Virtual Crystals and Nakajima Monomials Virtual Crystals and Nakajima
Monomials

Ben SALISBURY † and Travis SCRIMSHAW ‡ B. Salisbury and T. Scrimshaw

† Department of Mathematics, Central Michigan University, Mount Pleasant, MI
48859, USA<EMAIL_ADDRESS>http://people.cst.cmich.edu/salis1bt/

‡ School of Mathematics and Physics, The University of Queensland,
‡ St. Lucia, QLD 4072, Australia<EMAIL_ADDRESS>https://people.smp.uq.edu.au/TravisScrimshaw/

Received March 28, 2018, in final form September 20, 2018; Published online
September 26, 2018

An explicit description of the virtualization map for the (modified) Nakajima
monomial model for crystals is given. We give an explicit description of the
Lusztig data for modified Nakajima monomials in type $A_{n}$.

crystal; Nakajima monomial; virtualization; PBW basis; Kostant partition

05E10; 17B37

##  1 Introduction

Diagram folding is an effective technique used to study a non-simply-laced
Kac–Moody algebra $\mathfrak{g}$ by embedding $\mathfrak{g}$ inside of a
simply-laced Kac–Moody algebra $\widehat{\mathfrak{g}}$. On the level of
crystals, Kashiwara [28] introduced a method for embedding highest weight
$U_{q}(\mathfrak{g})$-crystals inside highest weight
$U_{q}(\widehat{\mathfrak{g}})$-crystals, provided the Dynkin diagram of
$\mathfrak{g}$ can be obtained from the Dynkin diagram of
$\widehat{\mathfrak{g}}$ using a diagram folding. The embedding is called a
virtualization map and resulting image with an induced crystal structure is
called a virtual crystal.

It is known that simply-laced crystals have a nice local characterization
[72], a description which is lacking in the non-simply-laced types (see [6,
73] for partial results in this direction). One can develop the theory of
crystals using a combination of the Stembridge axioms and the theory of
virtual crystals, which is the approach taken in the recent text of Bump and
Schilling [4]. Therefore, developing an explicit virtualization map on
different models for highest weight crystals and $B(\infty)$, the crystal of
the lower half of the quantum group $U_{q}^{-}(\mathfrak{g})$, for the correct
model could result in an extension of the Stembridge axioms to all
symmetrizable types.

In terms of Kashiwara–Nakashima tableaux [32], which only is valid for
classical types,111Type $G_{2}$ was given by Kang–Misra tableaux [24]. a
virtualization map has been explicitly described in [1, 71]. However, it is a
little complicated with specific rules depending on the type. To give a more
uniform description of a virtualization map, we can use the polyhedral model
of [52, 53], which is described by iterating the Kashiwara embedding. This was
the model used by Kashiwara in his construction [28], where he also gave
precise conditions on when a virtualization map exists from type
$\mathfrak{g}$ into type $\widehat{\mathfrak{g}}$. This virtualization is
natural in the sense that is commutes with the Kashiwara embedding (in a
suitable sense) and can be computed directly from the diagram folding (and
using that virtual crystals form a tensor category [57]).

The virtualization map has also been explicitly described for a number of
other (uniform) models for highest weight crystals. In [59], the
virtualization map for the Littelmann path model [41, 42] was completely
described, which naturally extends to a virtualization map of the Littelmann
path model for $B(\infty)$ of [40]. This was also examined by Naito and Sagaki
in [44], but the different type of diagram automorphism allowed can have an
edge to be fixed under the automorphism but not the corresponding vertices.
This was used to construct virtual crystals for certain Kirillov–Reshetikhin
crystals [45, 47], a class of finite $U_{q}^{\prime}(\mathfrak{g})$-crystals
for $\mathfrak{g}$ of affine type.

If $\mathfrak{g}$ is of finite type, then there is an analog of a PBW basis of
the universal enveloping algebra for the quantum group (see, e.g., [43]).
Lusztig constructed a canonical basis for $U_{q}^{-}(\mathfrak{g})$ [43],
where it was shown to be equivalent to Kashiwara’s crystal basis [7].
Moreover, the crystal basis is unitriangular with respect to the PBW basis and
precisely one monomial survives in the $q\to 0$ limit. The exponents of this
monomial are called the Lusztig datum of the element. The transition functions
of [3] between these PBW bases can be used to describe the crystal structure,
but one needs to apply numerous piecewise-linear equations in order to do so.
Recently, this was simplified for certain reduced expressions, where a
bracketing rule was given on Lusztig data [62, 63].

Mirković–Vilonen (MV) polytopes are the image of the moment map of MV cycles
and are a model for $B(\infty)$ for finite types [17, 18]. Furthermore, MV
polytopes encode all of the Lusztig data of an element, and so it is an
equivalent model to the PBW bases. Furthermore, the virtualization map on MV
polytopes was described explicitly in [15, 46], and hence, extends to the
Lusztig data/PBW bases. Note that the virtualization map is natural as it is
induced from the corresponding embeddings of word, which comes from the
diagram folding. Moreover, for certain choices of reduced words of the long
element of the corresponding Weyl group, this naturally extends to the
bracketing rules on Lusztig data.

Rigged configurations are a combinatorial model for highest weight crystals
and $B(\infty)$ stemming from statistical mechanics [64, 65, 70, 71]. The
virtualization map on rigged configurations is also natural and explicitly
determined [64, 71]. It was recently shown that the $*$-involution has a
natural interpretation on rigged configurations [66], and it is
straightforward to see that the $*$-involution commutes with the
virtualization map.

The main model we focus on is given by Nakajima monomials, which were
originally used to describe the $t$-analog of $q$-characters [48, 49, 50, 51].
We will primarily be using the crystal structure given by Kashiwara [31],
which is distinct from that given by Nakajima [50]. This can be considered a
generalization of the polyhedral model, which follows from looking at the
tensor product rule [25] and using the $A_{i,k}$ variables (corresponding to
simple roots). Therefore, our explicit virtualization map can be considered a
generalization of that of [28, Theorem 5.1], but it also reflects the
naturality of the map as the virtualization map can also be described using
the $Y_{i,k}$ variables (corresponding to fundamental weights). Our results
allow us to explicitly determine the conditions necessary to have a
virtualization map, giving another proof of the results in [28, 59].

We note that the naturality of the virtualization is a reflection of the close
connection of Nakajima monomials to geometry; in particular, quiver varieties
[51, 67]. This gives a further relation with the results of [68]. Our results
are further evidence that there should be a simple, explicit method to
construct a bijection between Nakajima monomials and other models such as
Littelmann paths or rigged configurations. Finding such a bijection between
rigged configurations and Nakajima monomials would give a connection between
the $*$-involution, the crystal commutor, and quiver varieties [11, 19, 69].

Let $\mathfrak{g}$ be of affine type. We note that our virtualization map
extends to the crystal structure on Nakajima monomials as given by Nakajima
[50]. As an application of our embedding, we obtain a virtualization map for
KR crystals $B^{r,1}$, for certain $r\in I_{0}$ that also depends on the type,
given by Nakajima monomials described in [12]. These are analogous results to
[59], along with some of the cases of [44, 45, 47]. Thus, our results give an
alternative proof of certain special cases of [57, Conjecture 3.7], which was
recently settled for non-exceptional types in [54] using type-specific
arguments.

For type $A_{n}$, we give an explicit map between the modified Nakajima
monomial model and the Kostant partition crystal (or equivalently, the PBW
crystal) of [63] for a specific reduced word of the long element $w_{0}$ (the
dual BZL word). Thus, we provide an explicit way to obtain the Lusztig data
for the specific word from the Nakajima monomial model. A goal of this paper
was to obtain similar data for types $B_{n}$ and $C_{n}$ using our
virtualization map on Nakajima monomials. However, we have a certain aligned
condition, which we can consider as compatible orientations on the
corresponding Dynkin diagrams, that is not compatible with the reduced
expression for $w_{0}$. We can combine this with the transition maps of [3],
but this is not combinatorial or simple to describe. So our approach will need
to be modified in order to obtain the Lusztig data from Nakajima monomials in
types $B_{n}$ and $C_{n}$.

This paper is organized as follows. In Section 2, we recall the definition of
(modified) Nakajima monomials. In Section 3, we recall the definition of
virtual crystals. In Section 4, we construct the virtualization map on
Nakajima monomials. In Section 5, we extend our virtualization map to certain
KR crystals as given by [12]. In Section 6, we give an explicit map between
Nakajima monomials and PBW crystals or Lusztig data. In Section 7, we give
some open problems from our work.

##  2 Crystals and Nakajima monomials

Since we will only be working with one particular model for crystals, we
explicitly describe the properties of that model and leave the generalities
for the interested reader to pursue. (See, for example, [4, 13].)

Let $C=(C_{ij})_{i,j\in I}$ be a generalized Cartan matrix for a symmetrizable
Kac–Moody algebra $\mathfrak{g}$ with index set $I$, weight lattice $P$,
fundamental weights $\\{\Lambda_{i}\colon i\in I\\}$, and simple roots
$\\{\alpha_{i}\colon i\in I\\}$. Let $P^{\vee}$ denote the coroot lattice. For
$\lambda\in P$ and $h\in P^{\vee}$, let $\langle h,\lambda\rangle=\lambda(h)$
be the canonical pairing. Recall that $\Lambda_{j}(h_{i})=\delta_{ij}$
(Kronecker delta) and $\alpha_{j}(h_{i})=C_{ij}$, where $\\{h_{i}\colon i\in
I\\}$ is the set of simple coroots in $P^{\vee}$. We identity simple roots in
the weight space $\mathbf{Q}\otimes_{\mathbf{Z}}P$ via the Cartan matrix; that
is, in $\mathbf{Q}\otimes_{\mathbf{Z}}P$, we have
$\alpha_{i}=\sum\limits_{j\in I}C_{ji}\Lambda_{i}$. We write $i\sim j$ if
$i\neq j$ and $C_{ij}\neq 0$; i.e., if $i$ and $j$ are adjacent in the Dynkin
diagram of $\mathfrak{g}$.

