# Nakajima’s quiver varieties and triangular bases of bipartite cluster
algebras

Li Li  Department of Mathematics and Statistics, Oakland University,
Rochester, MI 48309-4479, USA<EMAIL_ADDRESS>
###### Abstract.

Berenstein and Zelevinsky introduced quantum cluster algebras [2] and the
triangular bases [3]. The support conjecture proposed in [8], which asserts
that the support of each triangular basis element for a rank-2 cluster algebra
is bounded by an explicitly described region, was established in [9] for skew-
symmetric rank-2 cluster algebras. In this paper we extend this result by
proving a bound on the support of each triangular basis element for bipartite
cluster algebras.

###### 2010 Mathematics Subject Classification:

Primary 13F60; Secondary 14F06, 16G20, 32S60

###### Contents

  1. 1 Introduction
  2. 2 Quantum cluster algebras and triangular bases
    1. 2.1 Quantum cluster algebras
    2. 2.2 Triangular bases
  3. 3 Nakajima’s graded quiver varieties
  4. 4 Decomposition theorem
    1. 4.1 Algebraic Transversal Slice Theorem
    2. 4.2 Proof of trivial local systems
  5. 5 Dual canonical basis
    1. 5.1 Definition of $L(w)$, $M(w)$, $\mathbf{R}^{\mathrm{finite}}_{t}$, $\chi$
    2. 5.2 $\\{\overline{\chi(M(w))}\\}$ and the standard monomial basis
    3. 5.3 $\\{\chi(L(w))\\}$ and the triangular basis
  6. 6 The proof of the main result

##  1\. Introduction

Cluster algebras and quantum cluster algebras were introduced by Fomin-
Zelevinsky [4] and Berenstein-Zelevinsky [2], respectively. A primary
objective behind the inception of quantum cluster algebras is to understand
good bases arising from the representation theory of certain non-associative
algebras. Mimicking the construction of the dual canonical basis from a PBW
basis [10], Berenstein and Zelevinsky [3] constructed the triangular bases for
acyclic quantum cluster algebras. These triangular bases exhibit many
desirable properties. It particular, it is worth mentioning the work by Qin
[13, 14, 15], where he has constructed triangular bases for injective-
reachable cluster algebras using a different approach, and showed that his
triangular bases coincide with the bases by Berenstein and Zelevinsky for the
seeds associated with acyclic quivers.

In [8], Lee, Rupel, Zelevinsky and the author proposed a conjecture of a
precisely described region to constrain the support of each triangular basis
element of any rank-2 quantum cluster algebra. The proof for the skew-
symmetric rank-2 case was established in [9] using Nakajima’s quiver
varieties. In this work, we prove a similar result for any bipartite cluster
algebra.

The definition of quantum cluster algebra and (Berenstein-Zelevinsky’s)
triangular basis will be reviewed in §2. Our focus here is on bipartite
quantum cluster algebras, that is, those come from bipartite quiver.
Additionally, for the sake of clarity and simplicity, we confine our study to
quantum cluster algebras with principal quantization.

Consider a rank-$n$ bipartite cluster algebra with principal quantization
determined by an $n\times n$ skew-symmetric matrix $B=[b_{ij}]$. Let the
initial seed be $(\Lambda,\tilde{B},\tilde{X})$, where

$\tilde{B}=\begin{bmatrix}B\\\ {\bf
I}_{n}\end{bmatrix},\quad\Lambda=\begin{bmatrix}0&-{\bf I}_{n}\\\ {\bf
I}_{n}&-B\end{bmatrix},\quad\tilde{X}=(X_{1},\dots,X_{2n})$

where ${\bf I}_{n}$ the $n\times n$ identity matrix. In this setup, the first
$n$ variables of $\tilde{X}$ are called cluster variables, while the last $n$
variables are called frozen variables.

For $c\in\mathbb{R}$, we denote $[c]_{+}=\max(c,0)$; the vector $b_{k}$ is the
$k$-th column of $\tilde{B}$; for a vector $b$, we use $[b]_{+}$ to denote the
vector obtained from $b$ by applying $[\;]_{+}$ to each coordinate.

The triangular basis $\\{C_{a}\\}_{a\in\mathbb{Z}^{2n}}$ is of the following
form:

(1.1) $C_{a}=\sum_{v}e_{v}X^{a+\tilde{B}v},\quad
e_{v}\in\mathbb{Z}[\mathbf{v}^{\pm}].$

We define $\deg(e_{v})$ to be largest exponent of $\mathbf{v}$ in $e_{v}$.
Define a quiver $\mathcal{Q}^{\rm op}=(I,\Omega)$ with vertex set
$I=\\{1,\dots,n\\}=I_{0}\sqcup I_{1}$ (where vertices in $I_{0}$ are sinks,
vertices in $I_{1}$ are sources) and arrow set $\Omega$ determined by $B$ such
that there are $b_{ij}$ arrows $h:j\to i$ whenever $b_{ij}>0$. We denote the
source and target of $h$ by $s(h)$ and $t(h)$, respectively. See §2 for
details.

######  Theorem 1.1.

Let $a=(a_{1},\dots,a_{2n})\in\mathbb{Z}^{2n}$. Each $e_{v}$ in $C_{a}$ as in
(1.1) is symmetric and unimodal, and

$\deg(e_{v})\leq
f(v):=-\sum_{i=1}^{n}(a_{i}+v_{i})v_{i}+\sum_{h\in\Omega}v_{s(h)}v_{t(h)}.$

In particular, $e_{v}\neq 0$ only if $f(v)\geq 0$.

Together with (3.3), the support

${\rm Supp}(C_{a}):=\\{v\in\mathbb{Z}_{\geq 0}^{n}\;|\;e_{v}\neq 0\\}$

satisfies the following condition:

$f(v)\geq 0,\quad 0\leq v_{\alpha}\leq[-a_{\alpha}]_{+}\;(\forall\alpha\in
I_{0}),\;\textrm{ and }\;0\leq v_{\beta}\leq[a_{\beta}]_{+}+\sum_{h:\
s(h)=\beta}v_{t(h)}\;(\forall\beta\in I_{1}).$

Note that not every integer point satisfying the above condition are in ${\rm
Supp}(C_{a})$. In an ongoing project, the author is looking for a precise
bound of ${\rm Supp}(C_{a})$ to full generalize the results in [9].

The proof of the theorem follows the idea presented in [9]. While a
significant portion of the methodology used in [9] can be extended to the
bipartite case, many adjustments are necessary, and the proofs and
computations become notably more intricate. Certain results are already
established in the work of Nakajima [11] and Qin [13]; however, our approach
endeavors to provide a self-contained treatment, with minimal reliance on
representation theory and predominantly using elementary algebraic geometry,
in alignment with the approach adopted in [9].

The paper is organized as follows. In §2 we revisit the definition and
properties of quantum cluster algebra and triangular basis. In §3 we review
some facts on Nakajima’s quiver varieties. In §4 we prove an algebraic version
of the transversal slice theorem, and show that the local systems appeared in
the BBDG decomposition theorem are trivial. The BBDG Decomposition theorem [1]
serves as the primary advanced tool used in this paper. It asserts that, for a
proper map of algebraic varieties $f:X\to Y$, the pushforward of the
intersection cohomology complex $IC_{X}$ can be decomposed into a direct sum
of $IC_{V}(L)[d]$ for a collection of irreducible subvarieties $V\subseteq Y$,
local systems $L$ on (an open dense subset of) $V$, and integers $d$. While
the local systems are often trivial, as is the case in our scenario, the proof
is not always straightforward. Here, we present a proof using the Algebraic
Transversal Slice Theorem. In §5 we identify the dual canonical basis and
triangular basis in our specific case. In §6 we prove the main theorem and
provide some examples.

Acknowledgments. The author thanks the organizers, Yiqiang Li and Changlong
Zhong, of the AMS Special Session held at the University of Buffalo in
September 2023, during which numerous inspiring conversations took place.
Computer calculations were performed using SageMath [16].

##  2\. Quantum cluster algebras and triangular bases

In this section we recall the definition and facts of quantum cluster algebras
with principal quantization. See [2, 3] for details.

###  2.1. Quantum cluster algebras

Let $n$ be a positive integer and $m=2n$. Let $B$ be a skew-symmetrizable
integer $n\times n$ matrix, that is, there exists a diagonal matrix $D$ such
that $DB$ is a skew-symmetric matrix. Denote the $n\times n$ identity matrix
by ${\bf I}_{n}$, or simply by ${\bf I}$ if $n$ is clear from the context. Let
$\tilde{B}:=\begin{bmatrix}B\\\ {\bf I}_{n}\end{bmatrix}$ where ${\bf I}_{n}$
the $n\times n$ identity matrix. Let $\Lambda:=\begin{bmatrix}0&-D\\\
D&-DB\end{bmatrix}$, viewed as a bilinear form by defining $\Lambda({\bf
u},{\bf u}^{\prime})={\bf u}^{T}\Lambda{\bf u}^{\prime}$ for column vectors
${\bf u},{\bf u}^{\prime}\in\mathbb{Z}^{m}$. Let the quantum torus
$\mathcal{T}$ be the $\mathbb{Z}[\mathbf{v}^{\pm 1}]$-algebra with basis
$\\{X^{\bf e}:{\bf e}\in\mathbb{Z}^{m}\\}$, and the multiplication given by
$X^{\bf e}X^{{\bf e}^{\prime}}=\mathbf{v}^{\Lambda({\bf e},{\bf
e}^{\prime})}X^{{\bf e}+{\bf e}^{\prime}}$ for ${\bf e},{\bf
e}^{\prime}\in\mathbb{Z}^{m}$. Let $\mathcal{F}$ be the skew-field of
fractions of $\mathcal{T}$. Let $e_{1},\dots,e_{m}\in\mathbb{Z}^{m}$ be the
standard basis, that is, $e_{i}$ has 1 in the $i$-th coordinate and 0
elsewhere. Let $\tilde{X}=\\{X_{1},\dots,X_{m}\\}$ where $X_{i}=X^{e_{i}}$ are
called (initial) cluster variables if $1\leq i\leq n$, or frozen variables if
$n+1\leq i\leq m$. The _bar-involution_ is the $\mathbb{Z}$-linear anti-
automorphism of $\mathcal{T}$ determined by $\overline{\mathbf{v}^{i}X^{\bf
e}}=\mathbf{v}^{-i}X^{\bf e}$ for ${\bf e}\in\mathbb{Z}^{m}$. Note that
$\overline{XY}=\overline{Y}\;\overline{X}$ for $X,Y\in\mathcal{T}$. An element
in $\mathcal{T}$ which is invariant under the bar-involution is said to be
_bar-invariant_.

Fix an $n$-regular tree $\mathbb{T}_{n}$ with root $t_{0}$ and label each edge
by a number in $\\{1,\dots,n\\}$ such that any two edges sharing an endpoint
are labelled differently. Attach to $t_{0}$ the tuple
$(\Lambda,\tilde{B},\tilde{X})$ as above, called the initial seed. Given an
edge joining $t$ and $t^{\prime}$ with label $k$, we denote
$t^{\prime}=\mu_{k}(t)$ (and $t=\mu_{k}(t^{\prime})$); if the seed
$(\Lambda,\tilde{B},\tilde{X})$ at $t$ is constructed, we construct a seed
$(\Lambda^{\prime},\tilde{B}^{\prime},\tilde{X}^{\prime})$ attached to
$t^{\prime}$ by mutation, where $\tilde{B}^{\prime}=[b^{\prime}_{ij}]$ is
obtained from $\tilde{B}=[b_{ij}]$ by

$b^{\prime}_{ij}=\begin{cases}&-b_{ij},\textrm{ if $i=k$ or $j=k$};\\\
&b_{ij}+[b_{ik}]_{+}[b_{kj}]_{+}-[-b_{ik}]_{+}[-b_{kj}]_{+},\textrm{
otherwise.}\\\ \end{cases}$

and $\tilde{X}^{\prime}$ is obtained from $\tilde{X}$ by replacing $X_{k}$
with $X^{\prime}_{k}=X^{-e_{k}+[b_{k}]_{+}}+X^{-e_{k}+[-b_{k}]_{+}}$. The
bilinear form $\Lambda^{\prime}$ is determined by

$\begin{cases}&\Lambda^{\prime}(e_{i},e_{j})=\Lambda(e_{i},e_{j}),\textrm{ for
$i,j\neq k$;}\\\
&\Lambda^{\prime}(e_{k},e_{j})=\Lambda(-e_{k}+[b_{k}]_{+},e_{j}),\textrm{ for
$j\neq k$.}\\\ \end{cases}$

Denote
$\mathbb{Z}P[\mathbf{v}\pm]:=\mathbb{Z}[\mathbf{v}^{\pm}][X_{n+1}^{\pm},\dots,X_{m}^{\pm}]$.
The _quantum cluster algebra_ $\mathcal{A}$ is the
$\mathbb{Z}P[\mathbf{v}^{\pm 1}]$-subalgebra of $\mathcal{F}$ generated by all
cluster variables.

###  2.2. Triangular bases

Assume that the seed $(\Lambda,\tilde{B},\tilde{X})$ is acyclic, that is,
there exists a linear order $\lhd$ on $\\{1,\dots,n\\}$ such that $b_{ij}\leq
0$ if $i\lhd j$. Define $r(a)=\sum_{k=1}^{n}[-a_{k}]_{+}$ for
$a\in\mathbb{Z}^{m}$ and define a partial order $\prec$ on $\mathbb{Z}^{m}$ by
$a^{\prime}\prec a\Leftrightarrow r(a^{\prime})<r(a)$.

The construction of the triangular basis starts with the standard monomial
basis $\\{E_{a}\,:\,a\in\mathbb{Z}^{m}\\}$. Let
$X^{\prime}_{k}=\mu_{k}(X_{k})$. For every $a=(a_{1},\dots,a_{m})$, the
_standard monomial_ $E_{a}$ is defined as

(2.1)
$E_{a}=\mathbf{v}^{v(a)}X^{\sum_{i=n+1}^{m}a_{i}e_{i}+\sum_{j=1}^{n}[a_{j}]_{+}e_{j}}\prod_{1\leq
k\leq n}^{\lhd}(X_{k}^{\prime})^{[-a_{k}]_{+}}\ .$

where $\prod_{1\leq k\leq n}^{\lhd}$ means taking the product in increasing
order with respect to $\lhd$, and $v({a})\in\mathbb{Z}$ is determined by the
condition that the leading term of $E_{a}$, which is the term obtained by
replacing $X_{k}^{\prime}$ by $X^{-e_{k}-[b_{k}]_{+}}$ in (2.1), is bar-
invariant. (In [3], $X^{-e_{k}+[b_{k}]_{+}}$ is used, but as they remarked, it
can be replaced by $X^{-e_{k}-[b_{k}]_{+}}$ and the value of $v(a)$ remains
the same. We use the latter, because then $a$ gives the $g$-vector of $E_{a}$.
This is also the choice in [13].)

It is known that $\\{E_{a}\\}$ is a $\mathbb{Z}[\mathbf{v}^{\pm 1}]$-basis of
the cluster algebra $\mathcal{A}$. However, these $E_{a}$ are not bar-
invariant in general and do not contain all the cluster monomials, and they
are inherently dependent on the choice of an initial cluster. These drawbacks
provide a motivation to consider the triangular basis (which is introduced in
[3] and recalled below) constructed from the standard monomial basis with a
built-in bar-invariance property.

######  Definition 2.1.

[3] The triangular basis $\\{C_{a}\,:\,{a}\in\mathbb{Z}^{2n}\\}$ is the unique
collection of elements in $\mathcal{A}(\tilde{X},\tilde{B})$ satisfying:

  1. (P1) Each $C_{a}$ is bar-invariant;

  2. (P2) For each $a$, $\displaystyle C_{a}-E_{a}\in\bigoplus_{a^{\prime}}\mathbf{v}\mathbb{Z}[\mathbf{v}]E_{a^{\prime}}.$

Moreover, those $a^{\prime}$ satisfy $a^{\prime}\prec a$.

It is shown in [3] that the triangular basis exists uniquely, does not depend
on the choice of an initial (acyclic) seed, and contains all cluster monomials
associated to acyclic seeds.

Note: In the rest of the paper we assume $B$ is skew-symmetric and take
$D=I_{n}$.

##  3\. Nakajima’s graded quiver varieties

In this section we recall Nakajima’s graded quiver varieties. The references
for this section are [11, 13]. There is some inconsistency in these
references. In this paper, We will start with a quiver $\mathcal{Q}^{op}$
(pardon the notation), since it is used essentially in the paper, for
Nakajima’s graded quiver varieties. Its opposite quiver $\mathcal{Q}$ is to
match the exchange matrix $B$ for quantum cluster algebras in the usual
convention.

Let $\mathcal{Q}^{\rm op}$ be a bipartite quiver with vertex set
$I=\\{1,\dots,n\\}=I_{0}\sqcup I_{1}$ such that vertices in $I_{0}$ (resp.
$I_{1}$) are sinks (resp. sources), and let $\Omega=\\{h_{1},\dots,h_{r}\\}$
be the set of arrows in $\mathcal{Q}^{\rm op}$, and denote
$\alpha_{j}:=t(h_{j})$, $\beta_{j}:=s(h_{j})$, that is, $h_{j}$ is an arrow
$\beta_{j}\to\alpha_{j}$. Let $\widetilde{\mathcal{Q}^{\rm op}}$ be the
decorated quiver obtained from $\mathcal{Q}^{\rm op}$ by adding a new vertex
$i^{\prime}$ and an arrow $i^{\prime}\to i$ if $i\in I_{0}$ (resp. an arrow
$i\to i^{\prime}$ if $i\in I_{1}$).

