$, let
$J^{\prime}=\\{w\beta\mid\beta\in J\\}$ be the corresponding subset of
$I_{\kappa\mu}:=\\{\beta\text{ weight of
}R(d)^{\mu}\mid\langle\kappa,\beta\rangle>0\\}$, see Proposition 5.5. There is
an isomorphism

$\mathcal{O}_{(\mathcal{X}(d)^{\mu})^{\kappa\leqslant
0}}\otimes\Gamma_{(GL(d)^{\mu})^{\kappa\leqslant
0}}\left(w*(\chi-N^{\varphi}_{\lambda}-\sigma_{J})\right)[-|J^{\varphi}_{\lambda}|]\cong\\\
q_{\kappa^{-1}\mu}^{*}\left(\Psi\left(\mathcal{O}_{\mathcal{X}(d)^{\nu}}\otimes\Gamma_{GL(d)^{\nu}}\left(\chi-N^{\varphi}_{\lambda}\right)[-|J^{\varphi}_{\lambda}|]\right)\otimes\mathcal{O}(-\sigma_{J^{\prime}})\right).$

For $\mathcal{B}$ a complex in $D^{b}(\mathcal{X}(d)^{\nu})$, by the
definition of $\widetilde{\text{sw}}_{C}$ we have that

(5.15)
$\widetilde{\text{sw}}_{C}(\mathcal{B}):=\Psi\left(\mathcal{B}\otimes\mathcal{O}\left(-N^{\varphi}_{\lambda}\right)\right)[|I^{\varphi}_{\lambda}|-|J^{\varphi}_{\lambda}|].$

We next want to use Proposition 2.1 for the map $\iota_{\kappa^{-1}\mu}$. For
this, it is convenient to use the following notation

$\mathrm{F}\left(\mathcal{O}_{(R(d)^{\mu})^{\kappa\leqslant
0}}\otimes\Gamma_{(G(d)^{\mu})^{\kappa\leqslant
0}}(\theta)\right):=\mathcal{O}_{R(d)^{\mu}}\otimes\Gamma_{(G(d)^{\mu})^{\kappa\leqslant
0}}(\theta)$

for $\theta$ a dominant weight of $T(d)$. We consider the Koszul resolution:

(5.16)
$\iota_{\kappa^{-1}\mu*}q_{\kappa^{-1}\mu}^{*}\left(\widetilde{\text{sw}}_{C}\left(\widetilde{\Delta}_{\nu\lambda}\left(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi)\right)\right)\right)\cong\\\
\left(\bigoplus_{J^{\prime}\subset
I_{\kappa\mu}}\mathrm{F}q_{\kappa^{-1}\mu}^{*}\left(\widetilde{\text{sw}}_{C}\left(\widetilde{\Delta}_{\nu\lambda}\left(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi)\right)\right)\right)\otimes\mathcal{O}(-\sigma_{J^{\prime}})[-|J^{\prime}|],d\right),$

where the differential $d$ is induced by multiplication with generators of the
polynomial ring $\mathbb{C}\left[\left(R(d)^{\mu}\right)^{\kappa<0}\right]$.
There are isomorphisms

$\displaystyle\widetilde{\Delta}_{B}\pi_{\lambda^{-1}*}\left(\bigoplus_{J\subset
A^{\varphi}_{\lambda}}\mathcal{O}_{R(d)}\otimes\Gamma_{GL(d)^{\lambda\leqslant
0}}(\chi)\otimes\mathcal{O}\left(-N^{\varphi}_{\lambda}-\sigma_{J}\right)[|I^{\varphi}_{\lambda}|-|J|],d\right)$
$\displaystyle\overset{(1)}{\cong}\pi_{\kappa^{-1}\mu*}\left(\bigoplus_{J^{\prime}\subset
I_{\kappa\mu}}\mathrm{F}q_{\kappa^{-1}\mu}^{*}\left(\widetilde{\text{sw}}_{C}\left(\widetilde{\Delta}_{\kappa\lambda}(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi))\right)\right)\otimes\mathcal{O}(-\sigma_{J^{\prime}})[-|J^{\prime}|],d\right)$
$\displaystyle\overset{(2)}{\cong}\pi_{\kappa^{-1}\mu*}\iota_{\kappa^{-1}\mu*}q_{\kappa^{-1}\mu}^{*}\left(\widetilde{\text{sw}}_{C}\left(\widetilde{\Delta}_{\nu\lambda}(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi))\right)\right)$
$\displaystyle\overset{(3)}{\cong}\widetilde{m}_{\kappa\mu}\left(\widetilde{\text{sw}}_{C}\left(\widetilde{\Delta}_{\nu\lambda}(\mathcal{O}_{\mathcal{X}(d)^{\lambda}}\otimes\Gamma_{GL(d)^{\lambda}}(\chi))\right)\right).$