Let $\mathcal{M}=\\{Y_{i,k}\colon i\in I,\,k\in\mathbf{Z}\\}\cup\\{\bm{1}\\}$
be a set of commuting variables and in which $\bm{1}$ denotes the
multiplicative identity. Choose an array ${\bm{c}}=(c_{ij}\colon i,j\in
I,\,i\neq j)$ of integers which satisfy the condition $c_{ij}+c_{ji}=1$.
Define $\mathcal{M}_{{\bm{c}}}$ as $\mathcal{M}$ with a
$U_{q}(\mathfrak{g})$-crystal structure as follows. For a monomial
$M=\prod\limits_{i\in I}\prod\limits_{k\in\mathbf{Z}}Y_{i,k}^{y_{i}(k)}$,
define

$\displaystyle\mathrm{wt}(M)=\sum_{i\in
I}\bigg{(}\sum_{k\in\mathbf{Z}}y_{i}(k)\bigg{)}\Lambda_{i},$ (2.1a)
$\displaystyle\varphi_{i}(M)=\max\bigg{\\{}\sum_{j\leq k}y_{i}(j)\colon
k\in\mathbf{Z}\bigg{\\}},$ (2.1b)
$\displaystyle\varepsilon_{i}(M)=\varphi_{i}(M)-\langle
h_{i},\mathrm{wt}(M)\rangle.$ (2.1c) $\displaystyle
f_{i}M=\begin{cases}MA^{-1}_{i,k_{i}^{f}}&\text{if }\varphi_{i}(M)>0,\\\
0&\text{if }\varphi_{i}(M)=0,\end{cases}$ (2.1d) $\displaystyle
e_{i}M=\begin{cases}MA_{i,k_{i}^{e}}&\text{if }\varepsilon_{i}(M)>0,\\\
0&\text{if }\varepsilon_{i}(M)=0,\end{cases}$ (2.1e)

where

$\displaystyle A_{i,k}=Y_{i,k}Y_{i,k+1}\prod_{j\neq
i}Y_{j,k+c_{j,i}}^{C_{ji}},$ $\displaystyle
k_{i}^{f}=k_{i}^{f}(M)=\min\bigg{\\{}k\colon\varphi_{i}(M)=\sum_{j\leq
k}y_{i}(j)\bigg{\\}},$ $\displaystyle
k_{i}^{e}=k_{i}^{e}(M)=\max\bigg{\\{}k\colon\varphi_{i}(M)=\sum_{j\leq
k}y_{i}(j)\bigg{\\}}.$

######  Remark 2.1.

We can also define $\varepsilon_{i}$ by

$\displaystyle\varepsilon_{i}(M)=\max\bigg{\\{}{-}\sum_{j>k}y_{i}(j)\colon
k\in\mathbf{Z}\bigg{\\}}.$

Let $\mathcal{M}(M)_{{\bm{c}}}$ denote the closure of a monomial $M$ under the
crystal operators given above, and let $B(\lambda)$ denote the
$U_{q}(\mathfrak{g})$-crystal associated to the irreducible highest weight
representation $V(\lambda)$ of $U_{q}(\mathfrak{g})$ with highest weight
$\lambda\in P$.

######  Theorem 2.2 (Kashiwara [31]).

Let $M$ be a monomial such that $e_{i}M=0$ for all $i\in I$. Then
$\mathcal{M}(M)_{{\bm{c}}}\cong B\bigl{(}{\rm wt}(M)\bigr{)}$.

In particular, denote
$\mathcal{M}(\lambda)_{{\bm{c}}}:=\mathcal{M}(Y_{\lambda})_{{\bm{c}}}$, where

$\displaystyle Y_{\lambda}=\prod_{i\in I}Y_{i,0}^{\langle
h_{i},\lambda\rangle}.$

This is referred to as the $B(\lambda)$ model using _Nakajima monomials_.

Next, suppose $c_{ij}\in\mathbf{Z}_{\geq 0}$ for all $i$, $j$. We define
$\mathcal{M}(\infty)_{{\bm{c}}}$ as the closure of $\bm{1}$ under the modified
Kashiwara operators

$\displaystyle\overline{f}_{i}M=MA_{i,\overline{k}_{i}^{f}}^{-1},$

where

$\displaystyle\overline{k}_{i}^{f}=\min\bigg{\\{}k\geq
0\colon\varphi_{i}(M)=\sum_{0\leq j\leq k}y_{i}(j)\bigg{\\}},$

with the remaining crystal structure being the same as for
$\mathcal{M}_{{\bm{c}}}$. Note if $M\in\mathcal{M}(\infty)_{\bm{c}}$, then $M$
has the form

$\displaystyle M=\prod_{i\in I}\prod_{k\geq 0}Y_{i,k}^{y_{i}(k)}.$ (2.2)

When there is no danger of confusion, we will simply write the modified
Kashiwara operator as $f_{i}$. The crystal $\mathcal{M}(\infty)_{{\bm{c}}}$ is
called the crystal of _modified Nakajima monomials_.

######  Theorem 2.3 (Kang–Kim–Shin [23]).

Let $\mathfrak{g}$ be of symmetrizable type. Then
$\mathcal{M}(\infty)_{\bm{c}}\cong B(\infty)$.

Let $\lambda\in P$ be a dominant integral weight. Define
$R_{\lambda}=\\{r_{\lambda}\\}$ to be the one-element abstract
$U_{q}(\mathfrak{g})$-crystal whose operations, for all $i\in I$, are defined
as

$\displaystyle
e_{i}r_{\lambda}=f_{i}r_{\lambda}=0,\qquad\varepsilon_{i}(r_{\lambda})=-\langle
h_{i},\lambda\rangle,\qquad\varphi_{i}(r_{\lambda})=0,\qquad{\rm
wt}(r_{\lambda})=\lambda.$

By [26], there is a strict crystal embedding
$B(\lambda)\lhook\joinrel\longrightarrow B(\infty)\otimes R_{\lambda}$, where
$\otimes$ denotes the crystal tensor product defined in [25]. We will not
require the general definition of tensor product of crystals, but note that
the connected component containing $u_{\infty}\otimes r_{\lambda}$, where
$u_{\infty}$ is the highest weight element of $B(\infty)$, is isomorphic to
$B(\lambda)$. The crystal operators of the tensor product $M\otimes
r_{\lambda}\in\mathcal{M}(\infty)_{{\bm{c}}}\otimes R_{\lambda}$ precisely
corresponds to taking $Y_{\lambda}M$ for any
$M\in\mathcal{M}(\infty)_{{\bm{c}}}$ and using the (unmodified) crystal
structure of (2.1). Going in the opposite direction, we can construct
$\mathcal{M}(\infty)_{{\bm{c}}}$ as the direct limit

$\displaystyle\mathcal{M}(\infty)_{{\bm{c}}}=\varinjlim_{\lambda\to\infty}Y_{\lambda}^{-1}\mathcal{M}(Y_{\lambda}),$

where we consider the partial order $\lambda\leq\mu$ if and only if $\langle
h_{i},\lambda\rangle\leq\langle h_{i},\mu\rangle$ for all $i\in I$. We refer
the reader to [29] for a precise definition.

##  3 Virtual crystals

A _diagram folding_ is a surjective map $\phi\colon\widehat{I}\longrightarrow
I$ between index sets of Kac–Moody algebras and a set
$(\gamma_{i}\in\mathbf{Z}_{>0}\colon i\in I)$ of _scaling factors_. One may
induce a map from $\phi$ on the corresponding weight lattices
$\widetilde{\phi}\colon P\longrightarrow\widehat{P}$ by asserting

$\displaystyle\Lambda_{i}\mapsto\gamma_{i}\sum_{i^{\prime}\in\phi^{-1}(i)}\widehat{\Lambda}_{i^{\prime}}.$
(3.1)

If there exists an embedding
$\iota\colon\mathfrak{g}\lhook\joinrel\longrightarrow\widehat{\mathfrak{g}}$
of symmetrizable Kac–Moody algebras that induces $\widetilde{\phi}$, then
$\iota$ induces an injection $v\colon B(\lambda)\lhook\joinrel\longrightarrow
B(\widehat{\lambda})$ as sets, where
$\widehat{\lambda}:=\widetilde{\phi}(\lambda)$. Suppose the crystal structure
on $B(\lambda)$ is denoted by $(e_{i},f_{i},\varepsilon_{i},\varphi_{i},{\rm
wt})$ and the crystal structure on $B(\widehat{\lambda})$ is denoted by
$(\widehat{e}_{i},\widehat{f}_{i},\widehat{\varepsilon}_{i},\widehat{\varphi}_{i},\widehat{{\rm
wt}})$. Then there is additional structure on the image under $v$ as a
_virtual crystal_ , where the action $e_{i}$ and $f_{i}$ are defined on the
image as

$\displaystyle
e^{v}_{i}=\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{e}_{i^{\prime}}^{\,\gamma_{i}}\quad\quad\text{
and }\quad\quad
f^{v}_{i}=\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{f}_{i^{\prime}}^{\,\gamma_{i}},$
(3.2)

respectively, and commute with $v$ [1, 57, 58]; i.e., $e_{i}^{v}\circ v=v\circ
e_{i}$ for all $i\in I$, etc. These are known as the _virtual Kashiwara
(crystal) operators_. It is shown in [28] that for any $i\in I$ and
$i_{1}^{\prime},i_{2}^{\prime}\in\phi^{-1}(i)$ we have
$\widehat{e}_{i_{1}^{\prime}}\widehat{e}_{i_{2}^{\prime}}=\widehat{e}_{i_{2}^{\prime}}\widehat{e}_{i_{1}^{\prime}}$
and
$\widehat{f}_{i_{1}^{\prime}}\widehat{f}_{i_{2}^{\prime}}=\widehat{f}_{i_{2}^{\prime}}\widehat{f}_{i_{1}^{\prime}}$
as operators (recall that the nodes indexed by $i_{1}^{\prime}$ and
$i_{2}^{\prime}$ in the Dynkin diagram of $\widehat{\mathfrak{g}}$ are not
connected), so both $e^{v}_{i}$ and $f^{v}_{i}$ are well-defined. The
inclusion map $v$ also satisfies the relation $\widetilde{\phi}\circ{\rm
wt}=\widehat{{\rm wt}}\circ v$. In [1], it was shown that this defines a
$U_{q}(\mathfrak{g})$-crystal structure on the image of $v$. More generally,
we define a virtual crystal as follows.