Following the notation in [11, §5], denote dimension vectors
$w=(w_{i},w^{\prime}_{i})_{i\in I}$, $v=(v_{i})_{i\in I}$, where all the
components are nonnegative integers.

Fix $\mathbb{C}$-vector spaces $W_{i},W^{\prime}_{i},V_{i}$ with dimensions
$w_{i},w^{\prime}_{i},v_{i}$ for $i\in I$, respectively. Define

$W:=\bigoplus_{i\in I}(W_{i}\oplus W_{i}^{\prime}),\quad V:=\bigoplus_{i\in
I}V_{i}.$

A quiver representation of $\widetilde{\mathcal{Q}^{\rm op}}$ consists linear
maps $x_{i}:W^{\prime}_{i}\to W_{i}$ ($i\in I_{0}$), $x_{i}:W_{i}\to
W_{i}^{\prime}$ ($i\in I_{1}$), and $y_{h}:W_{s(h)}\to W_{t(h)}$
($h\in\Omega$). Below is an example with $I_{0}=\\{3\\}$ and
$I_{1}=\\{1,2\\}$:

(3.1)
$\textstyle{W_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x_{1}}$$\scriptstyle{y_{1}}$$\scriptstyle{\phantom{X}y_{2}}$$\textstyle{W^{\prime}_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x_{3}}$$\textstyle{W_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x_{2}}$$\scriptstyle{y_{4}}$$\scriptstyle{\phantom{XX}y_{3}}$$\textstyle{W_{1}^{\prime}}$$\textstyle{W_{3}}$$\textstyle{W_{2}^{\prime}}$

######  Definition 3.1.

For each sink $\alpha\in I_{0}$, let $r_{1}<\cdots<r_{s}$ be the indices such
that $h_{r_{1}},\dots,h_{r_{s}}$ are all the arrows with target $\alpha$, and
define a linear map

$\displaystyle A_{\alpha}:=x_{\alpha}+\sum_{h:\ t(h)=\alpha}y_{h}:$
$\displaystyle\;W_{\alpha}^{\prime}\oplus\bigoplus_{h:\
t(h)=\alpha}W_{s(h)}\to W_{\alpha},$
$\displaystyle(a,b_{1},\dots,b_{s})\mapsto
x_{\alpha}(a)+y_{r_{1}}(b_{1})+\cdots+y_{r_{s}}(b_{r})$

For each source $\beta\in I_{1}$, let $r^{\prime}_{1}<\cdots<r^{\prime}_{t}$
be the indices such that $h_{r^{\prime}_{1}},\dots,h_{r^{\prime}_{t}}$ are all
the arrows with source $\beta$, and define a linear map

$\displaystyle A_{\beta}:=x_{\beta}\oplus\bigoplus_{h:\ s(h)=\beta}y_{h}:$
$\displaystyle\;W_{\beta}\to W_{\beta}^{\prime}\oplus\bigoplus_{h:\
s(h)=\beta}W_{t(h)},$ $\displaystyle
b\mapsto(x_{\beta}(b),y_{r^{\prime}_{1}}(b),\dots,y_{r^{\prime}_{t}}(b))$

Note that $s,r_{1},\dots,r_{s}$ depend on $\alpha$ so, strictly speaking, we
should write them as $s(\alpha)$, $r_{1}(\alpha)$, etc., but we omit $\alpha$
for simplicity of notation; similarly, $t,r^{\prime}_{1},\dots,r^{\prime}_{t}$
depend on $\beta$.

Alternatively, we can express these maps by matrices. Fix bases for $W_{i}$
and $W^{\prime}_{i}$ , and by abuse of notation we use $x_{i}$ (resp. $y_{h}$)
to denote the matrix representing the corresponding linear map. Then the above
$A_{\alpha}$ and $A_{\beta}$ are represented by the following block matrices:

(3.2) $\displaystyle
A_{\alpha}=[x_{\alpha},y_{r_{1}},\dots,y_{r_{s}}]_{\text{hor}}:=[x_{\alpha}|y_{r_{1}}|y_{r_{2}}|\ldots|y_{r_{s}}],$
$\displaystyle
A_{\beta}=[x_{\beta},y_{r^{\prime}_{1}},y_{r^{\prime}_{2}}\dots,y_{(r^{\prime}_{t})}]_{\text{vert}}:=\begin{bmatrix}x_{\beta}\\\
y_{r^{\prime}_{1}}\\\ y_{r^{\prime}_{2}}\\\ \vdots\\\
y_{r^{\prime}_{t}}\end{bmatrix}.$

Below is the list of sizes of various matrices, where $\alpha\in I_{0}$,
$\beta=I_{1}$:

$x_{\alpha}$ | $x_{\beta}$ | $t_{h}$ | $A_{\alpha}$ | $A_{\beta}$
---|---|---|---|---
$w_{\alpha}\times w^{\prime}_{\alpha}$ | $w_{\beta}^{\prime}\times w_{\beta}$ | $w_{t(h)}\times w_{s(h)}$ | $\displaystyle w_{\alpha}\times(w_{\alpha}^{\prime}+\sum_{h:\;t(h)=\alpha}w_{s(h)})$ | $\displaystyle(w^{\prime}_{\beta}+\sum_{h:\;s(h)=\beta}w_{t(h)})\times w_{\beta}$

For a vector space $E$ of dimension $e$, denote by $Gr(d,E)$ or $Gr(d,e)$ the
Grassmannian space parametrizing all $d$-dimensional linear subspaces of $E$.
Let $S_{Gr(d,e)}$ be the tautological subbundle (i.e., the universal
subbundle) on the Grassmannian variety $Gr(d,e)$.

######  Definition 3.2.

[11, §4] Let dimension vectors $w=(w_{i},w^{\prime}_{i})_{i\in I}$,
$v=(v_{i})_{i\in I}$, where all the components are nonnegative integers.

(a) The variety $\mathbf{E}_{w}$ is defined as the space of quiver
representations of the quiver $\widetilde{\mathcal{Q}^{\rm op}}$:

$\displaystyle\mathbf{E}_{w}$ $\displaystyle:=\bigoplus_{\alpha\in
I_{0}}\mathrm{Hom}(W^{\prime}_{\alpha},W_{\alpha})\oplus\bigoplus_{\beta\in
I_{1}}\mathrm{Hom}(W_{\beta},W^{\prime}_{\beta})\oplus\bigoplus_{h\in\Omega}\mathrm{Hom}(W_{s(h)},W_{t(h)})$
$\displaystyle\cong\mathbb{C}^{\sum_{i\in
I}w_{i}w_{i}^{\prime}+\sum_{h\in\Omega}w_{s(h)}w_{t(h)}}.$

We denote elements of $\mathbf{E}_{w}$ as $(x_{i},y_{h})_{i\in
I,\,h\in\Omega}$.

(b) The projective variety $\mathcal{F}_{v,w}$ is a closed subvariety of

$\prod_{\alpha\in I_{0}}Gr(v_{\alpha},W_{\alpha})\times\prod_{\beta\in
I_{1}}Gr\Big{(}v_{\beta},W^{\prime}_{\beta}\oplus\bigoplus_{h:\
s(h)=\beta}W_{t(h)}\Big{)}$

defined as

$\displaystyle\mathcal{F}_{v,w}:=\big{\\{}(X_{i})_{i\in I}\;|$
$\displaystyle\dim X_{i}=v_{i}\,(\forall i\in I),\;X_{\alpha}\subseteq
W_{\alpha}(\forall\alpha\in I_{0}),$ $\displaystyle X_{\beta}\subseteq
W_{\beta}^{\prime}\oplus\bigoplus_{h:\ s(h)=\beta}X_{t(h)}(\forall\beta\in
I_{1})\big{\\}}.$

(c) _Nakajima’s nonsingular graded quiver variety_ $\tilde{\mathcal{F}}_{v,w}$
is given by

$\displaystyle\tilde{\mathcal{F}}_{v,w}:=\big{\\{}(x_{i},y_{h},X_{i})_{i\in
I,h\in\Omega}\;|$ $\displaystyle(x_{i},y_{h})_{i\in
I,h\in\Omega}\in\mathbf{E}_{w},(X_{i})_{i\in I}\in\mathcal{F}_{v,w},$
$\displaystyle{\mathrm{im}}A_{\alpha}\subseteq X_{\alpha}\;(\forall\alpha\in
I_{0}),\;{\mathrm{im}}A_{\beta}\subseteq X_{\beta}\;(\forall\beta\in
I_{1})\big{\\}}.$

(Note that $A_{\alpha}$, $A_{\beta}$ are determined by $(x_{i},y_{h})_{i\in
I,h\in\Omega}$.)

(d) _Nakajima’s affine graded quiver variety_ is

$\displaystyle\mathbf{E}_{v,w}$ $\displaystyle:=\\{(x_{i},y_{h})_{i\in
I,h\in\Omega}\in\mathbf{E}_{w}\,|\,\mathrm{rank}A_{i}\leq v_{i}\;(\forall i\in
I)\\}.$

Nakajima showed in [11] that the above definition of Nakajima’s nonsingular
(resp. affine) graded quiver variety is equivalent to the original definition
given by the GIT quotient (resp. algebro-geometric quotient). The following
statement is also proved by Nakajima.

######  Lemma 3.3.

Given dimension vectors $w=(w_{i},w^{\prime}_{i})_{i\in I}\in\mathbb{Z}_{\geq
0}^{2n}$, $v=(v_{i})_{i\in I}\in\mathbb{Z}_{\geq 0}^{n}$.

(a) The variety $\mathbf{E}_{v,w}$ contains the origin, so is nonempty. The
variety $\mathcal{F}_{v,w}$ (thus $\tilde{\mathcal{F}}_{v,w}$) is nonempty if
and only if

(3.3) $0\leq v_{\alpha}\leq w_{\alpha}\;(\forall\alpha\in I_{0}),\;\textrm{
and }\;0\leq v_{\beta}\leq w_{\beta}^{\prime}+\sum_{h:\
s(h)=\beta}v_{t(h)}\;(\forall\beta\in I_{1}).$

(b) If (3.3) holds, then the natural projection

$\pi^{\prime}:\tilde{\mathcal{F}}_{v,w}\to\mathcal{F}_{v,w},\;(x_{i},y_{h},X_{i})_{i\in
I,h\in\Omega}\mapsto(X_{i})_{i\in I}$

gives a vector bundle of rank $\sum_{\alpha\in
I_{0}}v_{\alpha}w^{\prime}_{\alpha}+\sum_{\beta\in I_{1}}v_{\beta}w_{\beta}$.
Meanwhile, the natural projection

(3.4)
$\pi:\tilde{\mathcal{F}}_{v,w}\to\mathbf{E}_{v,w},\quad(x_{i},y_{h},X_{i})_{i\in
I,h\in\Omega}\mapsto(x_{i},y_{h})_{i\in I,h\in\Omega}$

has the zero fiber $\pi^{-1}(0)\cong\mathcal{F}_{v,w}$.

###### Proof.

Part (a) and the second statement of (b) are obvious. For the first statement
of (b), following a similar proof of [9, Lemma 4.3], we have

(3.5) $\tilde{\mathcal{F}}_{v,w}\cong\iota^{*}\Big{(}\bigoplus_{\alpha\in
I_{0}}S_{Gr(v_{\alpha},W_{\alpha})}^{\oplus
w^{\prime}_{\alpha}}\oplus\bigoplus_{\beta\in
I_{1}}S_{Gr(v_{\beta},W^{\prime}_{\beta}\oplus\bigoplus_{h:\
s(h)=\beta}{W_{t(h)}})}^{\oplus w_{\beta}}\Big{)}$

where $\iota$ is the natural embedding

$\iota:\mathcal{F}_{v,w}\to\prod_{\alpha\in
I_{0}}Gr(v_{\alpha},W_{\alpha})\times\prod_{\beta\in
I_{1}}Gr(v_{\beta},W^{\prime}_{\beta}\oplus\bigoplus_{h:\
s(h)=\beta}{W_{t(h)}}).$

It follows that $\tilde{\mathcal{F}}_{v,w}$ is a vector bundle over
$\mathcal{F}_{v,w}$ of rank $\sum_{\alpha\in
I_{0}}v_{\alpha}w^{\prime}_{\alpha}+\sum_{\beta\in I_{1}}v_{\beta}w_{\beta}$.
∎

Consider the following stratification
$\mathbf{E}_{w}=\bigcup_{v}\mathbf{E}^{\circ}_{v,w}$ where

(3.6) $\mathbf{E}^{\circ}_{v,w}:=\\{(x_{i},y_{h})_{i\in
I,h\in\Omega}\in\mathbf{E}_{w}\,|\,\mathrm{rank}A_{i}=v_{i}\;(\forall i\in
I)\\}.$

For any two vectors $u=(u_{1},\dots,u_{n})$ and $v=(v_{1},\dots,v_{n})$, we
denote $u\leq v$ if $u_{i}\leq v_{i}$ for all $i$. We say that $j$ is adjacent
to $i$ if there is an arrow $h\in\Omega$ of the form $i\to j$ or $i\leftarrow
j$; we denote this arrow as $h:i-j$ whenever the direction of the arrow is
irrelevant. Recall the $q$-analog $C_{q}$ of the Cartan matrix in [11, (3.2)]:

(3.7) $\displaystyle C_{q}v=(u_{i},u^{\prime}_{i})_{i\in I},\textrm{where
}\;u_{i}=v_{i},\;u^{\prime}_{i}=v_{i}-\sum_{h:i-j}v_{j}$

where $h$ runs over all arrows with an endpoint $i$.

######  Definition 3.4.

Given dimension vectors $w=(w_{i},w^{\prime}_{i})_{i\in I}\in\mathbb{Z}_{\geq
0}^{2n}$, $v=(v_{i})_{i\in I}\in\mathbb{Z}_{\geq 0}^{n}$. We say that $(v,w)$
is $l$-_dominant_ if $w-C_{q}v\geq 0$, or equivalently, if

(3.8) $\textrm{ For all $\forall i\in I$,}\quad w_{i}\geq v_{i},\quad
w^{\prime}_{i}\geq v_{i}-\sum_{h:i-j}v_{j}.$

(Note that the condition (3.8) implies (3.3).) Define

$\mathrm{Dom}(w):=\\{v\in\mathbb{Z}_{\geq 0}^{n}\ |\ (v,w)\textrm{ is
$l$-dominant}\\}.$ $\mathrm{Dom}:=\\{(v,w)\in\mathbb{Z}_{\geq
0}^{n}\times\mathbb{Z}_{\geq 0}^{2n}\ |\ (v,w)\textrm{ is $l$-dominant}\\}.$

In the following, we define $\bar{v}$ with the property that $(\bar{v},w)$ is
$l$-dominant and $\mathbf{E}_{v,w}=\mathbf{E}_{\bar{v},w}$ (which will be
proved in Proposition 3.7).

######  Lemma 3.5.

Let $w=(w_{i},w_{i}^{\prime})_{i\in I}\in\mathbb{Z}_{\geq 0}^{2n}$,
$v=(v_{i})_{i\in I}\in\mathbb{Z}_{\geq 0}^{n}$,
$V=\\{v^{\prime}\in\mathbb{Z}_{\geq 0}^{n}\ |\ v^{\prime}\leq v\\}$. Then
$V\cap\mathrm{Dom}(w)$ has a unique maximal element
$\bar{v}=(\bar{v}_{i})_{i\in I}$ in the sense that every $v^{\prime}\in
V\cap\mathrm{Dom}(w)$ satisfies $v^{\prime}\leq\bar{v}$. More explicitly,

(3.9)
$\bar{v}_{i}=\min\big{(}v_{i},w_{i},w^{\prime}_{i}+\sum_{h:i-j}\min(v_{j},w_{j})\big{)}.$

In particular:

– if $(v,w)$ satisfies (3.3), then for $\beta\in I_{1}$ we have
$\bar{v}_{\beta}=\min(v_{\beta},w_{\beta})$;

– if $v\in\mathrm{Dom}(w)$, then $\bar{v}=v$.

###### Proof.

It suffices to prove (3.9). It is obvious true if “$=$” is replaced by
“$\leq$”. We are left to prove that $\bar{v}$ defined in (3.9) satisfies
(3.8). By definition, $w_{i}\geq\bar{v}_{i}$ for all $i\in I$. So we only need
to show $w^{\prime}_{i}\geq\bar{v}_{i}-\sum_{h:i-j}\bar{v}_{j}$ for all $i\in
I$.

We prove by contradiction. Assume for some $i\in I$,
$w^{\prime}_{i}<\bar{v}_{i}-\sum_{h:i-j}\bar{v}_{j}$. If $i$ is an isolated
vertex (that is, not incident to any arrow in $\Omega$), then
$w^{\prime}_{i}\geq\min(v_{i},w_{i},w^{\prime}_{i})=\bar{v}_{i}=\bar{v}_{i}-\sum_{h:i-j}\bar{v}_{j}$,
a contradiction. So we can assume that $i$ is adjacent to at least one vertex.
Let $m_{i}=\min(v_{i},w_{i})$ for $i\in I$. Then
$\bar{v}_{i}=\min(m_{i},w^{\prime}_{i}+\sum_{h:j-i}m_{j})$, and

(3.10) $\displaystyle\min(m_{i}-w^{\prime}_{i},\sum_{h:i-j}m_{j})$
$\displaystyle=\bar{v}_{i}-w^{\prime}_{i}>\sum_{h:i-j}\bar{v}_{j}=\sum_{h:i-j}\min(m_{j},w^{\prime}_{j}+\sum_{h^{\prime}:j-k}m_{k})$
$\displaystyle\geq\sum_{h:i-j}\min(m_{j},w^{\prime}_{j}+m_{i}),$

where the last inequality is because $\sum_{h^{\prime}:j-k}m_{k}\geq m_{i}$
(since $h$ is one of the possible $h^{\prime}$).