The isomorphism (1) follows from (5.14) and (5.15) and it respects the
differentials. The isomorphism (2) follows from (5.16). The isomorphism (3)
follows from the definition of $\widetilde{m}_{\kappa\mu}$. ∎

######  Proposition 5.4.

Let $\lambda$ and $\mu$ be dominant cocharacters of $GL(d)$ and let $w\in W$.
Assume that $\chi$ is a dominant weight with $\chi+\rho\in F(\lambda)$ and
that $I\subset A_{\lambda}$ such that $w*(\chi-\sigma_{I})+\rho\in F(\mu)$.
Let $\varphi:=w^{-1}\mu$. Then

(5.17) $I=\\{\beta\in A_{\lambda}\mid\langle\varphi,\beta\rangle=0\\}\sqcup J$

for a subset $J\subset A^{\varphi}_{\lambda}$.

Conversely, for all $I\subset A_{\lambda}$ as in (5.17), we have
$w*(\chi-\sigma_{I})+\rho\in F(\mu)$.

###### Proof.

For two cocharacters $\tau$ and $\tau^{\prime}$, we use the notations

$N^{\tau>0}:=\sum_{A_{\tau}}\beta,\
N^{\tau^{\prime}=0,\tau>0}=\sum_{A^{\tau^{\prime}}_{\tau}}\beta.$

Let $v=\langle 1_{d},\chi\rangle$. Write

$\displaystyle\chi-v\tau_{d}+\rho$
$\displaystyle=\frac{1}{2}N^{\lambda>0}+\psi,$ $\displaystyle
w*(\chi-v\tau_{d}-\sigma_{I})+\rho$
$\displaystyle=\frac{1}{2}N^{\mu>0}+\phi^{\prime},$
$\displaystyle\chi-v\tau_{d}-\sigma_{I}+\rho$
$\displaystyle=\frac{1}{2}N^{\varphi>0}+\phi,$

where $\psi\in\textbf{W}(\lambda)_{0}$ and $\phi\in\textbf{W}(\varphi)_{0}$,
see [Pădb, Proposition 3.4]. Then

(5.18)
$\sigma_{I}=N_{\lambda}^{\varphi}+\left(\frac{1}{2}N^{\varphi=0,\lambda>0}-\phi\right)+\left(\psi-\frac{1}{2}N^{\lambda=0,\varphi>0}\right).$

The weight $\widetilde{\phi}:=\frac{1}{2}N^{\varphi=0,\lambda>0}-\phi$ is a
sum with nonnegative coefficients of weights $\beta$ such that
$\langle\varphi,\beta\rangle=0$ and $\langle\lambda,\beta\rangle>0$ and
weights $\beta^{\prime}$ such that
$\langle\varphi,\beta^{\prime}\rangle=\langle\lambda,\beta^{\prime}\rangle=0$.
The weight $\widetilde{\psi}:=\psi-\frac{1}{2}N^{\lambda=0,\varphi>0}$ is a
sum with nonnegative coefficients of weights $\beta$ such that
$\langle\lambda,\beta\rangle=0$ and $\langle\varphi,\beta\rangle<0$ and
weights $\beta^{\prime}$ such that
$\langle\varphi,\beta^{\prime}\rangle=\langle\lambda,\beta^{\prime}\rangle=0$.
We denote by $\sigma^{\lambda+}_{\varphi 0}$ a sum with nonnegative
coefficients of weights $\beta$ such that $\langle\lambda,\beta\rangle>0$ and
$\langle\varphi,\beta\rangle=0$ etc. Then we can write