######  Definition 3.1.

Consider any symmetrizable types $\mathfrak{g}$ and $\widehat{\mathfrak{g}}$
with index sets $I$ and $\widehat{I}$, respectively. Let
$\phi\colon\widehat{I}\longrightarrow I$ be a surjection such that
$\widehat{C}_{i_{1}^{\prime}i_{2}^{\prime}}=0$ for all
$i_{1}^{\prime},i_{2}^{\prime}\in\phi^{-1}(i)$ and $i\in I$. Let $\widehat{B}$
be a $U_{q}(\widehat{\mathfrak{g}})$-crystal and $V\subseteq\widehat{B}$. Let
$\gamma=(\gamma_{i}\in\mathbf{Z}_{>0}\colon i\in I)$ be the scaling factors. A
_virtual crystal_ is the quadruple $(V,\widehat{B},\phi,\gamma)$ such that $V$
has an abstract $U_{q}(\mathfrak{g})$-crystal structure defined using the
Kashiwara operators $e_{i}^{v}$ and $f_{i}^{v}$ from (3.2) above,

$\displaystyle\varepsilon_{i}^{v}(x):=\frac{\widehat{\varepsilon}_{i^{\prime}}(x)}{\gamma_{i}},\qquad\varphi_{i}^{v}(x):=\frac{\widehat{\varphi}_{i^{\prime}}(x)}{\gamma_{i}},\qquad\text{for
all}\quad i^{\prime}\in\phi^{-1}(i)\quad\text{and}\quad x\in V,$

and ${\rm wt}^{v}:=\widetilde{\phi}^{-1}\circ\widehat{{\rm wt}}$.

We say $B$ _virtualizes_ in $\widehat{B}$ if there exists a
$U_{q}(\mathfrak{g})$-crystal isomorphism $v\colon B\longrightarrow V$. The
resulting isomorphism is called the _virtualization map_. We denote the
quadruple $(V,\widehat{B},\phi,\gamma)$ simply by $V$ when there is no risk of
confusion.

##  4 The virtualization map

Assume the notations of the previous sections. We will say
$\widehat{{\bm{c}}}=(\widehat{c}_{i^{\prime}j^{\prime}})_{i^{\prime}\neq
j^{\prime}}$ is _compatible_ with ${\bm{c}}$ when
$\widehat{c}_{i^{\prime}j^{\prime}}=c_{ij}$ for all
$i^{\prime}\in\phi^{-1}(i)$ and $j^{\prime}\in\phi^{-1}(j)$ satisfying
$i^{\prime}\sim j^{\prime}$ and $i\sim j$. For this section assume that
$\widehat{{\bm{c}}}$ is compatible with ${\bm{c}}$. Let
$\widehat{\mathcal{M}}_{\widehat{{\bm{c}}}}$ be the set of modified Nakajima
monomials of type $\widehat{\mathfrak{g}}$ in the variables
$\\{\widehat{Y}_{i,k}\colon i\in\widehat{I},\,k\in\mathbf{Z}\\}$ and identity
$\widehat{\bm{1}}$ equipped with $U_{q}(\widehat{\mathfrak{g}})$-crystal
structure denoted by $\widehat{e}_{i}$, $\widehat{f}_{i}$,
$\widehat{\varepsilon}_{i}$, $\widehat{\varphi}_{i}$, and $\widehat{{\rm
wt}}$. With these definitions, define a map

$\displaystyle v\colon\
\mathcal{M}_{\bm{c}}\longrightarrow\widehat{\mathcal{M}}_{\widehat{{\bm{c}}}}\qquad\text{by}\qquad
Y_{i,k}\mapsto\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{Y}_{i^{\prime},k}^{\gamma_{i}}$

and extend multiplicatively. By abuse of notation, we also consider
$v\colon\mathcal{M}(\infty)_{{\bm{c}}}\longrightarrow\widehat{\mathcal{M}}(\infty)_{\widehat{{\bm{c}}}}$.

######  Lemma 4.1.

Suppose

$\displaystyle\widetilde{\phi}(\alpha_{i})=\displaystyle\sum_{i^{\prime}\in\phi^{-1}(i)}\gamma_{i}\widehat{\alpha}_{i^{\prime}}$

for all $i\in I$. Then, for all $i\in I$ and $k\in\mathbf{Z}_{\geq 0}$,

$\displaystyle
v(A_{i,k})=\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{A}_{i^{\prime},k}^{\gamma_{i}}.$

######  Example 4.2.

Let

$\displaystyle C=\begin{pmatrix}2&-2\\\
-1&2\end{pmatrix},\\!\\!\\!\qquad{\bm{c}}=\begin{pmatrix}\bullet&0\\\
1&\bullet\end{pmatrix},\\!\\!\\!\qquad\text{and}\\!\\!\\!\qquad\widehat{C}=\begin{pmatrix}2&-1&0\\\
-1&2&-1\\\
0&-1&2\end{pmatrix},\\!\\!\\!\qquad\widehat{{\bm{c}}}=\begin{pmatrix}\bullet&0&0\\\
1&\bullet&1\\\ 1&0&\bullet\end{pmatrix}.$

Here, $C$ is a Cartan matrix of type $C_{2}$ and $\widehat{C}$ is a Cartan
matrix of type $A_{3}$. For arbitrary scaling factors $(\gamma_{i}\colon i\in
I)$, the condition
$\widetilde{\phi}(\alpha_{i})=\sum\limits_{j\in\phi^{-1}(i)}\gamma_{i}\widehat{\alpha}_{j}$
imposes a linear dependence relation amongst the $\gamma_{i}$. In this
example, we have

$\displaystyle\widetilde{\phi}(\alpha_{1})=\widetilde{\phi}(2\Lambda_{1}-\Lambda_{2})=2\widetilde{\phi}(\Lambda_{1})-\widetilde{\phi}(\Lambda_{2})=2\gamma_{1}(\widehat{\Lambda}_{1}+\widehat{\Lambda}_{3})-\gamma_{2}\widehat{\Lambda}_{2},$
$\displaystyle\gamma_{1}(\widehat{\alpha}_{1}+\widehat{\alpha}_{3})=\gamma_{1}(2\widehat{\Lambda}_{1}-\widehat{\Lambda}_{2}-\widehat{\Lambda}_{2}+2\widehat{\Lambda}_{3})=2\gamma_{1}(\widehat{\Lambda}_{1}-\widehat{\Lambda}_{2}+\widehat{\Lambda}_{3}),$

so the relation $2\gamma_{1}=\gamma_{2}$ is required to ensure
$\widetilde{\phi}(\alpha_{1})=\gamma_{1}(\widehat{\alpha}_{1}+\widehat{\alpha}_{3})$.
Similarly, for $k\geq 0$,

$\displaystyle v(A_{1,k})=v(Y_{1,k})v(Y_{1,k+1})\prod_{j\neq
i}v(Y_{j,k+c_{j1}})^{C_{j1}}=\widehat{Y}_{1,k}^{\gamma_{1}}\widehat{Y}_{3,k}^{\gamma_{1}}\widehat{Y}_{1,k+1}^{\gamma_{1}}\widehat{Y}_{3,k+1}^{\gamma_{1}}\widehat{Y}_{2,k+1}^{-\gamma_{2}},$
$\displaystyle\prod_{i^{\prime}\in\phi^{-1}(1)}\widehat{A}_{i^{\prime},k}^{\gamma_{i}}=\widehat{A}_{1,k}^{\gamma_{1}}\widehat{A}_{3,k}^{\gamma_{1}}=\widehat{Y}_{1,k}^{\gamma_{1}}\widehat{Y}_{1,k+1}^{\gamma_{1}}\widehat{Y}_{2,k+1}^{-\gamma_{1}}\widehat{Y}_{3,k}^{\gamma_{1}}\widehat{Y}_{3,k+1}^{\gamma_{1}}\widehat{Y}_{2,k+1}^{-\gamma_{1}}.$

These two are equal if and only if $2\gamma_{1}=\gamma_{2}$.