If there exists an arrow $h:i-j_{0}$ such that $m_{j_{0}}\geq
w^{\prime}_{j_{0}}+m_{i}$, then

$m_{i}-w^{\prime}_{i}\geq\min_{i}(m_{i}-w^{\prime}_{i},\sum_{h:i-j}m_{j})\stackrel{{\scriptstyle\eqref{eq:min
mi}}}{{>}}\sum_{h:i-j}\min(m_{j},w^{\prime}_{j}+m_{i})\geq\min(m_{j_{0}},w^{\prime}_{j_{0}}+m_{i})=w^{\prime}_{j_{0}}+m_{i}$

which gives a contradiction. So we can assume
$\min(m_{j},w^{\prime}_{j}+m_{i})=m_{j}$ for all $j$ in the last $\sum$ of
(3.10). Thus (3.10) becomes

$\min(m_{i}-w^{\prime}_{i},\sum_{h:i-j}m_{j})>\sum_{h:i-j}m_{j}$

which again gives a contradiction. ∎

######  Lemma 3.6.

Fix $w=(w_{i},w_{i}^{\prime})_{i\in I}\in\mathbb{Z}_{\geq 0}^{2n}$,
$v=(v_{i})_{i\in I}\in\mathbb{Z}_{\geq 0}^{n}$. Let
$\pi:\tilde{\mathcal{F}}_{v,w}\to\mathbf{E}_{v,w}$ be the map defined in
(3.4).

(a) For each $v^{\prime}\leq v$, the variety
$\mathbf{E}^{\circ}_{v^{\prime},w}$ defined in (3.6) is nonempty if and only
if $(v^{\prime},w)$ is $l$-dominant. In this case, it is nonsingular and
locally closed in $\mathbf{E}_{w}$ (so is also locally closed in
$\mathbf{E}_{v,w}$), and is irreducible and rational. Thus the variety
$\mathbf{E}_{v,w}$ has a stratification

$\mathbf{E}_{v,w}=\bigcup_{v^{\prime}}\mathbf{E}^{\circ}_{v^{\prime},w}\,,\quad\textrm{where
$v^{\prime}\leq v$ and $(v^{\prime},w)$ are $l$-dominant}.$

(b) For each $v^{\prime}\leq v$ with $(v^{\prime},w)$ being $l$-dominant, the
restricted projection
$\pi^{-1}(\mathbf{E}^{\circ}_{v^{\prime},w})\to\mathbf{E}^{\circ}_{v^{\prime},w}$
is a Zariski locally trivial $\mathcal{M}$-bundle, where $\mathcal{M}$ itself
is a

$\Big{(}\prod_{\beta\in
I_{1}}Gr(v_{\beta}-v^{\prime}_{\beta},w_{\beta}^{\prime}-v^{\prime}_{\beta}+\sum_{h:s(h)=\beta}v_{t(h)}\Big{)}\textrm{-bundle
over }\prod_{\alpha\in
I_{0}}Gr(v_{\alpha}-v^{\prime}_{\alpha},w_{\alpha}-v^{\prime}_{\alpha})$

defined as

$\displaystyle\mathcal{M}:=\Big{\\{}(X^{\prime}_{i})_{i\in I}\Big{|}$
$\displaystyle X_{\alpha}^{\prime}\in
Gr(v_{\alpha}-v_{\alpha}^{\prime},w_{\alpha}-v_{\alpha}^{\prime})\
(\forall\alpha\in I_{0}),$ $\displaystyle X_{\beta}^{\prime}\in
Gr\big{(}v_{\beta}-v_{\beta}^{\prime},\mathbb{A}^{w_{\beta}^{\prime}-v_{\beta}^{\prime}+\sum_{h:s(h)=\beta}v^{\prime}_{t(h)}}\oplus\bigoplus_{h:s(h)=\beta}(X_{t(h)}^{\prime}\big{)}\
(\forall\beta\in I_{1})\Big{\\}}.$

In particular, $\mathcal{F}_{v,w}$ is a $\prod_{\beta\in
I_{1}}Gr(v_{\beta},w_{\beta}^{\prime}+\sum_{h:s(h)=\beta}v_{t(h)})$-bundle
over $\prod_{\alpha\in I_{0}}Gr(v_{\alpha},w_{\alpha})$.

###### Proof.

The proof is similar to the proof of [9, Lemma 4.6].

(a) $\mathbf{E}^{\circ}_{v^{\prime},w}\subseteq\mathbf{E}_{w}$ is locally
closed because

$\displaystyle\mathbf{E}^{\circ}_{v^{\prime},w}=$
$\displaystyle\\{(x_{i},y_{h})_{i\in
I,h\in\Omega}\in\mathbf{E}_{w}\,|\,\mathrm{rank}A_{i}\leq
v^{\prime}_{i}(\forall i)\\}$ $\displaystyle-\bigcup_{j\in
I}\\{(x_{i},y_{h})_{i\in
I,h\in\Omega}\in\mathbf{E}_{w}\,|\,\mathrm{rank}A_{j}\leq
v^{\prime}_{j}-1,\;\mathrm{rank}A_{i}\leq v^{\prime}_{i}(\forall i)\\}$

To show $\mathbf{E}^{\circ}_{v^{\prime},w}$ is nonsingular, we consider the
following open covering

(3.11) $\mathbf{E}^{\circ}_{v^{\prime},w}=\bigcup U_{\mathbf{J}}$

where ${\bf J}=(J_{i})_{i\in I}$, $J_{i}\subseteq\\{1,\dots,w_{i}\\}$ and
$|J_{i}|=v_{i}^{\prime}$, $U_{\bf
J}\subseteq\mathbf{E}^{\circ}_{v^{\prime},w}$ consists of those
$(x_{i},y_{h})$ such that for all $i\in I_{0}$ (resp. $i\in I_{1}$), the rows
(resp. column) of $A_{i}$ indexed by $J_{i}$ are linearly independent. We will
show that $U_{\bf J}$ is isomorphic to an open subset
$\tilde{\mathbb{A}}^{\circ}$ of an affine space $\tilde{\mathbb{A}}$, which
implies that $\mathbf{E}^{\circ}_{v^{\prime},w}$ is nonsingular.

Without loss of generality, assume $J_{i}=\\{1,\dots,v_{i}^{\prime}\\}$.
Denote

$\displaystyle\tilde{\mathbb{A}}$ $\displaystyle:=\prod_{\alpha\in
I_{0}}\mathbb{A}^{(w_{\alpha}-v_{\alpha}^{\prime})\times
v_{\alpha}^{\prime}}\times\prod_{\beta\in
I_{1}}\mathbb{A}^{v_{\beta}^{\prime}\times(w_{\beta}-v_{\beta}^{\prime})}\times\prod_{\alpha\in
I_{0}}\mathbb{A}^{v_{\alpha}^{\prime}\times
w_{\alpha}^{\prime}}\times\prod_{\beta\in
I_{1}}\mathbb{A}^{w_{\beta}^{\prime}\times
v_{\beta}^{\prime}}\times\prod_{h\in\Omega}\mathbb{A}^{v^{\prime}_{t(h)}\times
v^{\prime}_{s(h)}}$ $\displaystyle\cong\mathbb{A}^{\sum_{i\in
I}v^{\prime}_{i}w_{i}+\sum_{h\in\Omega}v^{\prime}_{t(h)}v^{\prime}_{s(h)}}$

and write its elements as $(P_{i},Z_{i},C_{h})_{i\in I,h\in\Omega}$ where the
size of each matrix is given in the following table: (where $\alpha\in I_{0}$,
$\beta\in I_{1}$, $h\in\Omega$)

$P_{\alpha}$ | $P_{\beta}$ | $Z_{\alpha}$ | $Z_{\beta}$ | $C_{h}$
---|---|---|---|---
$(w_{\alpha}-v^{\prime}_{\alpha})\times v^{\prime}_{\alpha}$ | $v_{\beta}^{\prime}\times(w_{\beta}-v_{\beta}^{\prime})$ | $v_{\alpha}^{\prime}\times w_{\alpha}^{\prime}$ | $w_{\beta}^{\prime}\times v_{\beta}^{\prime}$ | $v^{\prime}_{t(h)}\times v^{\prime}_{s(h)}$

Then define a morphism

$\displaystyle\varphi:\tilde{\mathbb{A}}$ $\displaystyle\to\mathbf{E}_{w}$
$\displaystyle(P_{i},Z_{i},C_{h})_{i\in I,h\in\Omega}$
$\displaystyle\mapsto(x_{i},y_{h})_{i\in I,h\in\Omega}$

where, $x_{\alpha}=\begin{bmatrix}Z_{\alpha}\\\
P_{\alpha}Z_{\alpha}\end{bmatrix}$ for $\alpha\in I_{0}$;
$x_{\beta}=\begin{bmatrix}Z_{\beta}&Z_{\beta}P_{\beta}\end{bmatrix}$ for
$\beta\in I_{1}$; $y_{h}=\begin{bmatrix}C_{h}&C_{h}P_{\beta}\\\
P_{\alpha}C_{h}&P_{\alpha}C_{h}Q_{\beta}\end{bmatrix}$ for
$h:\beta\to\alpha\in\Omega$. It is easy to see that ${\rm
im}\varphi\subseteq\mathbf{E}_{v^{\prime},w}$.

For $\alpha\in I_{0}$, $\beta\in I_{1}$, let $r_{1},\dots,r_{s}$ and
$r^{\prime}_{1},\dots,r^{\prime}_{t}$ be given as in Definition 3.1. Define

(3.12) $A^{\prime}_{\alpha}:=[Z_{\alpha}\ C_{r_{1}},\ \cdots,\
C_{r_{s}}]_{\text{hor}},\quad A^{\prime}_{\beta}:=[Z_{\beta}\
C_{r^{\prime}_{1}},\ \cdots,\ C_{r^{\prime}_{t}}]_{\text{vert}}$

Note that the size of $A^{\prime}_{\alpha}$ is
$v^{\prime}_{\alpha}\times(w^{\prime}_{\alpha}+\sum_{h:\
t(h)=\alpha}v^{\prime}_{s(h)})$, so its number of rows $\leq$ its number of
columns when $(v^{\prime},w)$ is $l$-dominant; similarly, the size of
$A^{\prime}_{\beta}$ is $(w^{\prime}_{\beta}+\sum_{h:\
s(h)=\beta}v^{\prime}_{t(h)})\times v^{\prime}_{\beta}$, so its number of
columns $\leq$ its number of rows when $(v^{\prime},w)$ is $l$-dominant.

Define
$\tilde{\mathbb{A}}^{\circ}:=\big{\\{}(P_{i},Z_{i},C_{h})\in\tilde{\mathbb{A}}\;\Bigg{|}\;\mathrm{rank}A^{\prime}_{i}=v^{\prime}_{i}\;(\forall
i\in I)\big{\\}}$ to be an open subset of $\tilde{\mathbb{A}}$.

Similar to the proof of [9, Lemma 4.6], it can be shown that:

(i) $\varphi$ restricts to an isomorphism
$\varphi|_{\tilde{\mathbb{A}}^{\circ}}:\
\tilde{\mathbb{A}}^{\circ}\stackrel{{\scriptstyle\cong}}{{\to}}U_{\bf J}$;

(ii) $\mathbf{E}^{\circ}_{v^{\prime},w}$ is nonempty if and only if
$(v^{\prime},w)$ is $l$-dominant;

(iii) if $\mathbf{E}^{\circ}_{v^{\prime},w}$ is nonempty, it is irreducible.

It follows that $\mathbf{E}^{\circ}_{v^{\prime},w}$ is rational.

As pointed out in the proof of [9, Lemma 4.6], the proof of the “if” part of
(ii) and the proof of (iii) therein make use of some generic choice.
Alternatively, We give the new proofs below that do not use the generic
choice.

For the “if” part of (ii): it suffices to show that
$\tilde{\mathbb{A}}^{\circ}$ is nonempty by constructing a point on it. Note
that $P_{i}$ are irrelevant so we let them be zero matrices. We construct
$Z_{i},C_{h}$ as follows: start with all $Z_{i}$, $C_{h}$ being zero matrices,
then for each $A^{\prime}_{\alpha}$ and $A^{\prime}_{\beta}$ defined in
(3.12), change the diagonal entries (that is, the $(j,j)$\- entry for all $j$)
to 1 and make the corresponding change in $Z_{i}$ and $C_{h}$. We showed that
the resulting $(P_{i},Z_{i},C_{h})$ satisfies the equality
$\mathrm{rank}A^{\prime}_{i}=v^{\prime}_{i}\;(\forall i\in I)$. Without loss
of generality, we only need to show
$\mathrm{rank}A^{\prime}_{\alpha}=v^{\prime}_{\alpha}$ for $\alpha\in I_{0}$.
We claim the $A^{\prime}_{\alpha}$ is an “upper triangular matrix” in the
sense that its $(i,j)$-entries are $0$ whenever $i>j$; then together with the
fact that its diagonal entries are 1, we can conclude that
$\mathrm{rank}A^{\prime}_{\alpha}=v^{\prime}_{\alpha}$ (that is,
$A^{\prime}_{\alpha}$ has full row rank). Assume the contrary that some
$(i,j)$-entry of $A^{\prime}_{\alpha}$ is $1$ with $i>j$. Then it corresponds
to the $(i,j^{\prime})$-entry of $C_{h}$ for some arrow $h:\beta\to\alpha$ and
$j^{\prime}\leq j$; moreover it must correspond to a diagonal entry of
$A^{\prime}_{\beta}$. So it is simultaneously the
$(j^{\prime},j^{\prime})$-entry of $A^{\prime}_{\beta}$ and the
$(i,j^{\prime})$-entry of $A^{\prime}_{\alpha}$. In particular,
$i=j^{\prime}$. However, $i>j\geq j^{\prime}$, a contradiction.

For (iii), assuming $(v^{\prime},w)$ is $l$-dominant (so
$\mathbf{E}^{\circ}_{v^{\prime},w}$ is nonempty), we shall prove that
$\mathbf{E}^{\circ}_{v^{\prime},w}$ is irreducible. It suffices to show that
$U_{\bf J}\cap U_{\bf J^{\prime}}\neq\emptyset$ where ${\bf J}=(J_{i})_{i\in
I}$ and ${\bf J^{\prime}}=(J^{\prime}_{i})_{i\in I}$ satisfy the following
condition: there exists $i_{0}\in I$ such that $J_{i_{0}}$ and
$J^{\prime}_{i_{0}}$ only differ by one element, and $J_{i}=J^{\prime}_{i}$
for $i\neq i_{0}$. Without loss of generality, we can assume $i_{0}=\alpha\in
I_{0}$, $J_{\alpha}=\\{1,\dots,v^{\prime}_{\alpha}\\}$,
$J^{\prime}_{\alpha}=\\{1,\dots,v^{\prime}_{\alpha-1},v^{\prime}_{\alpha}\\}$.
Define $Z_{i}(i\in I),C_{h}(h\in\Omega)$ to be the same as the last paragraph,
$P_{i}(i\in I)$ be any matrices, under the condition that the first row of
$P_{\alpha}$ is $[0,\dots,0,1]$. Then in the $v^{\prime}_{\alpha}$-th row and
the $(v^{\prime}_{\alpha}+1)$-th row of $A^{\prime}_{\alpha}$ are identical.
It is then obvious that $\varphi((P_{i},Z_{i},C_{h}))$ is a point in $U_{\bf
J}\cap U_{\bf J^{\prime}}$, thus the latter is nonempty.

(b) Similar to the proof of [9, Lemma 4.6]: let

$\displaystyle\mathcal{X}:=\big{\\{}(X_{i},Y_{i})_{i\in I}\ |\ (Y_{i})_{i\in
I}\in\mathcal{F}_{v^{\prime},w},(X_{i})_{i\in
I}\in\mathcal{F}_{v,w},Y_{i}\subseteq X_{i}(\forall i\in I)\big{\\}},$
$\displaystyle f:(x_{i},y_{h})_{i\in I,h\in\Omega}\mapsto(X_{i})_{i\in
I},\;\textrm{ where }X_{i}=\text{im}A_{i},$ $\displaystyle
q:(X_{i},Y_{i})_{i\in I}\mapsto(Y_{i})_{i\in I}.$

Then
$\pi^{-1}(\mathbf{E}^{\circ}_{v^{\prime},w})\to\mathbf{E}^{\circ}_{v^{\prime},w}$
is the pullback of the locally trivial bundle
$q:\mathcal{X}\to\mathcal{F}_{v^{\prime},w}$:

$\textstyle{\pi^{-1}(\mathbf{E}^{\circ}_{v^{\prime},w})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\mathbf{E}^{\circ}_{v^{\prime},w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathcal{F}_{v^{\prime},w}}$

and the fibers of $q$ are isomorphic to $\mathcal{M}$. ∎

######  Proposition 3.7.

Let $v,w,\pi$ be the same as Lemma 3.6 and $\bar{v}$ be as in Lemma 3.5.