(5.19) $\sigma_{I}-N^{\varphi}_{\lambda}=\sigma^{\lambda+}_{\varphi
0}+\sigma^{\lambda+}_{\varphi+}-n^{\lambda+}_{\varphi-},$

where all the sums on the right hand side are further partial sums of weights
in $A_{\lambda}$. We can rewrite (5.18) as

(5.20) $\sigma^{\lambda+}_{\varphi
0}+\sigma^{\lambda+}_{\varphi+}=n^{\lambda+}_{\varphi-}+\widetilde{\phi}^{\lambda+}_{\varphi
0}+\widetilde{\phi}^{\lambda 0}_{\varphi 0}+\widetilde{\psi}^{\lambda
0}_{\varphi-}+\widetilde{\psi}^{\lambda 0}_{\varphi 0}.$

The $\varphi$-weight of the left hand is nonnegative, while the
$\varphi$-weight of the right hand side is nonpositive. Thus
$\sigma^{\lambda+}_{\varphi+}=n^{\lambda+}_{\varphi-}=\widetilde{\psi}^{\lambda
0}_{\varphi-}=0$. In particular, (5.19) becomes

$\sigma_{I}=N^{\varphi}_{\lambda}+\sigma^{\lambda+}_{\varphi 0},$

which implies the first direction. The converse follows in a similar way. ∎

######  Proposition 5.5.

For a subset $J\subset A^{\varphi}_{\lambda}$, the set
$J^{\prime}=\\{w\beta\mid\beta\in J\\}$ is a subset of
$I_{\kappa\mu}:=\\{\beta\text{ weight of
}R(d)^{\mu}\mid\langle\kappa,\beta\rangle>0\\}$. This transformation induces a
bijection of sets
$\mathcal{A}^{\varphi}_{\lambda}\xrightarrow{\sim}I_{\kappa\mu}$.

###### Proof.

It suffices to check the first claim. To construct an inverse of this
transformation, send $L\subset I_{\kappa\mu}$ to
$L^{\circ}=\\{w^{-1}\beta\mid\beta\in L\\}$.

Let $\beta\in A^{\varphi}_{\lambda}$. Recall that $\varphi=w^{-1}\mu$, so we
have that $\langle w^{-1}\mu,\beta\rangle=0$, and thus
$\langle\mu,w\beta\rangle=0$.

It suffices to show that if a weight $\beta_{i}-\beta_{j}$ is in
$A^{\varphi}_{\lambda}$, then
$\langle\kappa,\beta_{w(i)}-\beta_{w(j)}\rangle>0$. For simplicity, we discuss
the case when $k=s=2$. Rename $f_{11}=f_{1}$, $f_{12}=f_{2}$, $f_{21}=f_{3}$,
$f_{22}=f_{4}$. The permutation $w$ is

$\displaystyle w(i)=\begin{cases}i+f_{3},\text{ if }f_{1}+1\leqslant
i\leqslant f_{1}+f_{2},\\\ i-f_{2},\text{ if }f_{1}+f_{2}+1\leqslant
i\leqslant f_{1}+f_{2}+f_{3},\\\ i,\text{ otherwise}.\end{cases}$

We have that

$\langle\lambda,\beta_{i}-\beta_{j}\rangle>0\text{ and
}\langle\mu,\beta_{w(i)}-\beta_{w(j)}\rangle=0,$

so one of two possibilities happens:

$\displaystyle(1)\,$ $\displaystyle f_{1}+f_{2}+f_{3}+1\leqslant i\leqslant
f_{1}+f_{2}+f_{3}+f_{4}\text{ and }f_{1}+1\leqslant j\leqslant f_{1}+f_{2},$
$\displaystyle(2)\,$ $\displaystyle f_{1}+f_{2}+1\leqslant i\leqslant
f_{1}+f_{2}+f_{3}\text{ and }1\leqslant j\leqslant f_{1}.$

In case (1), we have that $i=w(i)$ and $f_{1}+f_{3}+1\leqslant w(j)\leqslant
f_{1}+f_{3}+f_{2}$, and then