###### Proof.

First, let’s examine the condition that
$\widetilde{\phi}(\alpha_{i})=\sum\limits_{i^{\prime}\in\phi^{-1}(i)}\gamma_{i}\widehat{\alpha}_{i^{\prime}}$.
Recall that the Cartan matrix is the change of basis matrix from the basis of
simple roots to the basis of fundamental weights. So, on the one hand, for
$i\in I$, we have

$\displaystyle\widetilde{\phi}(\alpha_{i})=\widetilde{\phi}\bigg{(}\sum_{j\in
I}C_{ji}\Lambda_{j}\bigg{)}=\sum_{j\in
I}C_{ji}\widetilde{\phi}(\Lambda_{j})=\sum_{j\in
I}C_{ji}\gamma_{j}\sum_{j^{\prime}\in\phi^{-1}(j)}\widehat{\Lambda}_{j^{\prime}},$

while, on the other hand, we have

$\displaystyle\widetilde{\phi}(\alpha_{i})=\sum_{i^{\prime}\in\phi^{-1}(i)}\gamma_{i}\widehat{\alpha}_{i^{\prime}}=\sum_{i^{\prime}\in\phi^{-1}(i)}\gamma_{i}\sum_{j^{\prime}\in\widehat{I}}\widehat{C}_{j^{\prime}i^{\prime}}\widehat{\Lambda}_{j^{\prime}}=\sum_{j^{\prime}\in\widehat{I}}\sum_{i^{\prime}\in\phi^{-1}(i)}\gamma_{i}\widehat{C}_{j^{\prime}i^{\prime}}\widehat{\Lambda}_{j^{\prime}}.$

Comparing the coefficient of $\widehat{\Lambda}_{j^{\prime}}$ from both of
these equations, we conclude, for all $i\in I$,

$\displaystyle\gamma_{j}C_{ji}=\sum_{i^{\prime}\in\phi^{-1}(i)}\gamma_{i}\widehat{C}_{j^{\prime}i^{\prime}}\qquad\text{for
any $j$ such that}\quad j^{\prime}\in\phi^{-1}(j).$ (4.1)

Let $i\in I$ and $k\geq 0$ be fixed. Then

$\displaystyle
v(A_{i,k})=\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{Y}_{i^{\prime},k}^{\gamma_{i}}\widehat{Y}_{i^{\prime},k+1}^{\gamma_{i}}\prod_{j\neq
i}\prod_{j^{\prime}\in\phi^{-1}(j)}\widehat{Y}_{j^{\prime},k+c_{ji}}^{\gamma_{j}C_{ji}}.$

Since $\widehat{c}_{i^{\prime}j^{\prime}}=c_{ij}$ for all
$i^{\prime}\in\phi^{-1}(i)$ and $j^{\prime}\in\phi^{-1}(j)$ such that $i\sim
j$ and $i^{\prime}\sim j^{\prime}$, we have

$\displaystyle\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{Y}_{i^{\prime},k}^{\gamma_{i}}\widehat{Y}_{i^{\prime},k+1}^{\gamma_{i}}\prod_{j\neq
i}\prod_{j^{\prime}\in\phi^{-1}(j)}\widehat{Y}_{j^{\prime},k+c_{ji}}^{\gamma_{j}C_{ji}}=\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{Y}_{i^{\prime},k}^{\gamma_{i}}\widehat{Y}_{i^{\prime},k+1}^{\gamma_{i}}\prod_{j\neq
i}\prod_{j^{\prime}\in\phi^{-1}(j)}\widehat{Y}_{j^{\prime},k+\widehat{c}_{j^{\prime}i^{\prime}}}^{\gamma_{j}C_{ji}}.$

By the calculation above,

$\displaystyle\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{Y}_{i^{\prime},k}^{\gamma_{i}}\widehat{Y}_{i^{\prime},k+1}^{\gamma_{i}}\prod_{j\neq
i}\prod_{j^{\prime}\in\phi^{-1}(j)}\widehat{Y}_{j^{\prime},k+\widehat{c}_{j^{\prime}i^{\prime}}}^{\gamma_{j}C_{ji}}=\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{Y}_{i^{\prime},k}^{\gamma_{i}}\widehat{Y}_{i^{\prime},k+1}^{\gamma_{i}}\prod_{j\neq
i}\prod_{j^{\prime}\in\phi^{-1}(j)}\widehat{Y}_{j^{\prime},k+\widehat{c}_{j^{\prime}i^{\prime}}}^{\gamma_{i}\widehat{C}_{j^{\prime}i^{\prime}}}.$

On the other hand,

$\displaystyle\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{A}_{i^{\prime},k}^{\gamma_{i}}=\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{Y}_{i^{\prime},k}^{\gamma_{i}}\widehat{Y}_{i^{\prime},k+1}^{\gamma_{i}}\prod_{j^{\prime}\neq
i^{\prime}}\widehat{Y}_{j^{\prime},k+\widehat{c}_{j^{\prime}i^{\prime}}}^{\gamma_{i}\widehat{C}_{j^{\prime}i^{\prime}}},$

which agrees with the calculation of $v(A_{i,k})$ by use of equation (4.1) and
the fact that $\widehat{C}_{j^{\prime}i^{\prime}}=0$ if
$i^{\prime},j^{\prime}\in\phi^{-1}(i)$. ∎

######  Theorem 4.3.

The map $v$ from Lemma 4.1 is a virtualization map.

###### Proof.

We show the case for $\mathcal{M}(\infty)_{{\bm{c}}}$ as the case for
$\mathcal{M}_{{\bm{c}}}$ is similar. By equation (2.2), write

$\displaystyle M=\prod_{i\in I}\prod_{k\geq
0}Y_{i,k}^{y_{i}(k)}\qquad\text{and}\qquad\widehat{M}:=v(M)=\prod_{i\in
I}\prod_{k\geq
0}\prod_{i^{\prime}\in\phi^{-1}(i)}\widehat{Y}_{i^{\prime},k}^{\gamma_{i}y_{i}(k)}.$

We need to show ${\rm wt}(M)={\rm wt}^{v}(\widehat{M})$,
$v(e_{i}M)=e_{i}^{v}\widehat{M}$, and $v(f_{i}M)=f_{i}^{v}\widehat{M}$ for all
$i\in I$. First

$\displaystyle\widetilde{\phi}\bigl{(}{\rm
wt}(M)\bigr{)}=\widetilde{\phi}\biggl{(}\sum_{i\in I}\bigg{(}\sum_{k\geq
0}y_{i}(k)\bigg{)}\Lambda_{i}\biggr{)}=\sum_{i\in I}\bigg{(}\sum_{k\geq
0}y_{i}(k)\bigg{)}\sum_{i^{\prime}\in\phi^{-1}(i)}\gamma_{i}\widehat{\Lambda}_{i^{\prime}}=\widehat{{\rm
wt}}(\widehat{M}),$

so $v$ commutes with the weight map.

Now fix $i\in I$ and suppose $k_{f}\geq 0$ is such that
$f_{i}M=MA^{-1}_{i,k_{f}}$. We need to show that
$f^{v}_{i}\widehat{M}=\widehat{M}\prod\limits_{i^{\prime}\in\phi^{-1}(i)}\widehat{A}_{i^{\prime},k_{f}}^{-\gamma_{i}}$,
but by Lemma 4.1 it suffices to show, for any $i^{\prime}\in\phi^{-1}(i)$,

$\displaystyle k_{f}=\min\bigg{\\{}k\geq
0\colon\widehat{\varphi}_{i^{\prime}}(\widehat{M})=\sum_{0\leq\ell\leq
k}\gamma_{i}y_{i}(\ell)\bigg{\\}}.$

By definition of $\widehat{\varphi}_{i^{\prime}}$ and the construction of
$\widehat{M}$, for all $i^{\prime}\in\phi^{-1}(i)$ we have

$\displaystyle\widehat{\varphi}_{i^{\prime}}(\widehat{M})=\max\bigg{\\{}\sum_{0\leq
r\leq s}\gamma_{i}y_{i}(r)\colon s\geq
0\bigg{\\}}=\gamma_{i}\max\bigg{\\{}\sum_{0\leq r\leq s}y_{i}(r)\colon s\geq
0\bigg{\\}}=\gamma_{i}\varphi_{i}(M).$

Hence
$\varphi_{i}^{v}(M)=\widehat{\varphi}_{i^{\prime}}(\widehat{M})/\gamma_{i}=\varphi_{i}(M)$
for all $i\in I$. Now

$\displaystyle\min\bigg{\\{}k\geq
0\colon\widehat{\varphi}_{i^{\prime}}(\widehat{M})=\sum_{0\leq\ell\leq
k}\gamma_{i}y_{i}(\ell)\bigg{\\}}$ $\displaystyle=\min\bigg{\\{}k\geq
0\colon\varphi_{i}^{v}(M)=\sum_{0\leq\ell\leq k}y_{i}(\ell)\bigg{\\}}$
$\displaystyle=\min\bigg{\\{}k\geq 0\colon\varphi_{i}(M)=\sum_{0\leq\ell\leq
k}y_{i}(\ell)\bigg{\\}}$ $\displaystyle=k_{f},$

as required. The case showing $e_{i}^{v}\widehat{M}=v(e_{i}M)$ is similar. ∎

######  Example 4.4.

We construct the highest weight $F_{4}$ crystal $B(\Lambda_{1})$ inside of
$B(\widehat{\Lambda}_{2})$ of type $E_{6}$ under the folding defined by

$E_{6}$$2$$4$$5$$6$$3$$1$$F_{4}$$1$$2$$3$$4$

Using the following code, the virtualization map can be given on a vertex-by-
vertex basis. See Table 1.

⬇

sage: LaE = RootSystem([’E’,6]).weight_lattice().fundamental_weights()

sage: LaF = RootSystem([’F’,4]).weight_lattice().fundamental_weights()

sage: new_c = matrix([[0,1,1,1,1,1],[0,0,1,0,1,1],[0,0,0,1,1,1],

....: [0,1,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]])

sage: ME = crystals.NakajimaMonomials(LaE[2], c=new_c)

sage: MF = crystals.NakajimaMonomials(LaF[4])

sage: phi = {1: [1,6], 2: [3,5], 3: [4], 4: [2]}

sage: sf = {i: 1 for i in MF.index_set()}

sage: v = MF.crystal_morphism(ME.module_generators, virtualization=phi,

....: scaling_factors=sf)