(a) $\mathbf{E}_{v,w}=\mathbf{E}_{\bar{v},w}$ is irreducible. Moreover,
$\mathbf{E}^{\circ}_{\bar{v},w}$ is its largest stratum; so
$\mathbf{E}_{v,w}=\overline{\mathbf{E}^{\circ}_{\bar{v},w}}$, the Zariski
closure of $\mathbf{E}^{\circ}_{\bar{v},w}$.

(b) Assume (3.3). Then $\tilde{\mathcal{F}}_{v,w}$ and $\mathcal{F}_{v,w}$ are
irreducible and nonsingular and $\pi$ is surjective. Moreover,

$\displaystyle\dim\mathcal{F}_{v,w}=d_{v,w}$ $\displaystyle:=\sum_{\beta\in
I_{1}}({w_{\beta}^{\prime}+\sum_{h:s(h)=\beta}v_{t(h)}}-v_{\beta}){v_{\beta}}+\sum_{\alpha\in
I_{0}}({w_{\alpha}}-v_{\alpha}){v_{\alpha}}$ $\displaystyle=\sum_{\alpha\in
I_{0}}w_{\alpha}v_{\alpha}+\sum_{\beta\in
I_{1}}w^{\prime}_{\beta}v_{\beta}-\sum_{i\in
I}v_{i}^{2}+\sum_{h\in\Omega}v_{s(h)}v_{t(h)}$
$\displaystyle\dim\tilde{\mathcal{F}}_{v,w}=\tilde{d}_{v,w}$
$\displaystyle:=\dim\mathcal{F}_{v,w}+\sum_{\alpha\in
I_{0}}v_{\alpha}w^{\prime}_{\alpha}+\sum_{\beta\in I_{1}}v_{\beta}w_{\beta}$
$\displaystyle=\sum_{\alpha\in
I_{0}}(w_{\alpha}+w^{\prime}_{\alpha})v_{\alpha}+\sum_{\beta\in
I_{1}}(w_{\beta}+w^{\prime}_{\beta})v_{\beta}-\sum_{i\in
I}v_{i}^{2}+\sum_{h\in\Omega}v_{s(h)}v_{t(h)}$
$\displaystyle\dim\mathbf{E}_{v,w}$
$\displaystyle=\dim\mathbf{E}_{\bar{v},w}=\tilde{d}_{\bar{v},w}$

(c) Further assume that $(v,w)$ is $l$-dominant. Then $\pi$ is birational. In
this case, it restricts to an isomorphism $\pi^{-1}({\bf
E}^{\circ}_{v,w})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\bf
E}^{\circ}_{v,w}$.

###### Proof.

Similar to the proof of [9, Proposition 4.8]. ∎

##  4\. Decomposition theorem

The following fact is proved by Nakajima in [12, Theorem 14.3.2] using the
representation theory of quantum affine algebras. We will give a proof similar
to [9, Theorem 5.1]; instead of representation theory, we use the algebraic
version of the Transversal Slice Theorem.

######  Theorem 4.1.

Assume ${\tilde{\mathcal{F}}_{v,w}}\neq\emptyset$. The local system appeared
in the BBDG decomposition for $\pi_{*}IC_{\tilde{\mathcal{F}}_{v,w}}$ are all
trivial. Thus,

(4.1)
$\pi_{*}(IC_{\tilde{\mathcal{F}}_{v,w}})=\bigoplus_{v^{\prime}}\bigoplus_{d}a^{d}_{v,v^{\prime};w}IC_{\mathbf{E}_{v^{\prime},w}}[d]$

where $v^{\prime}\leq v$ satisfies the condition that $(v^{\prime},w)$ is
$l$-dominant, and $a^{d}_{v,v^{\prime};w}\in\mathbb{Z}_{\geq 0}$.

The proof is delayed to §4.2. First, recall:

######  Theorem 4.2 (BBDG Decomposition Theorem).

[1] Let $f:X\to Y$ be a proper algebraic morphism between complex algebraic
varieties. Then there is a finite list of triples $(Y_{a},L_{a},n_{a})$, where
for each $a$, $Y_{a}$ is locally closed smooth irreducible algebraic
subvariety of $Y$, $L_{a}$ is a semisimple local system on $Y_{a}$, $n_{a}$ is
an integer, such that:

(4.2) $f_{*}IC_{X}\cong\bigoplus_{a}IC_{\overline{Y_{a}}}(L_{a})[n_{a}]$

Moreover, even though the isomorphism is not necessarily canonical, the direct
summands appeared on the right hand side are canonical.

###  4.1. Algebraic Transversal Slice Theorem

Nakajima proves an analytic transversal slice theorem [12, §3] and we
strengthen the result to an algebraic version for the varieties studied in
this paper using an explicit construction similar to the one in [9].

######  Lemma 4.3 (Algebraic Transversal Slice Theorem).

Let $p\in\mathbf{E}_{v,w}$ be a point in the stratum
$\mathbf{E}^{\circ}_{v^{0},w}$. So $v^{0}\leq v$. Define

$w^{\perp}=w-C_{q}v^{0}=(w^{\perp}_{i},{w^{\perp}_{i}}^{\prime})_{i\in
I},\quad v^{\perp}=v-v^{0}=(v^{\perp}_{i})_{i\in I},$

that is, $w^{\perp}_{i}=w_{i}-v_{i}^{0}$,
${w^{\perp}_{i}}^{\prime}=w^{\prime}_{i}-v_{i}^{0}+\sum_{h:j-i}v_{j}^{0}$,
$v^{\perp}_{i}=v_{i}-v_{i}^{0}$. Take
$U^{\perp}=\mathbf{E}_{v^{\perp},w^{\perp}}$. Then there exist Zariski open
neighborhoods $U\subseteq\mathbf{E}_{v,w}$ of $p$,
$U^{0}\subseteq\mathbf{E}^{\circ}_{v^{0},w}$ of $p$, and isomorphisms
$\varphi,\psi$ making the following diagram commute:

$\textstyle{\tilde{\mathcal{F}}_{v,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\supseteq}$$\textstyle{\pi^{-1}U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\varphi}$$\scriptstyle{\cong}$$\textstyle{U^{0}\times\pi^{-1}(U^{\perp})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\times\pi}$$\scriptstyle{\subseteq}$$\textstyle{\mathbf{E}^{\circ}_{v^{0},w}\times\tilde{\mathcal{F}}_{v^{\perp},w^{\perp}}}$$\textstyle{\mathbf{E}_{v,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\supseteq}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\cong}$$\textstyle{U^{0}\times
U^{\perp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{\mathbf{E}^{\circ}_{v^{0},w}\times\mathbf{E}_{v^{\perp},w^{\perp}}}$

We can further assume $U$ and $U^{0}$ are both invariant under the natural
$\mathbb{C}^{*}$-action (by multiplying all entries with the same scalar).

Moreover, $\psi(p)=(p,0)$, and the diagram is compatible with the
stratifications in the sense that $\psi({\bf E}^{\circ}_{u,w}\cap
U)=U_{0}\times{\bf E}^{\circ}_{u^{\perp},w^{\perp}}$ for each $u$ satisfying
$v^{0}\leq u\leq v$, where $u^{\perp}=u-v^{0}$.

###### Proof.

(1) We will define $\psi$. Recall that the point $p$ determines $n$ matrices
as in (3.2), denoted $A_{i}(p)$ ($i\in I$). Since
$p\in\mathbf{E}^{\circ}_{v^{0},w}$, without loss of generality, we can assume
that, for $i\in I_{0}$, the first $v_{i}^{0}$ row vectors of $A_{i}(p)$ are
linearly independent; and for $i\in I_{1}$, the first $v_{i}^{0}$ column
vectors of $A_{i}(p)$ are linearly independent. Then for $\alpha\in I_{0}$,
$\beta\in I_{1}$, there exist matrices with the indicated sizes:

$P_{\alpha}(p)$ | $P_{\beta}(p)$ | $Z_{\alpha}(p)$ | $Z_{\beta}(p)$ | $C_{h}(p)$
---|---|---|---|---
$(w_{\alpha}-v^{0}_{\alpha})\times v^{0}_{\alpha}$ | $v^{0}_{\beta}\times(w_{\beta}-v^{0}_{\beta})$ | $v^{0}_{\alpha}\times w_{\alpha}^{\prime}$ | $w_{\beta}^{\prime}\times v^{0}_{\beta}$ | $v^{0}_{t(h)}\times v^{0}_{s(h)}$

such that (where for simplicity we drop the parameter $p$ for $Z_{\alpha}$,
$C_{r_{1}}$, etc.)

$\displaystyle
A_{\alpha}(p)=\begin{bmatrix}Z_{\alpha}&C_{r_{1}}&C_{r_{1}}P_{\beta_{r_{1}}}&\cdots&C_{r_{s}}&C_{r_{s}}P_{\beta_{r_{s}}}\\\
P_{\alpha}Z_{\alpha}&P_{\alpha}C_{r_{1}}&P_{\alpha}C_{r_{1}}P_{\beta_{r_{1}}}&\cdots&P_{\alpha}C_{r_{s}}&P_{\alpha}C_{r_{s}}P_{\beta_{r_{s}}}\end{bmatrix},$
$\displaystyle A_{\beta}(p)=\begin{bmatrix}Z_{\beta}&Z_{\beta}P_{\beta}\\\
C_{r^{\prime}_{1}}&C_{r^{\prime}_{1}}P_{\beta}\\\
P_{\alpha_{r^{\prime}_{1}}}C_{r^{\prime}_{1}}&P_{\alpha_{r^{\prime}_{1}}}C_{r^{\prime}_{1}}P_{\beta}\\\
\vdots&\vdots\\\ C_{r^{\prime}_{t}}&C_{r^{\prime}_{t}}P_{\beta}\\\
P_{\alpha_{r^{\prime}_{t}}}C_{r^{\prime}_{t}}&P_{\alpha_{r^{\prime}_{t}}}C_{r^{\prime}_{t}}P_{\beta}\end{bmatrix},$

and that the corresponding $A^{\prime}_{\alpha}(p)$ of size
$v^{0}_{\alpha}\times(w^{\prime}_{\alpha}+\sum_{h:\ t(h)=\alpha}v^{0}_{s(h)})$
and $A^{\prime}_{\beta}(p)$ of size $(w^{\prime}_{\beta}+\sum_{h:\
s(h)=\beta}v^{0}_{t(h)})\times v^{0}_{\beta}$, as defined in (3.12), are full
rank.

Step 1: define $U$ and $U^{0}$. For $\alpha\in I_{0}$, define $M_{\alpha}$ to
be the index set of the columns of $A_{\alpha}(p)$ that correspond to the
columns of $A^{\prime}_{\alpha}(p)$. The submatrix
$A^{\prime}_{\alpha}(p):=A_{\alpha}(p)_{[1,\dots,v_{\alpha}^{0};M_{\alpha}]}$
has rank $v_{\alpha}^{0}$, thus there exists a subset $J_{\alpha}\subseteq
M_{\alpha}$ of cardinality $v_{\alpha}^{0}$ such that
$A_{\alpha}(p)_{[1,\dots,v_{\alpha}^{0};J_{\alpha}]}$ is an invertible square
matrix. Similarly, for $\beta\in I_{1}$, define $M_{\beta}$ to be the index
set of the rows of $A_{\beta}(p)$ that correspond to the rows of
$A^{\prime}_{\beta}(p)$, and there exists $J_{\beta}\subseteq M_{\beta}$ of
cardinality $v_{\beta}^{0}$ such that
$A_{\beta}(p)_{[J_{\beta};1,\dots,v_{\beta}^{0}]}$ is an invertible square
matrix. Define an open subset $U$ (which depends on $J_{\alpha}$) of
$\mathbf{E}_{v,w}$ as

$U:=\big{\\{}q\in\mathbf{E}_{v,w}\ \big{|}\ \textrm{
$A_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};J_{\alpha}]}$ ($\forall\alpha\in
I_{0}$) and $A_{\beta}(q)_{[J_{\beta};1,\dots,v_{\beta}^{0}]}$
($\forall\beta\in I_{1}$) are invertible}\big{\\}}.$

Define $U^{0}:=U\cap\mathbf{E}_{v^{0},w}$ which is an open subset of
$\mathbf{E}^{\circ}_{v^{0},w}$.So

$U^{0}=\big{\\{}q\in\mathbf{E}_{v^{0},w}\ \big{|}\ \textrm{
$A_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};J_{\alpha}]}$ ($\forall\alpha\in
I_{0}$) and $A_{\beta}(q)_{[J_{\beta};1,\dots,v_{\beta}^{0}]}$
($\forall\beta\in I_{1}$) are invertible}\big{\\}}.$

Obviously, both $U$ and $U^{0}$ are invariant under the natural
$\mathbb{C}^{*}$-action.

Step 2: in the rest we assume $q\in{\bf E}^{\circ}_{u,w}\cap U$. We construct
$q^{0}$ as follows. Denote the following (where we drop the parameter $q$ of
$Z_{\alpha}$, $Z^{\prime}_{\alpha}$, etc.)

$A_{\alpha}(q)=\begin{bmatrix}Z_{\alpha}&C_{r_{1}}&C_{r_{1}}^{\prime}&\cdots&C_{r_{s}}&C_{r_{s}}^{\prime}\\\
Z^{\prime}_{\alpha}&C_{r_{1}}^{\prime\prime}&C_{r_{1}}^{\prime\prime\prime}&\cdots&C_{r_{s}}^{\prime\prime}&C_{r_{s}}^{\prime\prime\prime}\end{bmatrix},\quad
A_{\beta}(q)=\begin{bmatrix}Z_{\beta}&Z^{\prime}_{\beta}\\\
C_{r^{\prime}_{1}}&C_{r^{\prime}_{1}}^{\prime}\\\
C_{r^{\prime}_{1}}^{\prime\prime}&C_{r^{\prime}_{1}}^{\prime\prime\prime}\\\
\vdots&\vdots\\\ C_{r^{\prime}_{t}}&C_{r^{\prime}_{t}}^{\prime}\\\
C_{r^{\prime}_{t}}^{\prime\prime}&C_{r^{\prime}_{t}}^{\prime\prime\prime}\end{bmatrix}$

where the sizes of the blocks are as follows:

block | $Z_{\alpha}(q)$ | $Z_{\alpha}^{\prime}(q)$ | $Z_{\beta}(q)$ | $Z_{\beta}^{\prime}(q)$
---|---|---|---|---
size | $v_{\alpha}^{0}\times w^{\prime}_{\alpha}$ | $(w_{\alpha}-v_{\alpha}^{0})\times w^{\prime}_{\alpha}$ | $w^{\prime}_{\beta}\times v_{\beta}^{0}$ | $w^{\prime}_{\beta}\times(w_{\beta}-v_{\beta}^{0})$
block | $C_{j}(q)$ | $C^{\prime}_{j}(q)$ | $C^{\prime\prime}_{j}(q)$ | $C^{\prime\prime\prime}_{j}(q)$
---|---|---|---|---
size | $v_{\alpha_{j}}^{0}\times v_{\beta_{j}}^{0}$ | $v_{\alpha_{j}}^{0}\times(w_{\beta_{j}}-v_{\beta_{j}}^{0})$ | $(w^{\prime}_{\alpha_{j}}-v_{\alpha_{j}}^{0})\times v_{\beta_{j}}^{0}$ | $(w^{\prime}_{\alpha_{j}}-v_{\alpha_{j}}^{0})\times(w_{\beta_{j}}-v_{\beta_{j}}^{0})$

Since $A_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};J_{\alpha}]}$ ($\forall\alpha\in
I_{0}$) and $A_{\beta}(q)_{[J_{\beta};1,\dots,v_{\beta}^{0}]}$
($\forall\beta\in I_{1}$) are invertible, there are unique
$(w_{\alpha}-v_{\alpha}^{0})\times v_{\alpha}^{0}$ -matrix $P_{\alpha}(q)$
($\forall\alpha\in I_{0}$) and $v_{\beta}^{0}\times(w_{\beta}-v_{\beta}^{0})$
-matrix $P_{\beta}(q)$ ($\forall\beta\in I_{1}$) such that (where we omit the
parameter $q$ of $P_{\alpha}$, $P_{\beta}$)

(4.3)
$A_{\alpha}(q)_{[-;J_{\alpha}]}=\begin{bmatrix}A_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};J_{\alpha}]}\\\
P_{\alpha}A_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};J_{\alpha}]}\end{bmatrix},\
A_{\beta}(q)_{[J_{\beta};-]}=\bigg{[}A_{\beta}(q)_{[J_{\beta};1,\dots,v_{\beta}^{0}]}\;\;A_{\beta}(q)_{[J_{\beta};1,\dots,v_{\beta}^{0}]}P_{\beta}\bigg{]}.$

Define $q^{0}=(x_{i}^{0},y_{h}^{0})_{i\in I,h\in\Omega}$ where

$x_{\alpha}^{0}=\begin{bmatrix}Z_{\alpha}\\\
P_{\alpha}Z_{\alpha}\end{bmatrix},x_{\beta}^{0}=\begin{bmatrix}Z_{\beta}&Z_{\beta}P_{\beta}\end{bmatrix},y_{j}^{0}=\begin{bmatrix}C_{j}&C_{j}P_{\beta_{j}}\\\
P_{\alpha_{j}}C_{j}&P_{\alpha_{j}}C_{j}P_{\beta_{j}}\end{bmatrix}$

(where we omitted the parameter $q$). Then $q^{0}$ is in $U^{0}$.

Note that the projection $U\to U^{0}$, $q\mapsto q^{0}$ is compatible with the
$\mathbb{C}^{*}$-action.

Step 3: we let $U^{\perp}=\mathbf{E}^{\circ}_{v^{\perp},w^{\perp}}$ and
construct $q^{\perp}=(x_{i}^{\perp},y_{h}^{\perp})_{i\in I,h\in\Omega}$ as
follows. Define

$\widetilde{P}_{\alpha}:=\begin{bmatrix}{\bf I}_{v_{\alpha}^{0}}&0\\\
-P_{\alpha}&{\bf I}_{w_{\alpha}-v_{\alpha}^{0}}\end{bmatrix}(\forall\alpha\in
I_{0}),\quad\widetilde{P}_{\beta}:=\begin{bmatrix}{\bf
I}_{v_{\beta}^{0}}&-P_{\beta}\\\ 0&{\bf
I}_{w_{\beta}-v_{\beta}^{0}}\end{bmatrix}(\forall\beta\in I_{1})$

and for $\alpha\in I_{0}$, define (where we omit the parameter $q$)

$\widetilde{A}_{\alpha}(q):=[\widetilde{P}_{\alpha}x_{\alpha},\widetilde{P}_{\alpha}y_{r_{1}}\widetilde{P}_{\beta_{r_{1}}},\cdots,\widetilde{P}_{\alpha}y_{r_{s}}\widetilde{P}_{\beta_{r_{s}}}]_{\rm
hor},$

It is easy to see that
$\mathrm{rank}\widetilde{A}_{\alpha}(q)=\mathrm{rank}A_{\alpha}(q)=u_{\alpha}$,
and
$\widetilde{A}_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};J_{\alpha}]}=\begin{bmatrix}A_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};J_{\alpha}]}\\\
0\end{bmatrix}$.

Note that the space spanned by the top $v_{\alpha}^{0}$ rows of
$\widetilde{A}_{\alpha}(q)$ intersects only at the origin with the space
spanned by the rest rows. Since
$\mathrm{rank}\widetilde{A}_{\alpha}(q)=\mathrm{rank}A_{\alpha}(q)=u_{\alpha}\leq
v_{\alpha}$, and that
$\mathrm{rank}\widetilde{A}_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};-]}=v_{\alpha}^{0}$
(because
$\widetilde{A}_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};J_{\alpha}]}=A_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};J_{\alpha}]}$
is invertible), we have

$\mathrm{rank}\widetilde{A}_{\alpha}(q)_{[v_{\alpha}^{0}+1,\dots,w_{\alpha};-]}=\mathrm{rank}\widetilde{A}_{\alpha}(q)-\mathrm{rank}\widetilde{A}_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};-]}=u_{\alpha}-v_{\alpha}^{0}=u_{\alpha}^{\perp}\leq
v_{\alpha}^{\perp}.$

We want $A_{\alpha}(q)^{\perp}$ to be
$\widetilde{A}_{\alpha}(q)_{[v_{\alpha}^{0}+1,\dots,w_{\alpha};\\{1,\dots,w_{\alpha}^{\prime}+\sum_{h:\;t(h)=\alpha}w_{s(h)}\\}\setminus
J_{\alpha}]}$ with an appropriate rearrangement of the columns; to be more
precise (recall that $M_{\alpha}$ is defined in Step 1):

$A_{\alpha}(q)^{\perp}=\Big{[}\widetilde{A}_{\alpha}(q)_{[v_{\alpha}^{0}+1,\dots,w_{\alpha};M_{\alpha}\setminus
J_{\alpha}]}\ \vline\
P_{\alpha}C_{r_{1}}P_{\beta_{r_{1}}}-P_{\alpha}C^{\prime}_{r_{1}}-C^{\prime\prime}_{r_{1}}P_{\beta_{r_{1}}}+C_{r_{1}}^{\prime\prime\prime}\
\vline\ \cdots\Big{]}$

For this, we denote $J^{\prime}_{\alpha}=\iota^{-1}(J_{\alpha})$ where
$\iota:\\{1,\dots,|M_{\alpha}|\\}\to M_{\alpha}$ is the order preserving
bijection, and define

$\displaystyle x_{\alpha}^{\perp}:=\text{the matrix obtained from
$[-P_{\alpha}Z_{\alpha}+Z^{\prime}_{\alpha},-P_{\alpha}C_{r_{1}}+C_{r_{1}}^{\prime\prime},\cdots,-P_{\alpha}C_{r_{s}}+C_{r_{s}}^{\prime\prime}]_{\rm
hor}$ }$ $\displaystyle\hskip 30.0pt\text{by deleting columns of indices in
$J^{\prime}_{\alpha}$ (which are zero columns)},$ $\displaystyle
y_{h}^{\perp}:=P_{\alpha}C_{h}P_{\beta}-P_{\alpha}C_{h}^{\prime}-C_{h}^{\prime\prime}P_{\beta}+C_{h}^{\prime\prime\prime}\quad\textrm{
for arrow }h:\beta\to\alpha.$

Note that the column spaces of $A_{\alpha}(q)^{\perp}$ and
$\widetilde{A}_{\alpha}(q)_{[v_{\alpha}^{0}+1,\dots,w_{\alpha};-]}$ are the
same, which implies

(4.4) $\mathrm{rank}A_{\alpha}(q)^{\perp}=u_{\alpha}^{\perp}$

Similarly, we define $x_{\beta}^{\perp}$ to be obtained from
$[-Z_{\beta}P_{\beta}+Z^{\prime}_{\beta},-C_{r^{\prime}_{1}}P_{\beta}+C^{\prime}_{r^{\prime}_{1}},\cdots,-C_{r^{\prime}_{t}}P_{\beta}+C^{\prime}_{r^{\prime}_{t}}]_{\rm
vert}$ by deleting rows of indices in $J_{\beta}$ where
$J^{\prime}_{\beta}=\iota^{-1}(J_{\beta})$ (here
$\iota:\\{1,\dots,|M_{\beta}|\\}\to M_{\beta}$ is the order preserving
bijection). We have

(4.5) $\mathrm{rank}A_{\beta}(q)^{\perp}=u_{\beta}^{\perp}$

This complete the construction of $q^{\perp}$. Note that the projection $U\to
U^{\perp}$, $q\mapsto q^{\perp}$ is also compatible with the
$\mathbb{C}^{*}$-action.

Note that $\psi$ is a morphism because it can be expressed in terms of matrix
additions, multiplications, and inverses, so can be expressed as rational
functions.

(2) We show that $\psi$ is an isomorphism by constructing its inverse
$\psi^{-1}$. Assume $q^{0},q^{\perp}$ are given. From $q^{0}$ we can recover
$Z_{i},P_{i},C_{h}$ by rational functions. From $q^{\perp}$ we can recover
$Z_{i}^{\prime},C_{h}^{\prime},C_{h}^{\prime\prime},C_{h}^{\prime\prime\prime}$
by rational functions. This uniquely determines $q$. It is routine to check
that $q$ is well-defined, that $\psi^{-1}$ is a morphism and that $\psi^{-1}$
is indeed the two-sided inverse of $\psi$.

(3) We define $\varphi$. Given $(q=(x_{i},y_{h}),X_{i})\in\pi^{-1}U$, define
$\varphi(q,X_{i}):=(q^{0},(q^{\perp},X_{i}^{\perp}))$, where $q^{0}$ and
$q^{\perp}$ are defined in (1), and $X_{i}^{\perp}$ are defined as follows:

We define $X_{\alpha}^{\perp}$ ($\alpha\in I_{0}$) as follows. For ${\bf u}\in
X_{\alpha}$, define

$\rho^{0}({\bf u}):=\begin{bmatrix}{\bf I}&0\\\ P_{\alpha}&0\end{bmatrix}{\bf
u},\quad\rho^{\perp}({\bf u}):={\bf u}-\rho^{0}({\bf u})=\begin{bmatrix}0&0\\\
-P_{\alpha}&{\bf I}\end{bmatrix}{\bf u}.$

Then the top $v_{\alpha}^{0}$ entries of $\rho^{\perp}({\bf u})$ are 0. Define

$X_{\alpha}^{0}:=\\{\rho^{0}({\bf u})\ |\ {\bf u}\in X_{\alpha}\\},\quad
X_{\alpha}^{\perp}:=\\{\rho^{\perp}({\bf u})\ |\ {\bf u}\in
X_{\alpha}\\}=\Big{\\{}\begin{bmatrix}0\\\ -P_{\alpha}{\bf u}_{1}+{\bf
u}_{2}\end{bmatrix}\;\Big{|}\;\begin{bmatrix}{\bf u}_{1}\\\ {\bf
u}_{2}\end{bmatrix}\in X_{\alpha}\Big{\\}}$

Then $\dim X_{\alpha}^{0}=v_{\alpha}^{0}$, $\dim
X_{\alpha}^{\perp}=v_{\alpha}^{\perp}$. Note that
$X_{\alpha}^{\perp}=X_{\alpha}\cap(0\oplus\mathbb{A}^{w_{\alpha}-v_{\alpha}^{0}})$
is the set of vectors in $X_{\alpha}$ with the first $v_{\alpha}^{0}$ entries
being 0.

We define $X_{\beta}^{\perp}$ ($\beta\in I_{1}$) as follows. Let
$r^{\prime}_{1},\dots,r^{\prime}_{t}$ be defined as in Definition 3.1. Define

$P^{*}_{\beta}:=\begin{bmatrix}{\bf I}&0&\cdots&0\\\
0&\widetilde{P}_{\alpha_{r^{\prime}_{1}}}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\widetilde{P}_{\alpha_{r^{\prime}_{t}}}\end{bmatrix},\quad\widetilde{X}_{\beta}:=P^{*}_{\beta}X_{\beta}.$

Denote $\gamma|_{J_{\beta}}\in\mathbb{A}^{v_{\beta}^{0}}$ to be the vector
obtained from $\gamma$ by only keeping its entries of indices in $J_{\beta}$,
where $J_{\beta}$ is defined in (1) Step 1. Define

$X_{\beta}^{\perp}:=\big{\\{}\gamma\in\widetilde{X}_{\beta}\ {\textrm{such
that}}\ \gamma|_{J_{\beta}}={\bf 0}\big{\\}}.$

We now show that $\dim X_{\beta}^{\perp}=v_{\beta}^{\perp}$. Note that
$\dim\widetilde{X}_{\beta}=\dim X_{\beta}=v_{\beta}$. Note that

$\\{\gamma|_{J_{\beta}}\textrm{ for
}\gamma\in\widetilde{X}_{\beta}\\}=\\{\gamma|_{J_{\beta}}\textrm{ for
}\gamma\in X_{\beta}\\}=\mathbb{A}^{v_{\beta}^{0}}.$

So there is a subspace $V$ of $\widetilde{X}_{\beta}$ of dimension
$v_{\beta}^{0}$ such that $\\{\gamma|_{J_{\beta}}\textrm{ for }\gamma\in V\\}$
has full dimension $v_{\beta}^{0}$. Then $V\cap X_{\beta}^{\perp}=0$ and
$V+X_{\beta}^{\perp}=\widetilde{X}_{\beta}$, so $\dim
X_{\beta}^{\perp}=v_{\beta}^{\perp}$.

Next we shall show that $X_{\beta}^{\perp}\subseteq
W_{\beta}^{\prime\perp}\oplus\bigoplus_{j=1}^{t}X^{\perp}_{\alpha_{r^{\prime}_{j}}}$,
where
$W_{\beta}^{\prime\perp}\cong\mathbb{A}^{w_{\beta}^{\prime}+\sum_{h:s(h)=\beta}v_{t(h)}^{0}-v_{\beta}^{0}}$
consists of all vectors in
$\mathbb{A}^{w^{\prime}_{\beta}+\sum_{h:\;s(h)=\beta}w_{t(h)}}$ whose $j$-th
entry are 0 for $j\in J_{\beta}$ or $j\notin M_{\beta}$. Take an arbitrary
element $P_{\beta}^{*}{\bf u}\in X_{\beta}^{\perp}$ for $u\in X_{\beta}$. We
can write

${\bf u}=\begin{bmatrix}{\bf u}_{0}\\\ {\bf u}_{1}\\\ \vdots\\\ {\bf
u}_{t}\end{bmatrix}\;\textrm{ where ${\bf u}_{0}\in W^{\prime}_{\beta}$, ${\bf
u}_{i}\in X_{\alpha_{r^{\prime}_{i}}}$ for $i=1,\dots,t$. }$

Then

$P_{\beta}^{*}{\bf u}=\begin{bmatrix}{\bf u}_{0}\\\
\widetilde{P}_{\alpha_{r^{\prime}_{1}}}{\bf u}_{1}\\\ \vdots\\\
\widetilde{P}_{\alpha_{r^{\prime}_{t}}}{\bf u}_{t}\end{bmatrix}={\bf
v}_{1}+{\bf v}_{2},\textrm{ where }{\bf v_{1}}=\begin{bmatrix}{\bf u}_{0}\\\
\begin{bmatrix}{\bf I}&0\\\ 0&0\end{bmatrix}{\bf u}_{1}\\\ \vdots\\\
\begin{bmatrix}{\bf I}&0\\\ 0&0\end{bmatrix}{\bf u}_{t}\end{bmatrix},\;{\bf
v}_{2}=\begin{bmatrix}0\\\ \begin{bmatrix}0&0\\\
-P_{\alpha_{r^{\prime}_{1}}}&{\bf I}\end{bmatrix}{\bf u}_{1}\\\ \vdots\\\
\begin{bmatrix}0&0\\\ -P_{\alpha_{r^{\prime}_{t}}}&{\bf I}\end{bmatrix}{\bf
u}_{t}\end{bmatrix}$

Note that ${\bf v}_{1}\in W_{\beta}^{\prime\perp}$ and ${\bf v}_{2}\in{\bf
0}\oplus\bigoplus_{j=1}^{t}X^{\perp}_{\alpha_{r^{\prime}_{j}}}$. (Indeed, the
second one is immediate from the definition of ${\bf v}_{2}$. To see the first
one: if $j\notin M_{\beta}$, the $j$-th entry of ${\bf v}_{1}$ is 0 by the
definition of ${\bf v}_{1}$; if $j\in J_{\beta}$, the $j$-th entry of ${\bf
v}_{2}$ is 0 and the $j$-th entry of ${\bf v}_{1}+{\bf v}_{2}$ is 0 since
$P^{*}u|_{J_{\beta}}={\bf 0}$, which implies that the $j$-th entry of ${\bf
v}_{1}$ is 0). This proves $X_{\beta}^{\perp}\subseteq
W_{\beta}^{\prime\perp}\oplus\bigoplus_{j=1}^{t}X^{\perp}_{\alpha_{r^{\prime}_{j}}}$.

Similar as in (1), we see that $\varphi$ is a morphism.

(4) To show $\varphi$ is an isomorphism, we construct
$\varphi^{-1}(q^{0},(q^{\perp},X_{i}^{\perp}))=(q,X_{i})$ as follows: we let
$q=\psi^{-1}(q^{0},q^{\perp})$, and for $\alpha\in I_{0}$, $\beta\in I_{1}$,
let

$\displaystyle X_{\alpha}:=(\text{ column space of
}A_{\alpha}({q^{0}}))+X_{\alpha}^{\perp}=(\text{ column space of
}\begin{bmatrix}{\bf I}\\\ P_{\alpha}\end{bmatrix})+X_{\alpha}^{\perp}$
$\displaystyle X_{\beta}:=(\text{ column space of
}A_{\beta}({q^{0}})+(P_{\beta}^{*})^{-1}X_{\beta}^{\perp}$

It is routine to check that $\varphi$ and $\varphi^{-1}$ are indeed inverse to
each other.

(5) To check that the diagram commutes:

$\psi\circ\pi(q,X_{i})=\psi(q)=(q^{0},q^{\perp})=(1\times\pi)(q^{0},(q^{\perp},X_{i}^{\perp}))=(1\times\pi)\circ\varphi(q,X_{i})$

(6) To check $\psi(p)=(p,0)$: let $q=p$, and use the same assumption as above,
we need to show $q^{0}=q$ and $q^{\perp}=0$. For $\alpha\in I_{0}$, since
$A_{\alpha}(q)$ has rank $v_{\alpha}^{0}$, all other rows are linear
combinations of its first $v_{\alpha}^{0}$ rows, that is,
$A_{\alpha}(q)=\begin{bmatrix}A_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};-]}\\\
P^{\prime}A_{\alpha}(q)_{[1,\dots,v_{\alpha}^{0};-]}\\\ \end{bmatrix}$ for
some matrix $P^{\prime}$. Comparing with (4.3) we see that
$P^{\prime}=P_{\alpha}$. Thus $Z_{\alpha}^{\prime}=P_{\alpha}Z_{\alpha}$,
$C_{r_{j}}^{\prime\prime}=P_{\alpha}C_{r_{j}}$,
$C_{r_{j}}^{\prime\prime\prime}=P_{\alpha}C_{r_{j}}^{\prime}$. Argue similarly
for $A_{\beta}(q)$. We then conclude that $q=q^{0}$. Next, note that
$\widetilde{A}_{\alpha}(q)=\begin{bmatrix}Z_{\alpha}&C_{r_{1}}&0&\cdots&C_{r_{s}}&0\\\
0&0&0&\cdots&0&0\end{bmatrix}$, so $A_{\alpha}(q)^{\perp}=0$, and similarly
$A_{\beta}(q)^{\perp}=0$. Thus $q^{\perp}=0$.