$\displaystyle Y_{4,0}$ $\displaystyle\mapsto Y_{2,0}$ $\displaystyle
Y_{3,1}Y_{4,1}^{-1}$ $\displaystyle\mapsto Y_{2,1}^{-1}Y_{4,1}$ $\displaystyle
Y_{2,2}Y_{3,2}^{-1}$ $\displaystyle\mapsto Y_{3,2}Y_{4,2}^{-1}Y_{5,2}$
$\displaystyle Y_{1,3}Y_{2,3}^{-1}Y_{3,2}$ $\displaystyle\mapsto
Y_{1,3}Y_{3,3}^{-1}Y_{4,2}Y_{5,3}^{-1}Y_{6,3}$ $\displaystyle
Y_{1,4}^{-1}Y_{3,2}$ $\displaystyle\mapsto Y_{1,4}^{-1}Y_{4,2}Y_{6,4}^{-1}$
$\displaystyle Y_{1,3}Y_{3,3}^{-1}Y_{4,2}$ $\displaystyle\mapsto
Y_{1,3}Y_{2,2}Y_{4,3}^{-1}Y_{6,3}$ $\displaystyle
Y_{1,4}^{-1}Y_{2,3}Y_{3,3}^{-1}Y_{4,2}$ $\displaystyle\mapsto
Y_{1,4}^{-1}Y_{2,2}Y_{3,3}Y_{4,3}^{-1}Y_{5,3}Y_{6,4}^{-1}$ $\displaystyle
Y_{2,4}^{-1}Y_{3,3}Y_{4,2}$ $\displaystyle\mapsto
Y_{2,2}Y_{3,4}^{-1}Y_{4,3}Y_{5,4}^{-1}$ $\displaystyle
Y_{3,4}^{-1}Y_{4,2}Y_{4,3}$ $\displaystyle\mapsto Y_{2,2}Y_{2,3}Y_{4,4}^{-1}$
$\displaystyle Y_{4,2}Y_{4,4}^{-1}$ $\displaystyle\mapsto Y_{2,2}Y_{2,4}^{-1}$
$\displaystyle Y_{3,3}Y_{4,3}^{-1}Y_{4,4}^{-1}$ $\displaystyle\mapsto
Y_{2,3}^{-1}Y_{2,4}^{-1}Y_{4,3}$ $\displaystyle
Y_{2,4}Y_{3,4}^{-1}Y_{4,4}^{-1}$ $\displaystyle\mapsto
Y_{2,4}^{-1}Y_{3,4}Y_{4,4}^{-1}Y_{5,4}$ $\displaystyle
Y_{1,5}Y_{2,5}^{-1}Y_{3,4}Y_{4,4}^{-1}$ $\displaystyle\mapsto
Y_{1,5}Y_{2,4}^{-1}Y_{3,5}^{-1}Y_{4,4}Y_{5,5}^{-1}Y_{6,5}$ $\displaystyle
Y_{1,6}^{-1}Y_{3,4}Y_{4,4}^{-1}$ $\displaystyle\mapsto
Y_{1,6}^{-1}Y_{2,4}^{-1}Y_{4,4}Y_{6,6}^{-1}$ $\displaystyle
Y_{1,5}Y_{3,5}^{-1}$ $\displaystyle\mapsto Y_{1,5}Y_{4,5}^{-1}Y_{6,5}$
$\displaystyle Y_{1,6}^{-1}Y_{2,5}Y_{3,5}^{-1}$ $\displaystyle\mapsto
Y_{1,6}^{-1}Y_{3,5}Y_{4,5}^{-1}Y_{5,5}Y_{6,6}^{-1}$ $\displaystyle
Y_{2,6}^{-1}Y_{3,5}$ $\displaystyle\mapsto Y_{3,6}^{-1}Y_{4,5}Y_{5,6}^{-1}$
$\displaystyle Y_{3,6}^{-1}Y_{4,5}$ $\displaystyle\mapsto Y_{2,5}Y_{4,6}^{-1}$
$\displaystyle Y_{4,6}^{-1}$ $\displaystyle\mapsto Y_{2,6}^{-1}$
$\displaystyle Y_{1,3}Y_{4,3}^{-1}$ $\displaystyle\mapsto
Y_{1,3}Y_{2,3}^{-1}Y_{6,3}$ $\displaystyle Y_{1,4}^{-1}Y_{2,3}Y_{4,3}^{-1}$
$\displaystyle\mapsto Y_{1,4}^{-1}Y_{2,3}^{-1}Y_{3,3}Y_{5,3}Y_{6,4}^{-1}$
$\displaystyle Y_{2,4}^{-1}Y_{3,3}^{2}Y_{4,3}^{-1}$ $\displaystyle\mapsto
Y_{2,3}^{-1}Y_{3,4}^{-1}Y_{4,3}^{2}Y_{5,4}^{-1}$ $\displaystyle
Y_{3,3}Y_{3,4}^{-1}$ $\displaystyle\mapsto Y_{4,3}Y_{4,4}^{-1}$ $\displaystyle
Y_{2,4}Y_{3,4}^{-2}Y_{4,3}$ $\displaystyle\mapsto
Y_{2,3}Y_{3,4}Y_{4,4}^{-2}Y_{5,4}$ $\displaystyle Y_{1,5}Y_{2,5}^{-1}Y_{4,3}$
$\displaystyle\mapsto Y_{1,5}Y_{2,3}Y_{3,5}^{-1}Y_{5,5}^{-1}Y_{6,5}$
$\displaystyle Y_{1,6}^{-1}Y_{4,3}$ $\displaystyle\mapsto
Y_{1,6}^{-1}Y_{2,3}Y_{6,6}^{-1}$ Table 1: The explicit virtualization map from
$B(\Lambda_{1})$ of type $F_{4}$ to $B(\widehat{\Lambda}_{2})$ of type
$E_{6}$.

######  Example 4.5.

We consider the folding of type $D_{5}^{(1)}$ onto $C_{3}^{(1)}$ given by

$D_{5}^{(1)}$$0$$1$$2$$3$$5$$4$$C_{3}^{(1)}$$0$$1$$2$$3$

We embed $B(\Lambda_{0})$ into
$B(\widehat{\Lambda}_{0}+\widehat{\Lambda}_{1})$ and construct the
virtualization map $v$ by

⬇

sage: PC = RootSystem([’C’,3,1]).weight_lattice(extended=True)

sage: LaC = PC.fundamental_weights()

sage: PD = RootSystem([’D’,5,1]).weight_lattice(extended=True)

sage: LaD = PD.fundamental_weights()

sage: MC = crystals.NakajimaMonomials(LaC[0])

sage: MD = crystals.NakajimaMonomials(LaD[0]+LaD[1])

sage: sf = {i: 1 for i in MC.index_set()}

sage: phi = {0: [0,1], 1: [2], 2: [3], 3: [4,5]}

sage: v = MC.crystal_morphism(MD.module_generators, virtualization=phi,

....: scaling_factors=sf)

The resulting first four levels of $B(\Lambda_{0})$ and its image under $v$
are given in Fig. 1.

$Y_{0,1}Y_{1,1}^{-1}Y_{2,0}Y_{2,1}^{-1}Y_{3,0}$$Y_{0,1}Y_{1,1}^{-2}Y_{2,0}^{2}$$Y_{0,0}$$Y_{1,0}Y_{1,1}^{-1}Y_{2,0}$$Y_{1,0}Y_{2,1}Y_{3,1}^{-1}$$Y_{0,2}^{-1}Y_{2,0}^{2}$$Y_{0,1}^{-1}Y_{1,0}^{2}$$Y_{1,0}Y_{2,1}^{-1}Y_{3,0}$$3$$1$$0$$2$$1$$0$$1$$2$
$Y_{0,0}Y_{1,0}$$Y_{2,0}Y_{2,1}^{-1}Y_{3,0}$$Y_{2,0}Y_{3,1}Y_{4,1}^{-1}Y_{5,1}^{-1}$$Y_{0,1}^{-1}Y_{1,1}^{-1}Y_{2,0}^{2}$$Y_{2,0}Y_{3,1}^{-1}Y_{4,0}Y_{5,0}$$Y_{0,1}Y_{1,1}Y_{2,1}^{-1}Y_{3,0}Y_{3,1}^{-1}Y_{4,0}Y_{5,0}$$Y_{0,2}^{-1}Y_{1,2}^{-1}Y_{3,0}^{2}$$Y_{0,1}Y_{1,1}Y_{2,1}^{-2}Y_{3,0}^{2}$$2$$1$$1$$0$$1$$3$$0$$2$
Figure 1: The first four levels of $B(\Lambda_{0})$ of type $C_{3}^{(1)}$
(left) and the image under the virtualization map with $D_{5}^{(1)}$ (right).

######  Remark 4.6.

Consider an array ${\bm{c}}$ for $\mathcal{M}(\infty)_{{\bm{c}}}$ such that
there exists a total order $\prec$ on $I$ such that $i\prec j$ whenever
$c_{ij}=1$ and $i\sim j$. Let $I_{\prec}:=\\{i_{1}\prec i_{2}\prec\cdots\prec
i_{n}\\}$. From [26], there exists a (strict) embedding

$\displaystyle
B(\infty)\lhook\joinrel\longrightarrow\cdots\otimes\mathcal{B}_{i_{2}}\otimes\mathcal{B}_{i_{1}}\otimes\mathcal{B}_{i_{n}}\otimes\dotsm\otimes\mathcal{B}_{i_{2}}\otimes\mathcal{B}_{i_{1}},$
(4.2)

where $\mathcal{B}_{i}=\\{b_{i}(a)\colon a\in\mathbf{Z}\\}$ is the elementary
$i$-crystal. From the tensor product rule, the image of the embedding (4.2) is
isomorphic to $\mathcal{M}(\infty)_{{\bm{c}}}$ under the map

$\displaystyle\prod_{\begin{subarray}{c}i\in I\\\ k\in\mathbf{Z}_{\geq
0}\end{subarray}}\\!\\!A_{i,k}^{a_{i}(k)}\\!\mapsto\cdots\otimes
b_{i_{2}}\bigl{(}a_{i_{2}}(1)\bigr{)}\otimes
b_{i_{1}}\bigl{(}a_{i_{1}}(1)\bigr{)}\otimes
b_{i_{n}}\bigl{(}a_{i_{n}}(0)\bigr{)}\otimes\dotsm\otimes
b_{i_{2}}\bigl{(}a_{i_{2}}(0)\bigr{)}\otimes
b_{i_{1}}\bigl{(}a_{i_{1}}(0)\bigr{)},$

which can be considered a special case of the isomorphism to prove [23,
Theorem 3.1]. In other words, the values $k_{i}^{e}$ and $k_{i}^{f}$ determine
which tensor factor to act upon. Furthermore, the proof of [28, Theorem 5.1]
is a special case of the proof presented here.