The fact that $\psi({\bf E}^{\circ}_{u,w}\cap U)=U_{0}\times({\bf
E}^{\circ}_{u^{\perp},w^{\perp}}\cap U^{\perp})$, follows from (4.4) and
(4.5). ∎

###  4.2. Proof of trivial local systems

It is the same as the proof of [9, Theorem 5.1]. We reproduce it here for the
readers’ convenience. As pointed out in [9, Remark 5.5], for the proof to
work, it is essential to have the algebraic version, instead of the analytic
version, of the Transversal Slice Theorem (Lemma 4.3).

######  Lemma 4.4.

Given $f:X\to Y$, $Y_{a},L_{a},n_{a}$ and the decomposition in (4.2). Let $V$
be a nonsingular variety. Consider the following Cartesian diagram

$\textstyle{V\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\scriptstyle{1\times
f}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{V\times
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Y}$

where $p$ and $p^{\prime}$ are the natural projections. Define the pullback
$\tilde{L}_{a}:=p^{*}L_{a}$. Then

$(1\times f)_{*}IC_{V\times
X}\cong\bigoplus_{a}IC_{V\times\overline{Y_{a}}}(\tilde{L}_{a})[n_{a}]$

###### Proof.

Same as [9, Lemma 5.4]. ∎

###### Proof of Theorem 4.1.

Let $IC_{Z}(L)[n]$ be a direct summand that appears in the decomposition of
$\pi_{*}(IC_{\tilde{\mathcal{F}}_{v,w}})$. Take a general point $p$ of $Z$ is
in ${\bf E}^{\circ}_{v^{0},w}$ and apply Lemma 4.3. Then $U^{0}$ is a Zariski
open subset of ${\bf E}^{\circ}_{v^{0},w}$. By uniqueness of the BBDG
Decomposition, $IC_{Z}(L)[n]|_{U}$ is a direct summand of the decomposition of
$\pi_{*}(IC_{\pi^{-1}U})$. By Lemma 4.4, we have $IC_{Z}(L)[n]|_{U}\cong
IC_{U^{0}\times\overline{Y_{a}}}(\tilde{L}_{a})[n_{a}]$ for some $a$. Thus
$Z\cap U=U^{0}\times\overline{Y_{a}}$, $L\cong\tilde{L}_{a}$ on $Z\cap U$, and
$n=n_{a}$. But $Z\subseteq{\bf E}_{v^{0},w}$ implies $Z\cap U\subseteq
U^{0}\times\\{0\\}$. So we must have $Y_{a}=\\{0\\}$ and $Z\cap
U=U^{0}\times\\{0\\}$, thus $Z={\bf E}_{v^{0},w}$. Moreover, $L_{a}$ is the
trivial local system $\mathbb{Q}_{0}$. So the local system $\tilde{L}_{a}$ on
$U^{0}$ is also trivial since it is the pullback of $L_{a}$ under the map
$U^{0}\times\\{0\\}\to\\{0\\}$. Then $L$ is trivial on $Z\cap U$, and we see
that $IC_{Z}(L)[n]\cong IC_{{\bf E}_{v^{0},w}}[n]$. Let $v^{\prime}=v^{0}$.
Then $v^{\prime}\leq v$ as seen in Lemma 4.3, and $(v^{\prime},w)$ is
$l$-dominant because ${\bf E}^{\circ}_{v^{0},w}$ is nonempty. ∎

##  5\. Dual canonical basis

The identification between dual canonical basis and triangular basis is
explained to us by Fan Qin using results in [13, 7].

###  5.1. Definition of $L(w)$, $M(w)$, $\mathbf{R}^{\mathrm{finite}}_{t}$,
$\chi$

Let $\mathcal{D}(\mathbf{E}_{w})$ be the bounded derived category of
constructible sheaves of $\mathbb{Q}$-vector spaces on $\mathbf{E}_{w}$. For
simplicity of notation, denote $IC_{w}(v):=IC_{\mathbf{E}_{v,w}}$. Define
$\mathcal{P}_{w}:=\\{IC_{w}(v)\ |\ v\in\mathrm{Dom}(w)\\}$. Define a full
subcategory $\mathcal{Q}_{w}$ of $\mathcal{D}(\mathbf{E}_{w})$ whose objects
are finite direct sums of $IC_{w}(v)[k]$ for various $v\in\mathrm{Dom}(w)$,
$k\in\mathbb{Z}$.

Define an abelian group $\mathcal{K}_{w}$ to be generated by isomorphism
classes $(L)$ of objects of $\mathcal{Q}_{w}$ and quotient by the relations
$(L)+(L^{\prime})=(L^{\prime\prime})$ whenever $L\oplus L^{\prime}\cong
L^{\prime\prime}$. By abuse of notation we denote $(L)$ as $L$. We can view
$\mathcal{K}_{w}$ as a free $\mathbb{Z}[t^{\pm}]$-module with a basis
$\mathcal{P}_{w}$, by defining $t^{i}L=L[i]$ for $i\in\mathbb{Z}$. The duality
on $\mathcal{D}(\mathbf{E}_{w})$ induces the bar involution on
$\mathcal{K}_{w}$ satisfying
$\overline{tL}=t^{-1}\overline{L},\;\overline{IC_{w}(v)}=IC_{w}(v)$. By
Theorem 4.1, for arbitrary $v,w$,

(5.1)
$\pi_{w}(v):=\pi_{*}(IC_{\widetilde{\mathcal{F}}_{v,w}})=\sum_{v^{\prime}\in\mathrm{Dom}(w)}\sum_{d}a^{d}_{v,v^{\prime};w}IC_{w}(v^{\prime})[d]=\sum_{v^{\prime}\in\mathrm{Dom}(w)}a_{v,v^{\prime};w}IC_{w}(v^{\prime})$

where $a_{v,v^{\prime};w}:=\sum_{d}a^{d}_{v,v^{\prime};w}t^{d}$. By Lemma 4.3,

(5.2) $a_{v,v^{\prime};w}=a_{v^{\perp},v^{\prime\perp};w^{\perp}}\textrm{ for
any $v^{0}\in\mathrm{Dom}(w)$}.$

Then $\\{\pi_{w}(v)\ |\ v\in\mathrm{Dom}(w)\\}$ is also a
$\mathbb{Z}[t^{\pm}]$-basis for $\mathcal{K}_{w}$, and $a_{v,v;w}=1$ for
$(v,w)\in\mathrm{Dom}$. Define the dual $\mathcal{K}_{w}^{*}:={\rm
Hom}_{\mathbb{Z}[t^{\pm}]}(\mathcal{K}_{w},\mathbb{Z}[t^{\pm}])$, which is a
free $\mathbb{Z}[t^{\pm}]$-module with a basis $\\{L_{w}(v)\ |\
v\in\mathrm{Dom}(w)\\}$, the dual basis to $\mathcal{P}_{w}$, that is,
$\langle L_{w}(v),IC_{w}(v^{\prime})\rangle=\delta_{v,v^{\prime}}$. The
pairing
$\langle-,-\rangle:\mathcal{K}_{w}^{*}\times\mathcal{K}_{w}\to\mathbb{Z}[t^{\pm}]$
satisfies the condition $\langle L,C[1]\rangle=\langle L[-1],C\rangle=t\langle
L,C\rangle$, for all $L\in\mathcal{K}_{w}^{*},C\in\mathcal{K}_{w}$. Then
$\langle
L_{w}(v),IC_{w}(v^{\prime})\rangle=\delta_{v,v^{\prime}}=\delta_{v^{\perp},v^{\prime\perp}}=\langle
f_{w^{\perp}}(v^{\perp}),IC_{w^{\perp}}(v^{\prime\perp})\rangle\textrm{ for
any $v^{0}\leq v$}$, where $v^{0}$ is used to define
$w^{\perp}=w-C_{q}v^{0},v^{\perp}=v-v^{0},v^{\prime\perp}=v^{\prime}-v^{0}$.
Define

$\mathbf{R}_{t}:=\\{(f_{w})\in\prod_{w}\mathcal{K}_{w}^{*}\ |\ \langle
f_{w},IC_{w}(v)\rangle=\langle
f_{w^{\perp}},IC_{w^{\perp}}(v^{\perp})\rangle\textrm{ whenever
}v\in\mathrm{Dom}(w),v^{0}\leq v\\}.$

For $(f_{w})\in\prod_{w}\mathcal{K}_{w}^{*}$, define
$c_{wv}\in\mathbb{Z}[t^{\pm}]$ to satisfy
$f_{w}=\sum_{v\in\mathrm{Dom}(w)}c_{wv}L_{w}(v)$. Then $c_{wv}=\langle
f_{w},IC_{w}(v)\rangle$, and $(f_{w})\in\mathbf{R}_{t}\textrm{ if and only if
}c_{wv}=c_{w-C_{q}v,0}\textrm{ for every $l$-dominant pair $(w,v)$}$. So an
element in $\mathbf{R}_{t}$ is uniquely determined by $\\{c_{w0}\\}_{w}$.

Define $L(w)\in\mathbf{R}_{t}$ to be induced by $L_{w}(0)$, that is, for any
$(v^{\prime},w^{\prime})\in\mathrm{Dom}$, $\langle
L(w),IC_{w^{\prime}}(v^{\prime})\rangle=\delta_{w,w^{\prime}-C_{q}v^{\prime}}$.
Define $\mathbf{R}^{\mathrm{finite}}_{t}$ to be the
$\mathbb{Z}[t^{\pm}]$-submodule of $\mathbf{R}_{t}$ with the basis
$\\{L(w)\\}_{w}$.

Define $\\{M_{w}(v)\ |\ v\in\mathrm{Dom}(w)\\}\in\mathcal{K}_{w}^{*}$ to be
the functional

$(L)\mapsto\sum_{k}t^{\dim{\bf E}^{\circ}_{v,w}-k}\dim
H^{k}(i^{!}_{x_{v,w}}L)$

where $x_{v,w}$ is a point in ${\bf E}^{\circ}_{v,w}$, and
$i_{x_{v,w}}:x_{v,w}\to{\bf E}_{w}$ is the natural embedding.

Define $M(w)\in\mathbf{R}_{t}$ to be induced by $M_{w}(0)$, that is, denoting
$i_{0}:\\{0\\}\to{\bf E}_{w}$ to be the natural embedding, then for any
$(v^{\prime},w^{\prime})\in\mathrm{Dom}$,

(5.3) $\displaystyle\langle M(w),IC_{w^{\prime}}(v^{\prime})\rangle$
$\displaystyle=\begin{cases}&\langle
M_{w}(0),IC_{w}(v)\rangle=\sum\limits_{k}t^{-k}\dim
H^{k}(i^{!}_{0}IC_{w}(v)),\textrm{ if $\exists v\in\mathbb{Z}_{\geq
0}^{n}:w-w^{\prime}=C_{q}(v-v^{\prime})$;}\\\ &0,\textrm{ otherwise}.\\\
\end{cases}$

Similar to [9, §7], denote a partial order $w^{\prime}\leq_{\rm
w}w\Leftrightarrow w^{\prime}-w=C_{q}u$ for some $u\geq 0$ and denote
$w^{\prime}<_{\rm w}w$ if $w^{\prime}\leq_{\rm w}w$ and $w^{\prime}\neq w$.
Then $\\{M(w)\\}_{w}$ is a basis of $\mathbf{R}^{\mathrm{finite}}_{t}$, and

(5.4) $L(w)=\sum_{w^{\prime\prime}\leq_{\rm
w}w}b^{\prime}_{ww^{\prime\prime}}M(w^{\prime\prime})\in
M(w)+\sum_{w^{\prime\prime}<_{\rm
w}w}t^{-1}\mathbb{Z}[t^{-1}]M(w^{\prime\prime}).$

Define a map

(5.5) $\Phi:\mathbb{Z}^{2n}\mapsto\mathbb{Z}^{2n},\quad\Phi(w)=\sum_{\beta\in
I_{1}}(w_{\beta}-w^{\prime}_{\beta})e_{\beta}+\sum_{\alpha\in
I_{0}}(w_{\alpha}^{\prime}-w_{\alpha})e_{\alpha}$

Define a map $\chi:\mathbf{R}^{\mathrm{finite}}_{t}\to\mathcal{T}$ by defining
it on the basis $\\{M(w)\\}$ (note that we do not require
$v\in\mathrm{Dom}(w)$):

(5.6) $\displaystyle\chi(M(w))$ $\displaystyle=\sum_{v\in\mathbb{Z}_{\geq
0}^{n}}\big{(}\langle
M(w),\pi_{w}(v)\rangle_{t\to\mathbf{v}^{\delta}}\big{)}X^{\Phi(w)+\tilde{B}v}$
$\displaystyle=\sum_{v\in\mathbb{Z}_{\geq 0}^{n}}\sum_{k}\mathbf{v}^{-k}\dim
H^{k}(i_{0}^{!}\pi_{w}(v))X^{\Phi(w)+\tilde{B}v}$

and extend it by the rules $\chi(L_{1}+L_{2})=\chi(L_{1})+\chi(L_{2})$ and
$\chi(tL)=\mathbf{v}^{\delta}\chi(L)$, The following is shown in [9, §7.3]:

(5.7) $\chi(L(w))=\sum_{v\in\mathbb{Z}_{\geq
0}^{n}}a_{v,0;w}(\mathbf{v})X^{\Phi(w)+\tilde{B}v}$

###  5.2. $\\{\overline{\chi(M(w))}\\}$ and the standard monomial basis

All cohomology groups in this paper are taken with coefficient $\mathbb{Q}$,
thus we write $\dim H^{i}(X)$ for $\dim H^{i}(X,\mathbb{Q})$.

Denote $P_{\mathbf{v}}(X):=\sum_{i}\dim H^{i}(X)\mathbf{v}^{i}$. For
$n,k\in\mathbb{Z}_{\geq 0}$, define the $q$-binomial coefficient as follows
(where $q$ is replaced by $\mathbf{v}$).

$[n]:=\frac{\mathbf{v}^{n}-\mathbf{v}^{-n}}{\mathbf{v}-\mathbf{v}^{-1}},\quad{n\brack
k}:=\frac{[n][n-1]\cdots[n-k+1]}{[k][k-1]\cdots[1]}.$

The cohomology groups of a (complex) Grassmannian is well-known; from which we
have

$P_{\mathbf{v}}(Gr(k,n))=\mathbf{v}^{\dim_{\mathbb{C}}Gr(k,n)}{n\brack
k}=\mathbf{v}^{k(n-k)}{n\brack k}$

The following is also well-known.

######  Lemma 5.1.

Let $V$ be a complex vector bundle of rank $n$ over a complex variety $X$,
$1\leq d\leq n-1$, and $E=Gr_{X}(d,V)$ be the associated Grassmannian bundle
over $X$, $F\cong Gr(d,n)$ be a fiber. Then

$P_{\mathbf{v}}(E,\mathbb{Q})=P_{\mathbf{v}}(F,\mathbb{Q})\cdot
P_{\mathbf{v}}(X,\mathbb{Q}).$

###### Proof.

It follows the Leray–Hirsch theorem [6, §4.D] and the surjectivity of
$H^{*}(E)\to H^{*}(F)$. For the latter, recall that $A^{*}(X)$ is the Chow
ring of $X$ and $c_{k}(E)$ (for $k=1,\dots,n$) are the Chern classes of $E$.
By [5, Example 14.6.6],

$\displaystyle
A^{*}(E)=A^{*}(X)[a_{1},\dots,a_{d},b_{1},\dots,b_{n-d}]/(\sum_{i=0}^{k}a_{i}b_{k-i}-c_{k}(E))_{k=1,\dots,n},$
$\displaystyle
A^{*}(F)=\mathbb{Q}[a_{1},\dots,a_{d},b_{1},\dots,b_{n-d}]/(\sum_{i=0}^{k}a_{i}b_{k-i}-\delta_{k,0})_{k=1,\dots,n}$

thus the restriction $A^{k}(E)\to A^{k}(F)$ is surjective for any $k$. So the
composition $A^{k}(E)\to H^{2k}(E)\to H^{2k}(F)\cong A^{k}(F)$ is surjective.
Thus $H^{2k}(E)\to H^{2k}(F)$ is surjective. (Alternatively, one can prove an
analog of [5, Example 14.6.6] for $H^{*}$ and not use Chow rings.) ∎

######  Lemma 5.2.

(a) We have

$\displaystyle P_{\mathbf{v}}(\mathcal{F}_{v,w})$
$\displaystyle=\prod_{\beta\in
I_{1}}P_{\mathbf{v}}(Gr(v_{\beta},w_{\beta}^{\prime}+\sum_{h:s(h)=\beta}v_{t(h)}))\cdot\prod_{\alpha\in
I_{0}}P_{\mathbf{v}}(Gr(v_{\alpha},w_{\alpha}))$
$\displaystyle=\mathbf{v}^{d_{v,w}}\prod_{\beta\in
I_{1}}{w_{\beta}^{\prime}+\sum_{h:s(h)=\beta}v_{t(h)}\brack
v_{\beta}}\cdot\prod_{\alpha\in I_{0}}{w_{\alpha}\brack v_{\alpha}}$

(b) Recall that $d_{v,w}=\dim\mathcal{F}_{v,w}$,
$\tilde{d}_{v,w}=\dim\widetilde{\mathcal{F}}_{v,w}$. Then

(5.8) $\displaystyle\chi(M(w))$ $\displaystyle=\sum_{v\in\mathbb{Z}_{\geq
0}^{n}}\mathbf{v}^{d_{v,w}-\tilde{d}_{v,w}}\prod_{\beta\in
I_{1}}{w_{\beta}^{\prime}+\sum_{h:s(h)=\beta}v_{t(h)}\brack
v_{\beta}}\cdot\prod_{\alpha\in I_{0}}{w_{\alpha}\brack
v_{\alpha}}X^{\Phi(w)+\tilde{B}v}$ $\displaystyle=\sum_{v\in\mathbb{Z}_{\geq
0}^{n}}\mathbf{v}^{-\tilde{d}_{v,w}}P_{\mathbf{v}}(\mathcal{F}_{v,w})X^{\Phi(w)+\tilde{B}v}$

###### Proof.

(a) follows from iteratively applying Lemma 5.1.

(b) Denote the embedding $i_{0}:\\{0\\}\to\mathbf{E}_{v,w}$. Then

(5.9) $\displaystyle\langle
M(w),\pi_{w}(v)\rangle=\mathbf{v}^{2d_{v,w}-\tilde{d}_{v,w}}\sum_{k}\dim
t^{-k}H^{k}({{\mathcal{F}}_{v,w}})\ =\mathbf{v}^{2d_{v,w}-\tilde{d}_{v,w}}\dim
P_{\mathbf{v}^{-1}}({{\mathcal{F}}_{v,w}})$
$\displaystyle\stackrel{{\scriptstyle\rm(a)}}{{=}}\mathbf{v}^{2d_{v,w}-\tilde{d}_{v,w}}\mathbf{v}^{-\dim_{\mathbb{C}}\mathcal{F}_{v,w}}\prod_{\beta\in
I_{1}}{w_{\beta}^{\prime}+\sum_{h:s(h)=\beta}v_{t(h)}\brack
v_{\beta}}\cdot\prod_{\alpha\in I_{0}}{w_{\alpha}\brack v_{\alpha}}$
$\displaystyle=\mathbf{v}^{d_{v,w}-\tilde{d}_{v,w}}\prod_{\beta\in
I_{1}}{w_{\beta}^{\prime}+\sum_{h:s(h)=\beta}v_{t(h)}\brack
v_{\beta}}\cdot\prod_{\alpha\in I_{0}}{w_{\alpha}\brack
v_{\alpha}}=\mathbf{v}^{-\tilde{d}_{v,w}}P_{\mathbf{v}}(\mathcal{F}_{v,w}).$

This together with (5.6) imply (5.8). ∎

Without loss of generality, assume $I_{1}=\\{1,\dots,k\\}$,
$I_{0}=\\{k+1,\dots,n\\}$. We denote
$t^{\prime}=\mu_{I_{1}}(t_{0}):=\mu_{1}\cdots\mu_{k}(t_{0})$ and the attached
seed is $(\Lambda^{\prime},\tilde{B}^{\prime},\tilde{X}^{\prime})$. Thus
$\tilde{B}^{\prime}=\begin{bmatrix}-B\\\ J\end{bmatrix}$ where $J$ is the
diagonal matrix with the first $k$ diagonal entries being $-1$ and the rest
being $1$, $\tilde{\Lambda}^{\prime}=\begin{bmatrix}0&-J\\\
J&-B\end{bmatrix}$. The standard monomial basis elements with respect to this
initial seed is

$E_{a}(t^{\prime})=\mathbf{v}^{v^{\prime}(a)}X^{\sum_{i=n+1}^{2n}a_{i}e_{i}}\prod_{i=n,\dots,1}\Big{(}X_{i}(t^{\prime})^{[a_{i}]_{+}}X_{i}^{\prime}(t^{\prime})^{[-a_{i}]_{+}}\Big{)},\textrm{
for $a\in\mathbb{Z}^{2n}$.}$

where the index $i$ runs decreasingly from $n$ to $1$,
$X_{i}^{\prime}(t^{\prime})=\mu_{i}(X_{i}(t^{\prime}))$, and
$v^{\prime}(a)\in\mathbb{Z}$ is determined by the condition that the leading
term of $E_{a}(t^{\prime})$ in seed $t^{\prime}$ is bar-invariant. Then
$X^{\prime}_{i}(t^{\prime})=X_{i}$ for $i\in I_{1}$,
$X^{\prime}_{i}(t^{\prime})=X(t^{\prime})^{-e_{i}+[b^{\prime}_{i}]_{+}}+X(t^{\prime})^{-e_{i}+[-b^{\prime}_{i}]_{+}}$
for $i\in I_{0}$. Note that in the above expression of $E_{a}(t^{\prime})$,
the two factors in
$X_{i}(t^{\prime})^{[a_{i}]_{+}}X_{i}^{\prime}(t^{\prime})^{[-a_{i}]_{+}}$ can
be swapped because one of them is $1$.

We generalize the above expression into $E^{*}_{a}$ as follows: for $1\leq
i\leq n$, replace $[-a_{i}]_{+}$ by $w_{i}$, $[a_{i}]_{+}$ by
$w^{\prime}_{i}$, and define

$E^{*}_{w,a_{n+1},\dots,a_{2n}}=\mathbf{v}^{v^{\prime}}X^{\sum_{i=n+1}^{2n}a_{i}e_{i}}\prod_{i=n,\dots,k+1}\Big{(}X_{i}^{\prime}(t^{\prime})^{w_{i}}X_{i}(t^{\prime})^{w^{\prime}_{i}}\Big{)}\cdot\prod_{i=k,\dots,1}\Big{(}X_{i}(t^{\prime})^{w^{\prime}_{i}}X_{i}^{\prime}(t^{\prime})^{w_{i}}\Big{)}$

where $v^{\prime}$ is similar as above. Note a technical subtlety that in the
first product (for $i=n,\dots,k+1$) we have to write the two factors of
$X_{i}^{\prime}(t^{\prime})^{w_{i}}X_{i}(t^{\prime})^{w^{\prime}_{i}}$ in that
specific order; if we swap the two factors, then the power of $\mathbf{v}$ in
Lemma 5.4 will be incorrect.

Each $w\in\mathbb{Z}_{\geq 0}^{2n}$ can be uniquely written as
$w={}^{f}w+{}^{\phi}w$ where both ${}^{f}w$ and ${}^{\phi}w$ are in
$\mathbb{Z}_{\geq 0}^{2n}$ and ${}^{\phi}w$ satisfies
${}^{\phi}w_{i}{}^{\phi}w^{\prime}_{i}=0$ for all $1\leq i\leq n$. We have

(5.10) if $w={}^{\phi}w$, then
$E^{*}_{w,a_{n+1},\dots,a_{2n}}=E_{a}(t^{\prime})$.

The following is easy to show so omit skip the proof.

######  Lemma 5.3.

Assume $u_{i}\geq 0$ for $i\in I_{1}$. Then

$X(t^{\prime})^{u}=\sum_{v}(\prod_{i\in I_{1}}{u_{i}\brack
v_{i}})X^{\sum_{i\in I_{1}}(-u_{i}e_{i}+v_{i}b_{i})+\sum_{i\notin
I_{1}}u_{i}e_{i}}$

######  Lemma 5.4.

For any $w\in\mathbb{Z}_{\geq 0}^{2n}$, we have

$E^{*}_{w,0,\dots,0}=\overline{\chi(M(w))}=\sum_{v\in\mathbb{Z}_{\geq
0}^{n}}\mathbf{v}^{\tilde{d}_{v,w}-d_{v,w}}(\prod_{\alpha\in
I_{0}}{w_{\alpha}\brack v_{\alpha}}\prod_{\beta\in
I_{1}}{w^{\prime}_{\beta}+\sum_{\alpha\in
I_{0}}b_{\alpha\beta}v_{\alpha}\brack v_{\beta}})X^{\Phi(w)+\sum_{1\leq i\leq
n}v_{i}b_{i}}.$

where ${\Phi}$ is defined in (5.5). In particular, if $w={}^{\phi}w$, then
$\overline{\chi(M(w))}$ is the standard basis element
$E_{(w^{\prime}_{i}-w_{i})_{1\leq i\leq n}}(t^{\prime})$ with the initial seed
at $t^{\prime}=\mu_{I_{1}}(t_{0})$.

###### Proof.

For $i\in I_{0}$, we have $b^{\prime}_{i}\in\mathbb{Z}_{\geq 0}^{m}$, thus

$\displaystyle
X_{i}^{\prime}(t^{\prime})^{w_{i}}X_{i}(t^{\prime})^{w^{\prime}_{i}}$
$\displaystyle=\
\big{(}X(t^{\prime})^{-e_{i}+b^{\prime}_{i}}+X(t^{\prime})^{-e_{i}}\big{)}^{w_{i}}X(t^{\prime})^{w^{\prime}_{i}e_{i}}$
$\displaystyle=\sum_{v_{i}}{w_{i}\brack
v_{i}}X(t^{\prime})^{-w_{i}e_{i}+v_{i}b^{\prime}_{i}}X(t^{\prime})^{w^{\prime}_{i}e_{i}}$
$\displaystyle=\sum_{v_{i}}\mathbf{v}^{\Lambda^{\prime}(-w_{i}e_{i}+v_{i}b^{\prime}_{i},w^{\prime}_{i}e_{i})}{w_{i}\brack
v_{i}}X(t^{\prime})^{(w^{\prime}_{i}-w_{i})e_{i}+v_{i}b^{\prime}_{i}}$
$\displaystyle=\sum_{v_{i}}\mathbf{v}^{w^{\prime}_{i}v_{i}}{w_{i}\brack
v_{i}}X(t^{\prime})^{(w^{\prime}_{i}-w_{i})e_{i}+v_{i}b^{\prime}_{i}}$

For $i\in I_{1}$,
$X_{i}(t^{\prime})^{w^{\prime}_{i}}X_{i}^{\prime}(t^{\prime})^{w_{i}}=X(t^{\prime})^{w^{\prime}_{i}e_{i}}X_{i}^{w_{i}}$

By comparing the leading terms, it is easy to see that the power of
$\mathbf{v}$ to normalize $E^{*}_{w}$ in $t^{\prime}$ is the same as the one
to normalize $E^{*}_{w}$ in $t$. Thus we ignore factor of powers of
$\mathbf{v}$ in the following computation, and denote $f\sim\mathbf{v}^{i}f$
for $i\in\mathbb{Z}$.

$\displaystyle
E^{*}_{w,0,\dots,0}\sim(\prod_{i=n,\dots,k+1}\sum_{v_{i}}\mathbf{v}^{w^{\prime}_{i}v_{i}}{w_{i}\brack
v_{i}}X(t^{\prime})^{(w^{\prime}_{i}-w_{i})e_{i}+v_{i}b^{\prime}_{i}})\cdot(\prod_{i=k,\dots,1}X(t^{\prime})^{w^{\prime}_{i}e_{i}}\
X_{i}^{w_{i}})$
$\displaystyle\sim\sum_{v_{n},\dots,v_{k+1}}\mathbf{v}^{p_{1}}(\prod_{i=n,\dots,k+1}{w_{i}\brack
v_{i}})X(t^{\prime})^{\sum_{i=k+1}^{n}\big{(}(w^{\prime}_{i}-w_{i})e_{i}+v_{i}b^{\prime}_{i}\big{)}+\sum_{i=1}^{k}w^{\prime}_{i}e_{i}}\
\prod_{i=k,\dots,1}X_{i}^{w_{i}}$
$\displaystyle\sim\sum_{v_{n},\dots,v_{k+1}}\mathbf{v}^{p_{1}}(\prod_{i=n,\dots,k+1}{w_{i}\brack
v_{i}})X(t^{\prime})^{\sum_{i\in I_{1}}(w^{\prime}_{i}+\sum_{j\in
I_{0}}b_{ji}v_{j})e_{i}+\sum_{i\in
I_{0}}(w_{i}^{\prime}-w_{i})e_{i}+\sum_{i\in I_{0}}v_{i}e_{n+i}}\
\prod_{i=k,\dots,1}X_{i}^{w_{i}}$
$\displaystyle\sim\sum_{v}\mathbf{v}^{p_{1}}(\prod_{\alpha\in
I_{0}}{w_{\alpha}\brack v_{\alpha}}\prod_{\beta\in
I_{1}}{w^{\prime}_{\beta}+\sum_{\alpha\in
I_{0}}b_{\alpha\beta}v_{\alpha}\brack v_{\beta}})$ $\displaystyle\quad\cdot
X^{\sum_{\beta\in I_{1}}((-w^{\prime}_{\beta}-\sum_{\alpha\in
I_{0}}v_{\alpha}b_{\beta\alpha})e_{\beta}+v_{\beta}b_{\beta})+\sum_{\alpha\in
I_{0}}(w_{\alpha}^{\prime}-w_{\alpha})e_{\alpha}+\sum_{\alpha\in
I_{0}}v_{\alpha}e_{n+\alpha}}\
\prod_{\beta=k,\dots,1}X_{\beta}^{w_{\beta}}\textrm{ (Lemma \ref{lem:Xt'u}) }$
$\displaystyle\sim\sum_{v}\mathbf{v}^{p_{1}}(\prod_{\alpha\in
I_{0}}{w_{\alpha}\brack v_{\alpha}}\prod_{\beta\in
I_{1}}{w^{\prime}_{\beta}+\sum_{\alpha\in
I_{0}}b_{\alpha\beta}v_{\alpha}\brack v_{\beta}})$ $\displaystyle\quad\cdot
X^{\sum_{\beta\in I_{1}}(-w^{\prime}_{\beta}e_{\beta})+\sum_{\alpha\in
I_{0}}(w_{\alpha}^{\prime}-w_{\alpha})e_{\alpha}+\sum_{1\leq i\leq
n}v_{i}b_{i}}\ \prod_{\beta=k,\dots,1}X_{\beta}^{w_{\beta}}\textrm{ (used
$v_{\alpha}e_{n+\alpha}-\sum_{\beta\in
I_{1}}v_{\alpha}b_{\beta\alpha}e_{\beta}=v_{\alpha}b_{\alpha}$) }$
$\displaystyle\sim\sum_{v}\mathbf{v}^{p_{2}}(\prod_{\alpha\in
I_{0}}{w_{\alpha}\brack v_{\alpha}}\prod_{\beta\in
I_{1}}{w^{\prime}_{\beta}+\sum_{\alpha\in
I_{0}}b_{\alpha\beta}v_{\alpha}\brack v_{\beta}})X^{\sum_{\beta\in
I_{1}}(w_{\beta}-w^{\prime}_{\beta})e_{\beta}+\sum_{\alpha\in
I_{0}}(w_{\alpha}^{\prime}-w_{\alpha})e_{\alpha}+\sum_{1\leq i\leq
n}v_{i}b_{i}}$

where $p_{1}=\sum_{\alpha\in I_{0}}w^{\prime}_{\alpha}v_{\alpha}$,
$p_{2}=\sum_{\alpha\in I_{0}}w^{\prime}_{\alpha}v_{\alpha}+\sum_{\beta\in
I_{1}}w_{\beta}v_{\beta}=\tilde{d}_{v,w}-d_{v,w}$. Since the last expression
is already normalized, it gives a correct formula for $E_{w,0,\dots,0}^{*}$.
This completes the proof. ∎

######  Lemma 5.5.

Let $w\in\mathbb{Z}_{\geq 0}^{2n}$,
$a=(w^{\prime}_{1}-w_{1},\dots,w^{\prime}_{n}-w_{n},a_{n+1},\dots,a_{2n})\in\mathbb{Z}^{2n}$.

(a) If $\min(w_{\alpha},w^{\prime}_{\alpha})>0$ for some $\alpha\in I_{0}$,
then

$E^{*}_{w,a_{n+1},\dots,a_{2n}}=E^{*}_{w^{(1)},a_{n+1},\dots,a_{2n}}+\mathbf{v}^{p}E^{*}_{w^{(2)},a^{\prime}_{n+1},\dots,a_{n+\alpha}+1,\dots,a_{2n}}$

where $p$ is a positive integer, $w^{(1)}$ is obtained from $w$ by
$w_{\alpha}\mapsto w_{\alpha}-1$ and $w^{\prime}_{\alpha}\mapsto
w^{\prime}_{\alpha}-1$; $w^{(2)}$ is obtained from $w$ by $w_{\alpha}\mapsto
w_{\alpha}-1$, $w^{\prime}_{\alpha}\mapsto w^{\prime}_{\alpha}-1$,
$w^{\prime}_{\beta}\mapsto w^{\prime}_{\beta}+b^{\prime}_{\beta\alpha}$.

(b) If $\min(w_{\beta},w^{\prime}_{\beta})>0$ for some $\beta\in I_{0}$, then

$E^{*}_{w,a_{n+1},\dots,a_{2n}}=E^{*}_{w^{(1)},a_{n+1},\dots,a_{2n}}+\mathbf{v}^{p}E^{*}_{w^{(2)},a^{\prime}_{n+1},\dots,a_{n+\beta}+1,\dots,a_{2n}}$

where $p$ is a positive integer, $w^{(1)}$ is obtained from $w$ by
$w_{\beta}\mapsto w_{\beta}-1$ and $w^{\prime}_{\beta}\mapsto
w^{\prime}_{\beta}-1$; $w^{(2)}$ is obtained from $w$ by $w_{\beta}\mapsto
w_{\beta}-1$, $w^{\prime}_{\beta}\mapsto w^{\prime}_{\beta}-1$,
$w^{\prime}_{\alpha}\mapsto w^{\prime}_{\alpha}-b^{\prime}_{\alpha\beta}$.