##  5 Virtualization of Kirillov–Reshetikhin crystals

We first recall the definition of Nakajima monomials given by Nakajima [48,
49, 50, 51]. This definition of Nakajima monomials is necessary in order to
invoke the results of [12], and this modification is only necessary for this
section. In this case we have $c_{ij}=1$ for all $i,j\in I$ and instead use

$\displaystyle A^{\prime}_{i,k}=Y_{i,k}Y_{i,k+2}\prod_{j\neq
i}Y_{j,k+c_{ji}}^{C_{ji}}$

in place of $A_{i,k}$ in (2.1). In particular, Nakajima showed that if there
are no odd length cycles in the Dynkin diagram, then we have an abstract
$U_{q}(\mathfrak{g})$-crystal structure and the analog of Theorem 2.2.222For
type $A_{n}^{(1)}$, a common generalization of these two models was made in
[67]. We note that this restricted set of Cartan types includes all finite and
affine types except $A_{2k}^{(1)}$.

We note that by using Nakajima’s crystal structure, the compatibility
condition $\widehat{{\bm{c}}}$ and ${\bm{c}}$ is immediately satisfied.
Furthermore, it is straightforward to verify that Lemma 4.1 and Theorem 4.3
hold in this setting.

Now, let us consider $\mathfrak{g}$ of affine type and $\lambda$ be a _level-
zero dominant weight_ ; that is to say, we have
$\lambda=\sum\limits_{i=1}^{N}m_{i}(\Lambda_{r_{i}}-c_{r_{i}}\Lambda_{0})$ for
$m_{i}\in\mathbf{Z}_{\geq 0}$, where the positive integers $c_{r_{i}}$ are
such that $\Lambda_{r_{i}}-c_{r_{i}}\Lambda_{0}$ has level zero [16, Chapter
12]. We consider an extremal level-zero crystal $B(\lambda)$ coming from an
extremal level-zero module (see [30, Section 3] for more information).
Kashiwara showed that the crystal exists in [27], and there exists an
automorphism $\eta$ of $B(\lambda)$ such that

$\displaystyle
B(\lambda)/\eta\approx\bigotimes_{i=1}^{N}(B^{r_{i},1})^{\otimes m_{i}},$

where $B^{r,s}$ are certain finite crystals, in [30]. These finite crystals
were later realized to be _Kirillov–Reshetikhin $($KR$)$ crystals_, the
crystal basis corresponding to Kirillov–Reshetikhin modules [9, 10, 55].

In [12], a construction of the extremal level-zero crystal $B(\lambda)$ was
given in terms of Nakajima monomials as the closure of certain monomials and
the automorphism $\eta$. In particular, a power of the automorphism $\eta$ is
a shift map $\tau_{s}(Y_{i,k})=Y_{i,k+s}$. It is straightforward to see that
the defining monomials, the power of the automorphism, and the shift all
agree. Therefore, we obtain the analog results to [59, Theorem 4.2] and
partially [44, 45, 47] of a virtualization of KR crystals by using Nakajima
monomials.

######  Theorem 5.1.

Let $\mathfrak{g}$ be of affine type. Then $v\circ\eta=\eta\circ v$. Moreover,
the KR crystal $B^{r,1}$ virtualizes in
$\bigotimes_{r^{\prime}\in\phi^{-1}(r)}(B^{r^{\prime},1})^{\otimes\gamma_{r}}$.

Thus we have a proof of special cases of [57, Conjecture 3.7] using Nakajima
monomials.

##  6 Monomials to PBW data in type $\bm{A_{n}}$

Let $W$ be the Weyl group associated to the Cartan matrix $C=(C_{ij})$. In
this section, we are considering the Cartan matrix of type $A_{n}$, so $W$ is
isomorphic to the symmetric group $\mathfrak{S}_{n+1}$. In particular, $W$ is
a finite group generated by simple transpositions $s_{1},\dotsc,s_{n}$. Every
element $w\in W$ may be expressed in the form $w=s_{i_{1}}s_{i_{2}}\dotsm
s_{i_{\ell}}$; in this case, if $\ell$ is minimal among all such expressions
for $w$, then $\ell$ is called the length of $w$ and the expression above is
called reduced. An element $w\in W$ may have more than one reduced expression,
but all reduced expressions have the same length.

There is a unique element of longest length in $W$, which we denote as
$w_{0}$. Associated to a reduced expression $w_{0}=s_{i_{1}}s_{i_{2}}\dotsm
s_{i_{N}}$, where $N$ is the number of positive roots, one may construct a
realization of the crystal $B(\infty)$ using Lusztig’s PBW-type basis [43].
This construction can be complicated in large rank. However, in [62], an
efficient algorithm for calculating the crystal structure for certain reduced
expressions was developed. We now recall this efficient procedure for a
specifically chosen reduced expression for $w_{0}$.

Fix, once and for all, the reduced expression

$\displaystyle
w_{0}=s_{n}(s_{n-1}s_{n})(s_{n-2}s_{n-1}s_{n})\dotsm(s_{1}s_{2}\dotsm s_{n}).$
(6.1)

The reduced expression in equation (6.1) is called the _dual BZL word_.

######  Lemma 6.1 ([62]).

The word ${\mathbf{i}}=(n,n-1,n,\dots,1,2,\dots,n)$ is simply braided.

We will not need the explicit notion of a word being simply braided. For us,
this means that the crystal structure on $B(\infty)$ can be understood
efficiently using a bracketing rule on Kostant partitions.333This is the same
crystal structure as defined in, for example, [2], but the combinatorics
defining the rules is different. Let
$\Phi^{+}=\\{\alpha_{j,k}=\alpha_{j}+\alpha_{j+1}+\cdots+\alpha_{k}\colon
1\leq j\leq k\leq n\\}$ denote the set of positive roots of type $A_{n}$,
define $\mathcal{R}=\\{(\alpha)\colon\alpha\in\Phi^{+}\\}$, and set
$\mathrm{Kp}(\infty)$ to be the free $\mathbf{Z}_{\geq 0}$-span of
$\mathcal{R}$. An element of $\mathrm{Kp}(\infty)$ will be considered a
Kostant partition, and denoted by
$\bm{\alpha}=\sum\limits_{(\alpha)\in\mathcal{R}}c_{\alpha}(\alpha)$, where
$c_{\alpha}\in\mathbf{Z}_{\geq 0}$. For some fixed $i\in I$, in order to
compute $f_{i}\bm{\alpha}$ and $e_{i}\bm{\alpha}$, one must first calculate
the bracketing sequence associated to $i$ and $\bm{\alpha}$. Indeed, following
[62], define $S_{i}(\bm{\alpha})$ to be the string of brackets

$\displaystyle\underbrace{)\cdots)}_{c_{\alpha_{i,n}}}\
\underbrace{(\cdots(}_{c_{\alpha_{i+1,n}}}\
\underbrace{)\cdots)}_{c_{\alpha_{i,n-1}}}\
\underbrace{(\cdots(}_{c_{\alpha_{i+1,n-1}}}\ \ \ \cdots\ \ \
\underbrace{)\cdots)}_{c_{\alpha_{i,i+1}}}\
\underbrace{(\cdots(}_{c_{\alpha_{i+1,i+1}}}\
\underbrace{)\cdots)}_{c_{\alpha_{i,i}}}.$

Successively cancel $()$-pairs to obtain sequence of the form
$)\cdots)(\cdots($. We call the remaining brackets _uncanceled_.

######  Definition 6.2.

Let $i\in I$ and
$\bm{\alpha}=\sum\limits_{(\alpha)\in\mathcal{R}}c_{\alpha}(\alpha)\in\mathrm{Kp}(\infty)$.

  * •

Let $\beta$ be the root corresponding to the rightmost uncanceled ‘$)$’ in
$S_{i}(\bm{\alpha})$. Define

$\displaystyle e_{i}\bm{\alpha}=\bm{\alpha}-(\beta)+(\beta-\alpha_{i}).$

If $\beta=\alpha_{i}$, we interpret $(0)$ as the additive identity in
$\mathrm{Kp}(\infty)$. If no such ‘$)$’ exists, then $e_{i}\bm{\alpha}$ is
undefined.

  * •

Let $\gamma$ denote the root corresponding to the leftmost uncanceled ‘$($’ in
$S_{i}(\bm{\alpha})$. Define,

$\displaystyle f_{i}\bm{\alpha}=\bm{\alpha}-(\gamma)+(\gamma+\alpha_{i}).$

If no such ‘$($’ exists, set $f_{i}\bm{\alpha}=\bm{\alpha}+(\alpha_{i})$.

  * •

${\rm wt}(\bm{\alpha})=-\sum\limits_{\alpha\in\Phi^{+}}c_{\alpha}\alpha.$

  * •

$\varepsilon_{i}(\bm{\alpha})=\text{number of uncanceled `$)$' in the
bracketing sequence of $\bm{\alpha}$}$.

  * •

$\varphi_{i}(\bm{\alpha})=\varepsilon_{i}(\bm{\alpha})+\langle h_{i},{\rm
wt}(\bm{\alpha})\rangle$.

######  Example 6.3.

Let $n=3$ and define
$\bm{\alpha}=2(\alpha_{3})+3(\alpha_{2}+\alpha_{3})+2(\alpha_{2})+2(\alpha_{1}+\alpha_{2})+5(\alpha_{1})$.
Then

$\displaystyle S_{1}(\bm{\alpha})={(}\ (\ (\ )\ )\ (\ (\ )\ )\ )\ )\
)\,,\qquad S_{2}(\bm{\alpha})={)}\ )\ )\ (\ (\ )\ )\,,\qquad
S_{3}(\bm{\alpha})={)}\ )\,.$

Hence

$\displaystyle
f_{1}\bm{\alpha}=2(\alpha_{3})+3(\alpha_{2}+\alpha_{3})+2(\alpha_{2})+2(\alpha_{1}+\alpha_{2})+6(\alpha_{1}),$
$\displaystyle
f_{2}\bm{\alpha}=2(\alpha_{3})+3(\alpha_{2}+\alpha_{3})+3(\alpha_{2})+2(\alpha_{1}+\alpha_{2})+5(\alpha_{1}),$
$\displaystyle
f_{3}\bm{\alpha}=3(\alpha_{3})+3(\alpha_{2}+\alpha_{3})+2(\alpha_{2})+2(\alpha_{1}+\alpha_{2})+5(\alpha_{1}).$

######  Theorem 6.4 ([62]).

Let $u_{\infty}$ be the unique element of $B(\infty)$ with weight zero. The
map $u_{\infty}\mapsto(0)$ induces a $U_{q}(A_{n})$-crystal isomorphism
between $B(\infty)$ and $\mathrm{Kp}(\infty)$.

For the remainder of this section, define ${\bm{c}}=(c_{ij})_{i\neq j}$ by
$c_{ij}=0$ if $i>j$ and $c_{ij}=1$ if $i<j$. This is the same convention used
in [23, 33].

######  Theorem 6.5.

The map defined by

$\displaystyle\prod_{1\leq j\leq k\leq
n}\left(\prod_{p=j}^{k}A_{p,k-p}^{-1}\right)^{\ell_{j,k}}\mapsto\sum_{j=1}^{n}\sum_{k=j}^{n}\ell_{j,k}(\alpha_{j,k})$

is a $U_{q}(A_{n})$-crystal isomorphism from $\mathcal{M}(\infty)_{\bm{c}}$ to
Kostant partitions.

Before proving the theorem, it must be shown that such a map is well-defined.

######  Lemma 6.6.

Each $M\in\mathcal{M}(\infty)_{\bm{c}}$ can be written uniquely in the form

$\displaystyle M=\prod_{1\leq j\leq k\leq
n}\left(\prod_{p=j}^{k}A_{p,k-p}^{-1}\right)^{\ell_{j,k}},$

for some $\ell_{j,k}\in\mathbf{Z}_{\geq 0}$.

###### Proof.

In [33, Theorem 4.1], it was shown that

$\displaystyle\prod_{i\in
I}\prod_{q=0}^{n-i}A_{i,q}^{-a_{i,q}}\in\mathcal{M}(\infty)_{\bm{c}}$

if and only if

$\displaystyle 0\leq a_{1,i-1}\leq a_{2,i-2}\leq\cdots\leq a_{i,0}$ (6.2)

for all $1\leq i\leq n$. It is straightforward to see that we have

$\displaystyle a_{i,q}=\ell_{i,i+q}+\ell_{i-1,i+q}+\cdots+\ell_{1,i+q},$
$\displaystyle\ell_{i,i+q}=a_{i,q}-a_{i-1,q+1},$

and that $\ell_{i,k}\in\mathbf{Z}_{\geq 0}$ if and only if (6.2) holds. Hence,
the claim follows. ∎

###### Proof of Theorem 6.5.

Let $i\in I=\\{1,2,\dots,n\\}$. For notational brevity, once and for all in
this proof, denote

$\displaystyle M=\prod_{1\leq j\leq k\leq
n}\left(\prod_{p=j}^{k}A_{p,k-p}^{-1}\right)^{\ell_{j,k}},\qquad$
$\displaystyle f_{i}M=\prod_{1\leq j\leq k\leq
n}\left(\prod_{p=j}^{k}A_{p,k-p}^{-1}\right)^{\ell_{j,k}^{\prime}},$
$\displaystyle\bm{\alpha}=\sum_{1\leq j\leq k\leq
n}\ell_{j,k}(\alpha_{j,k}),\qquad$ $\displaystyle f_{i}\bm{\alpha}=\sum_{1\leq
j\leq k\leq n}\ell_{j,k}^{\prime\prime}(\alpha_{j,k}).$

Recall

$\displaystyle S_{i}(\bm{\alpha})=\underbrace{)\cdots)}_{\ell_{{i,n}}}\
\underbrace{(\cdots(}_{\ell_{{i+1,n}}}\
\underbrace{)\cdots)}_{\ell_{{i,n-1}}}\
\underbrace{(\cdots(}_{\ell_{{i+1,n-1}}}\ \ \ \cdots\ \ \
\underbrace{)\cdots)}_{\ell_{{i,i+1}}}\
\underbrace{(\cdots(}_{\ell_{{i+1,i+1}}}\
\underbrace{)\cdots)}_{\ell_{{i,i}}}.$

Suppose the leftmost uncanceled ‘(’ in $S_{i}(\bm{\alpha})$ occurs in the
position corresponding to the root $\alpha_{i+1,k_{0}}$, for some $i<k_{0}\leq
n$. Then
$f_{i}\bm{\alpha}=\bm{\alpha}-(\alpha_{i+1,k_{0}})+(\alpha_{i,k_{0}})$. Hence

$\displaystyle\ell_{j,k}^{\prime\prime}=\begin{cases}\ell_{i,k_{0}}+1&\text{if
}(j,k)=(i,k_{0}),\\\ \ell_{i,k_{0}}-1&\text{if }(j,k)=(i+1,k_{0}),\\\
\ell_{j,k}&\text{otherwise}.\end{cases}$

If there is no uncanceled ‘(’ in $S_{i}(\bm{\alpha})$, then set $k_{0}=i$;
therefore, $f_{i}\bm{\alpha}=\bm{\alpha}+(\alpha_{i,i})$ and

$\displaystyle\ell_{j,k}^{\prime\prime}=\begin{cases}\ell_{i,i}+1&\text{if
}(j,k)=(i,i),\\\ \ell_{j,k}&\text{otherwise}.\end{cases}$

We intend to show $k_{i}^{f}(M)=k_{0}-i$; in this case, a direct calculation
shows that $\ell_{j,k}^{\prime}=\ell_{j,k}^{\prime\prime}$ for all $1\leq
j\leq k\leq n$.

Expanding the given expression for $M$, we have

$\displaystyle M$ $\displaystyle=\prod_{1\leq j\leq k\leq
n}\left(\prod_{p=j}^{k}A_{p,k-p}^{-1}\right)^{\ell_{j,k}}$
$\displaystyle=\prod_{1\leq j\leq k\leq
n}\left(\prod_{p=j}^{k}Y_{p,k-p}^{-1}Y_{p,k-p+1}^{-1}Y_{p-1,k-p+1}Y_{p+1,k-p}\right)^{\ell_{j,k}}$
$\displaystyle=\prod_{1\leq j\leq k\leq
n}\big{(}Y_{j-1,k-j+1}Y_{j,k-j+1}^{-1}Y_{k,0}^{-1}Y_{k+1,0}\big{)}^{\ell_{j,k}}.$

If we express this same $M$ using the form $\prod\limits_{p\in
I}\prod\limits_{q\geq 0}Y_{p,q}^{y_{p}(q)}$, then, for $i\in I$,

$\displaystyle\sum_{r=0}^{s}y_{i}(r)=-\ell_{i,i}+\sum_{t=1}^{i}(\ell_{t,i-1}-\ell_{t,i})+\sum_{u=0}^{s-1}(\ell_{i+1,i+1+u}-\ell_{i,i+u}).$

If there is no uncanceled ‘(’ in $S_{i}(\bm{\alpha})$, then each term in the
third summand is nonpositive. Therefore

$\displaystyle\varphi_{i}(M)=\max\left\\{\sum_{r=0}^{s}y_{i}(r)\colon s\geq
0\right\\}=0$

and

$\displaystyle k_{i}^{f}(M)=\min\left\\{s\geq
0\colon\varphi_{i}(M)=\sum_{r=0}^{s}y_{i}(r)\right\\}=0=k_{0}-i.$

On the other hand, if there is an uncanceled ‘(’ in $S_{i}(\bm{\alpha})$
(which corresponds to $(\alpha_{i+1,k_{0}})$ by our definitions), then
$\ell_{i+1,i+1+t}\leq\ell_{i,i+t}$ for $t>k_{0}-i$. In particular, the maximum

$\displaystyle\max\left\\{\sum_{r=0}^{s}y_{i}(r)\colon s\geq 0\right\\}$

is attained when $s=k_{0}-i$. This maximum cannot be attained for any
$s_{0}<s$ since, if it did, it would contradict the position of the leftmost
uncanceled ‘(’ in $S_{i}(\bm{\alpha})$. Hence $k_{i}^{f}(M)=k_{0}-i$, as
required. ∎

An alternative approach to proving Theorem 6.5 would be to extend the map
given in [39, Proposition 6.12] to a map from $\mathcal{M}(\infty)$ to the
(marginally) large tableaux model [5, 14]. From there, one could construct the
Kostant partition using the BZL (or minimal lexicographic) word

$\displaystyle
w_{0}=s_{1}(s_{2}s_{1})(s_{3}s_{2}s_{1})\dotsm(s_{n}s_{n-1}\dotsm s_{1})$

and then change that into the dual BZL word to recover Theorem 6.5.

##  7 Open problems

An original goal of this paper was to use Theorems 4.3 and 6.5 to give a
method to extract Lusztig data from Nakajima monomials. However, because of
the compatibility condition of ${\bm{c}}$ and the specific reduced word (and
hence ${\bm{c}}$ that comes from the corresponding orientation of the Dynkin
diagram) needed for Theorem 6.5, our techniques do not apply. In this section,
we discuss these limitations in more detail.