(c) We have

$E^{*}_{w,a_{n+1},\dots,a_{2n}}\in E_{a}(t^{\prime})+\sum_{r(b)\leq
r(a)}\mathbf{v}\mathbb{Z}[\mathbf{v}]E_{b}(t^{\prime})$

###### Proof.

(a) If $\min(w_{\alpha},w^{\prime}_{\alpha})>0$ for $\alpha\in I_{0}$, then

$\displaystyle E^{*}_{w,a_{n+1},\dots,a_{2n}}\sim
X^{\sum_{i=n+1}^{2n}a_{i}e_{i}}\prod_{i=n,\dots,\alpha+1}\Big{(}X_{i}^{\prime}(t^{\prime})^{w_{i}}X_{i}(t^{\prime})^{w^{\prime}_{i}}\Big{)}\cdot
X_{\alpha}^{\prime}(t^{\prime})^{w_{\alpha}-1}\cdot\big{(}X_{\alpha}^{\prime}(t^{\prime})X_{\alpha}(t^{\prime})\big{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\cdot
X_{\alpha}(t^{\prime})^{w^{\prime}_{\alpha}-1}\cdot\prod_{i=\alpha-1,\dots,k+1}\Big{(}X_{i}^{\prime}(t^{\prime})^{w_{i}}X_{i}(t^{\prime})^{w^{\prime}_{i}}\Big{)}\prod_{i=k,\dots,1}\Big{(}X_{i}(t^{\prime})^{w^{\prime}_{i}}X_{i}^{\prime}(t^{\prime})^{w_{i}}\Big{)}$
$\displaystyle\sim
X^{\sum_{i=n+1}^{2n}a_{i}e_{i}}\prod_{i=n,\dots,\alpha+1}\Big{(}X_{i}^{\prime}(t^{\prime})^{w_{i}}X_{i}(t^{\prime})^{w^{\prime}_{i}}\Big{)}\cdot
X_{\alpha}^{\prime}(t^{\prime})^{w_{\alpha}-1}\cdot\big{(}1+\mathbf{v}X(t^{\prime})^{b^{\prime}_{\alpha}}\big{)}$
$\displaystyle\quad\quad\cdot
X_{\alpha}(t^{\prime})^{w^{\prime}_{\alpha}-1}\cdot\prod_{i=\alpha-1,\dots,k+1}\Big{(}X_{i}^{\prime}(t^{\prime})^{w_{i}}X_{i}(t^{\prime})^{w^{\prime}_{i}}\Big{)}\prod_{i=k,\dots,1}\Big{(}X_{i}(t^{\prime})^{w^{\prime}_{i}}X_{i}^{\prime}(t^{\prime})^{w_{i}}\Big{)}$
$\displaystyle\sim\mathbf{v}^{p_{1}}E^{*}_{w^{(1)},a_{n+1},\dots,a_{2n}}+\mathbf{v}^{p_{2}}E^{*}_{w^{(2)},a_{n+1},\dots,a_{n+\alpha}+1,\dots,a_{2n}}$

where $w^{(1)}$ is obtained from $w$ by $w_{\alpha}\mapsto w_{\alpha}-1$ and
$w^{\prime}_{\alpha}\mapsto w^{\prime}_{\alpha}-1$; $w^{(2)}$ is obtained from
$w$ by $w_{\alpha}\mapsto w_{\alpha}-1$, $w^{\prime}_{\alpha}\mapsto
w^{\prime}_{\alpha}-1$, $w^{\prime}_{\beta}\mapsto
w^{\prime}_{\beta}+b^{\prime}_{\beta\alpha}$. Note that
$(w^{(2)},a_{n+1},\dots,a_{n+\alpha}+1,\dots,a_{2n})$ corresponds to
$b=(a_{1}+b^{\prime}_{1\alpha},\dots,a_{k}+b^{\prime}_{k\alpha},a_{k+1},\dots,a_{n},a_{n+1},\dots,a_{n+\alpha}+1,\dots,a_{2n})$,
which satisfies $b\prec a$ or, unpleasantly, $r(b)=r(a)$. If the latter case
happens, at least we see that $\sum_{i}w_{i}$ strictly decreases.

Next, we show that $p>0$. Indeed, note that $e_{\alpha}^{\prime}=-e_{\alpha}$
for $\alpha\in I_{0}$, $e_{\beta}^{\prime}=-e_{\beta}+(-b_{\beta}^{\prime})$
for $\beta\in I_{0}$, and
$p_{1}=\Lambda^{\prime}\big{(}\sum_{n+1}^{2n}a_{i}e_{i},w_{n}e^{\prime}_{n},w_{n}^{\prime}e_{n},\dots,(w_{\alpha}-1)e^{\prime}_{\alpha}$,
$(w^{\prime}_{\alpha}-1)e_{\alpha},\dots$,
$w_{k+1}e^{\prime}_{k+1},w^{\prime}_{k+1}e_{k+1},w^{\prime}_{k}e_{k},w_{k}e^{\prime}_{k},\dots,w^{\prime}_{1}e_{1},w_{1}e^{\prime}_{1}\big{)}$,
while $p_{2}=1+\Lambda^{\prime}\big{(}\sum_{n+1}^{2n}a_{i}e_{i}$,
$w_{n}e^{\prime}_{n}$, $w_{n}^{\prime}e_{n}$, $\dots$,
$(w_{\alpha}-1)e^{\prime}_{\alpha},b^{\prime}_{\alpha},(w^{\prime}_{\alpha}-1)e_{\alpha},\dots,w_{k+1}e^{\prime}_{k+1},w^{\prime}_{k+1}e_{k+1},w^{\prime}_{k}e_{k},w_{k}e^{\prime}_{k},\dots,w^{\prime}_{1}e_{1},w_{1}e^{\prime}_{1}\big{)}$.
It is routine to verify that

$p=p_{2}-p_{1}=w_{\alpha}+w^{\prime}_{\alpha}-1+\sum_{\beta\in
I_{1}}w_{\beta}(-b^{\prime}_{\alpha\beta})\geq 1$

where the last inequality is because $b^{\prime}_{\alpha\beta}\leq 0$ for
$\alpha\in I_{0},\beta\in I_{1}$.

(b) is proved similarly to (a).

(c) is obtained by applying (a) and (b) recursively. ∎

###  5.3. $\\{\chi(L(w))\\}$ and the triangular basis

######  Lemma 5.6.

For any $w\in\mathbb{Z}_{\geq 0}^{2n}$, ${\chi(L(w))}=C_{\Phi(w)}$ is a
triangular basis element. In particular, $\chi(L({}^{\phi}w))=\chi(L(w))$
(therefore $a_{v,0;w}=a_{v,0;{}^{\phi}w}$ for any $v\in\mathbb{Z}_{\geq
0}^{n}$) and $\chi(L({}^{f}w))=1$.

###### Proof.

It follows from (5.7) that $\chi(L(w))$ is bar-invariant. Meanwhile,

$\displaystyle\chi(L(w))-E_{\Phi(w)}(t^{\prime})\in\chi(L(w))-E^{*}_{w,0,\dots,0}+\sum\mathbf{v}\mathbb{Z}[\mathbf{v}]E_{b}(t^{\prime})\quad\text{
(by Lemma \ref{M'' in M'}) }$
$\displaystyle\quad=\chi(L(w))-\overline{\chi(M(w))}+\sum\mathbf{v}\mathbb{Z}[\mathbf{v}]E_{b}(t^{\prime})\quad\text{
(by Lemma \ref{chiMbar=M''}) }$
$\displaystyle\quad\subseteq\overline{\chi(M(w))}+\sum\mathbf{v}\mathbb{Z}[\mathbf{v}]\overline{\chi(M(w^{\prime\prime}))}-\overline{\chi(M(w))}+\sum\mathbf{v}\mathbb{Z}[\mathbf{v}]E_{b}(t^{\prime})\quad\text{
(by \eqref{L=sum M})}$
$\displaystyle\quad=\sum\mathbf{v}\mathbb{Z}[\mathbf{v}]E^{*}_{w^{\prime\prime},0,\dots,0}+\sum\mathbf{v}\mathbb{Z}[\mathbf{v}]E_{b}(t^{\prime})\quad\text{
(by Lemma \ref{chiMbar=M''})}$
$\displaystyle\quad\subseteq\sum\mathbf{v}\mathbb{Z}[\mathbf{v}](\sum\mathbb{Z}[\mathbf{v}]E_{b^{\prime}}(t^{\prime})+\sum\mathbf{v}\mathbb{Z}[\mathbf{v}]E_{b}(t^{\prime})$
$\displaystyle\quad=\sum\mathbf{v}\mathbb{Z}[\mathbf{v}]E_{b}(t^{\prime})$

By [3, Theorem 1.1], $\chi(L(w))$ equals to a triangular basis element for the
seed at $t^{\prime}$. Since the triangular basis does not depend on the chosen
acyclic seed (also proved in [3]), $\chi(L(w))$ is also a triangular basis
element for initial seed at $t_{0}$. ∎

##  6\. The proof of the main result

We cite the following lemma in [9].

######  Lemma 6.1.

Let $f:Y\to X$ be a proper morphism between complex algebraic varieties, $Y$
be nonsingular, let $0$ be a point in $X$. Let $d=\dim Y$, $Y_{0}=f^{-1}(0)$
and $d_{0}=\dim Y_{0}$. Write the BBDG decomposition in the form

(6.1) $f_{*}IC_{Y}=\bigoplus_{b}IC_{0}^{\oplus
s_{0,b}}[b]\oplus\bigoplus_{V,L,b}IC_{V}^{\oplus s_{V,L,b}}(L)[b]$

where $V\neq 0$ are subvarieties of $X$, each $L$ is a local system on an open
dense subset of $V$, $s_{0,b},s_{V,L,b}\in\mathbb{Z}_{\geq 0}$ are
multiplicities of the corresponding $IC$-sheaves. Then $\\{s_{0,b}\\}$ satisfy
the following conditions:

i) $s_{0,b}=s_{0,-b}$ for every $b\in\mathbb{Z}$.

ii) $s_{0,b}\geq s_{0,b+2}$ for every $b\in\mathbb{Z}_{\geq 0}$.

iii) $s_{0,b}=0$ if $|b|>2d_{0}-d$. In particular, if $2d_{0}<d$, then
$s_{0,b}=0$ for all $b$.

###### Proof of Theorem 1.1.

For the given $a=(a_{i})\in\mathbb{Z}^{2n}$, define
$w=(w_{i},w^{\prime}_{i})_{1\leq i\leq n}\in\mathbb{Z}_{\geq 0}^{2n}$ by
$w^{\prime}_{i}=[a_{i}]_{+},w_{i}=[-a_{i}]_{+}$. Then
$\Phi(w)=(a_{1},\dots,a_{n},0,\dots,0)$. By (5.7), $e_{v}=a_{v,0;w}$. Apply
Lemma 6.1 to the projection $\pi:\tilde{\mathcal{F}}_{v,w}\to\mathbf{E}_{v,w}$
defined in (3.4), we conclude that $e_{v}$ is symmetric, unimodal, and

$\deg(e_{v})=\deg a_{v,0;w}\leq 2d_{v,w}-\tilde{d}_{v,w}=f(v)$

by Proposition 3.7 (b). ∎

We end the paper with two examples.

######  Example 6.2.

(a) Consider the initial seed determined by $B=\begin{bmatrix}0&-3\\\
3&0\end{bmatrix}$.

The corresponding quiver $\mathcal{Q}^{\rm op}$ is $1\mathrel{\mathop{\vbox{
\offinterlineskip\halign{\hbox to\dimexpr\@tempdima+2em{#}\cr
0.00003pt{\rightarrowfill\cr\kern 0.86108pt\cr
0.00003pt{\rightarrowfill\cr\kern 0.86108pt\cr
0.00003pt{\rightarrowfill\cr}}}\limits^{\\!}}2}}}$.

We compute the triangle basis element at $a=(9,-4,0,0)$. The corresponding
$w=(w_{1},w^{\prime}_{1},w_{2},w^{\prime}_{2})=(0,9,4,0)$. Then

$E_{(9,-4,0,0)}=X^{(-3,-4,0,4)}+(\mathbf{v}^{2}+1+\mathbf{v}^{-2})X^{(-3,-1,1,4)}+(\mathbf{v}^{2}+1+\mathbf{v}^{-2})X^{(-3,2,2,4)}+X^{(-3,5,3,4)}+(\mathbf{v}^{3}+\mathbf{v}+\mathbf{v}^{-1}+\mathbf{v}^{-3})X^{(0,-4,0,3)}+(\mathbf{v}^{2}+1+\mathbf{v}^{-2})X^{(0,-1,1,3)}+(\mathbf{v}^{4}+\mathbf{v}^{2}+2+\mathbf{v}^{-2}+\mathbf{v}^{-4})X^{(3,-4,0,2)}+X^{(3,-1,1,2)}+(\mathbf{v}^{3}+\mathbf{v}+\mathbf{v}^{-1}+\mathbf{v}^{-3})X^{(6,-4,0,1)}+X^{(9,-4,0,0)}$,

$f(v)=-(9+v_{1})v_{1}-(-4+v_{2})v_{2}+3v_{1}v_{2}=-v_{1}^{2}+3v_{1}v_{2}-v_{2}^{2}-9v_{1}+4v_{2}$.

${\rm
Supp}(C_{a})=\\{(0,0),((0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)\\}$.

Figure 1. Left: the support of $C_{(9,-4,0,0)}$ in (a). Right: the support of
$C_{(4,3,-3,0,0,0)}$ in (b). The points in ${\rm Supp}(C_{a})$ are in red
color.

See Figure 1 Left. The inequality $f(v)\geq 0$ gives the upper left side of
the blue curve. This triangular basis element corresponds to $C[3,4]$ given in
[9, §10.1(a)]. Note that the $d$-vector $(3,4)$ corresponds to $g$-vector
$(9,-4)$.

(b) Consider the initial seed determined by $=\begin{bmatrix}0&0&-2\\\
0&0&-2\\\ 2&2&0\end{bmatrix}$.

The corresponding quiver $\mathcal{Q}^{\rm op}$ is $1\mathrel{\mathop{\vbox{
\offinterlineskip\halign{\hbox to\dimexpr\@tempdima+2em{#}\cr
0.00003pt{\rightarrowfill\cr\kern 0.86108pt\cr
0.00003pt{\rightarrowfill\cr}}}\limits^{\\!}_{\\!}}3\mathrel{\mathop{\vbox{
\offinterlineskip\halign{\hbox to\dimexpr\@tempdima+2em{#}\cr
0.00003pt{\leftarrowfill\cr\kern 0.86108pt\cr
0.00003pt{\leftarrowfill\cr}}}\limits^{\\!}_{\\!}}2}}}}$.

We compute the triangle basis element at $a=(4,3,-3,0,0,0)$. The corresponding
$w=(w_{1},w^{\prime}_{1},w_{2},w^{\prime}_{2},w_{3},w^{\prime}_{3})=(0,4,0,3,3,0)$.
Then

$E_{(4,3,-3,0,0,0)}=X^{(-2,-3,-3,0,0,3)}+(\mathbf{v}^{2}+1+\mathbf{v}^{-2})X^{(-2,-3,-1,0,1,3)}+(\mathbf{v}+\mathbf{v}^{-1})*X^{(-2,-3,-1,1,0,3)}+(\mathbf{v}^{2}+1+\mathbf{v}^{-2})X^{(-2,-3,1,0,2,3)}+(\mathbf{v}^{3}+2\mathbf{v}+2\mathbf{v}^{-1}+\mathbf{v}^{-3})X^{(-2,-3,1,1,1,3)}+X^{(-2,-3,1,2,0,3)}+X^{(-2,-3,3,0,3,3)}+(\mathbf{v}^{3}+2\mathbf{v}+2\mathbf{v}^{-1}+\mathbf{v}^{-3})X^{(-2,-3,3,1,2,3)}+(\mathbf{v}^{2}+1+\mathbf{v}^{-2})X^{(-2,-3,3,2,1,3)}+(\mathbf{v}+\mathbf{v}^{-1})X^{(-2,-3,5,1,3,3)}+(\mathbf{v}^{2}+1+\mathbf{v}^{-2})X^{(-2,-3,5,2,2,3)}+X^{(-2,-3,7,2,3,3)}+(\mathbf{v}^{2}+1+\mathbf{v}^{-2})X^{(0,-1,-3,0,0,2)}+(\mathbf{v}^{2}+2+\mathbf{v}^{-2})X^{(0,-1,-1,0,1,2)}+(\mathbf{v}+\mathbf{v}^{-1})X^{(0,-1,-1,1,0,2)}+X^{(0,-1,1,0,2,2)}+(\mathbf{v}+\mathbf{v}^{-1})X^{(0,-1,1,1,1,2)}+(\mathbf{v}^{2}+1+\mathbf{v}^{-2})X^{(2,1,-3,0,0,1)}+X^{(2,1,-1,0,1,1)}+X^{(4,3,-3,0,0,0)}$,

$f(v)=-(4+v_{1})v_{1}-(3+v_{2})v_{2}-(-3+v_{3})v_{3}+2v_{1}v_{3}+2v_{2}v_{3}=-v_{1}^{2}-v_{2}^{2}-v_{3}^{2}+2v_{1}v_{3}+2v_{2}v_{3}-4v_{1}-3v_{2}+3v_{3}$.

${\rm
Supp}(C_{a})=\\{(0,0,0),(0,0,1),(0,0,2),(0,0,3),(0,1,1),(0,1,2),(0,1,3),(0,2,2),(0,2,3)$,
$(0,3,3),(1,0,2),(1,0,3),(1,1,2),(1,1,3),(1,2,3),(1,3,3),(2,0,3),(2,1,3),(2,2,3),(2,3,3)\\}$.

See Figure 1 Right. The inequality $f(v)\geq 0$ gives the upper region of the
blue surface. Note that the $d$-vector $(2,3,3)$ corresponds to $g$-vector
$(4,3,-3)$.

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