We first note that equation (3.1) gives a virtualization of Lusztig data in
types $B_{n}$ and $C_{n}$ in terms of type $A_{2n-1}$, and this corresponds to
the virtualization map given in [15, 46] (in terms of MV polytopes). Indeed,
let ${\mathbf{i}}$ denote a reduced expression for $w_{0}\in W$ corresponding
to a finite type $\mathfrak{g}$, and fix a diagram automorphism $\phi$. Then

$\displaystyle\widehat{{\mathbf{i}}}:=\overrightarrow{\prod_{i\in{\mathbf{i}}}}\prod_{j\in\phi^{-1}(i)}j$

is a reduced expression for $\widehat{w}_{0}\in\widehat{W}$. Let
$\mathcal{L}=(L_{i})_{i\in{\mathbf{i}}}$ denote the Lusztig data for an
element in the set of MV polytopes of type $\mathfrak{g}$. Then we define

$\displaystyle
v(\mathcal{L})=(\gamma_{\phi(j)}L_{\phi(j)})_{j\in\phi^{-1}({\mathbf{i}})}.$
(7.1)

However, there is no reduced word ${\mathbf{i}}$ for $w_{0}$ in type $B_{n}$
(equiv. $C_{n}$) such that $\widehat{{\mathbf{i}}}$ is the reduced word given
by Lemma 6.1, which is used in Theorem 6.5. Thus, one way to solve this
problem would be determine an explicit isomorphism

$\displaystyle\psi_{\widehat{{\bm{c}}},\widehat{{\bm{c}}}^{\prime}}\colon\
\widehat{\mathcal{M}}(\infty)_{\widehat{{\bm{c}}}}\longrightarrow\widehat{\mathcal{M}}(\infty)_{\widehat{{\bm{c}}}^{\prime}}.$

We recall a method to mutate $\mathcal{M}_{{\bm{c}}}$ into another with a
different array $\mathcal{M}_{{\bm{c}}^{\prime}}$ from [28]. Fix some
${\bm{m}}=(m_{i}\in\mathbf{Z}\colon i\in I)$, which we will call a _mutation_
, and define the array ${\bm{c}}^{\prime}=\\{c^{\prime}_{ij}\colon i,j\in I$,
$i\neq j\\}$ by $c^{\prime}_{ij}=c_{ij}+m_{i}-m_{j}$. This induces a crystal
isomorphism
$\mu_{{\bm{m}}}\colon\mathcal{M}_{{\bm{c}}}\longrightarrow\mathcal{M}_{{\bm{c}}^{\prime}}$
given by $Y_{i,k}\mapsto Y_{i,k+m_{i}}$. Therefore, we have an explicit
isomorphism

$\displaystyle\mu_{{\bm{m}}}\colon\
\mathcal{M}(M)_{{\bm{c}}}\longrightarrow\mathcal{M}\bigl{(}\mu_{{\bm{m}}}(M)\bigr{)}_{{\bm{c}}^{\prime}}$
(7.2)

for any fixed monomial $M$ by the restriction of $\mu_{{\bm{m}}}$ to
$\mathcal{M}(M)$.

First, note that there exists $\widehat{{\bm{m}}}$ such that
$v\circ\mu_{{\bm{m}}}=\mu_{\widehat{{\bm{m}}}}\circ v$ by considering
$\widehat{m}_{j}=m_{i}$ for all $j\in\phi^{-1}(i)$. Next, from [28], we can
always find a mutation ${\bm{m}}$ such that $\widehat{{\bm{c}}}$ is compatible
with ${\bm{c}}$ when the Dynkin diagram has no loops. Indeed, if we consider a
linear Dynkin diagram oriented towards an endpoint $v_{h}$, we can flip the
orientation of a single edge $v_{1}\to v_{0}$ by considering the mutation
sequence $m_{v_{1}}=\cdots=m_{v_{t}}=1$, where $v_{t}$ is the other endpoint
of the Dynkin diagram, and all other $m_{i}=0$. We can iterate this (which
corresponds to adding the mutations considered as vectors) for edges
successively closer to $v_{t}$, thus we can obtain the Dynkin diagram oriented
with both endpoints being sinks. Note that there may be values
$c_{ij}\notin\mathbf{Z}_{\geq 0}$, but these will all correspond to $i\not\sim
j$.

######  Example 7.1.

Consider a Dynkin diagram that is an edge decorated path of length $3$; e.g.,
type $B_{4}$ or $C_{3}^{(1)}$. Then we have

$\displaystyle{\bm{c}}:=\begin{bmatrix}\bullet&1&1&1\\\ 0&\bullet&1&1\\\
0&0&\bullet&1\\\ 0&0&0&\bullet\end{bmatrix}\xrightarrow[\hskip
60.0pt]{\mu_{(0,0,1,2)}}\begin{bmatrix}\bullet&1&0&-1\\\ 0&\bullet&0&-1\\\
1&1&\bullet&0\\\ 2&2&1&\bullet\end{bmatrix}=:{\bm{c}}^{\prime}.$

We note that the only entries $c_{ij}$ with $i\sim j$ correspond to those on
the superdiagonal and subdiagonal. We can compute this example with the
following code.

⬇

sage: def mutate(c, m):

....: cp = copy(c)

....: for i in range(c.nrows()):

....: for j in range(c.ncols()):

....: cp[i,j] += m[i] - m[j]

....: return cp

sage: c = matrix([[0,1,1,1],[0,0,1,1],[0,0,0,1],[0,0,0,0]])

sage: mutate(c, [0,0,1,2])

[ 0 1 0 -1]

[ 0 0 0 -1]

[ 1 1 0 0]

[ 2 2 1 0]

Recall that $\mathcal{M}(\infty)_{\bm{c}}$ can be defined as
$\varinjlim\limits_{\lambda\to\infty}Y_{\lambda}^{-1}\mathcal{M}(Y_{\lambda})$.
However, trying to take the corresponding virtualization of highest weight
crystals means we have to do a renormalization. From [28], there exists an
abstract isomorphism of crystals
$\mathcal{M}(M)_{{\bm{c}}}\cong\mathcal{M}(M^{\prime})_{{\bm{c}}^{\prime}}$,
where $M$ and $M^{\prime}$ are highest weight monomials with ${\rm wt}(M)={\rm
wt}(M^{\prime})$. However, no combinatorial description of this isomorphism is
known beyond equation (7.2). Therefore, we cannot renormalize to have the
highest weight monomial be $Y_{\lambda}$ for any choice of
$\widehat{{\bm{c}}}^{\prime}$. Instead, we currently must describe
$\psi_{\widehat{{\bm{c}}},\widehat{{\bm{c}}}^{\prime}}$ using a sequence of
transition maps on the polyhedral model from [3] to connect the Nakajima
monomials with the Lusztig data in types $B_{n}$ or $C_{n}$ using the
virtualization map.

To give a little more detail, fix a reduced word ${\mathbf{i}}$ for $w_{0}$ of
type $B_{n}$ (or $C_{n}$) and array ${\bm{c}}$. Let
$\widehat{{\bm{c}}}^{\prime}$ and $\widehat{{\mathbf{i}}}^{\prime}$ be the
array and reduced word for Theorem 6.5. Consider the commutative diagram:

$\mathcal{M}(\infty)_{\bm{c}}$$\widehat{\mathcal{M}}(\infty)_{\widehat{{\bm{c}}}}$$\widehat{\mathcal{M}}(\infty)_{\widehat{{\bm{c}}}^{\prime}}$$\mathrm{Kp}(\infty)_{{\mathbf{i}}}$$\widehat{\mathrm{Kp}}(\infty)_{\widehat{{\mathbf{i}}}}$$\widehat{\mathrm{Kp}}(\infty)_{\widehat{{\mathbf{i}}}^{\prime}}$,$v$$\psi_{\widehat{{\bm{c}}},\widehat{{\bm{c}}}^{\prime}}$$\zeta$$\xi$$\Upsilon$$v$

where $\mathrm{Kp}(\infty)_{{\mathbf{i}}}$ is the crystal of Kostant
partitions with respect to the reduced word ${\mathbf{i}}$ and $\Upsilon$ is
the isomorphism from Theorem 6.5. The maps $\xi$ and $\zeta$ are abstract
crystal isomorphisms for which we currently do not know an explicit
description other than by using transition maps of [3] and the diagram above.
Therefore, determining an explicit map without using a sequence of transitions
maps for $\psi_{\widehat{{\bm{c}}},\widehat{{\bm{c}}}^{\prime}}$ is equivalent
to an explicit description of $\xi$. Furthermore, a description of
$\psi_{\widehat{{\bm{c}}},\widehat{{\bm{c}}}^{\prime}}$ gives an explicit map
for $\zeta$ by Theorem 4.3 and equation (7.1). Note that having an explicit
map for $\zeta$ only partially gives a map for
$\psi_{\widehat{{\bm{c}}},\widehat{{\bm{c}}}^{\prime}}$ or $\xi$ (i.e., when
restricted to the virtual crystal).

######  Problem 7.2.

Determine an analog of Theorem 6.5 for other types.

For type $A_{n}^{(1)}$, there exists an explicit crystal isomorphism from
Nakajima monomials and generalized young walls given by composition of [23,
Theorem 5.1] and [34, Theorem 3.1], which uses the path model of [20]. (See
also [33, Remark 2.4].) Kim and Shin gave an extension of generalized Young
walls to other types using zigzag strip bundles [35, 36, 37, 38]. Another
potential application of our results is to give a virtualization map from
zigzag strip bundles to generalized Young walls through the explicit crystal
isomorphism with Nakajima monomials. However, this has the same technical
limitation as before.

Similarly, we can apply our virtualization map to the KR crystals of [8], but
that requires ${\bm{c}}$ to correspond to orienting the Dynkin diagram around
the cycle. Thus, this is a similar technical limitation as before, but has a
more severe limitation because there is not a way to apply mutations to the
cycle to obtain a compatible ${\bm{c}}$. Additionally, one needs to construct
$B^{n,s}$ using Nakajima monomials, but this should be similar to $B^{1,s}$
for a $\widehat{{\bm{c}}}$. Yet, it is likely that a solution to the Nakajima
monomial virtualization for KR crystals is equivalent to virtualization for
zigzag strip bundles via the Kyoto path model [21, 22, 56] and [8, Theorem
4.2].

### Acknowledgements

B.S. was partially supported by CMU Early Career grant #C62847 and by Simons
Foundation grant #429950. T.S. was partially supported by the National Science
Foundation RTG grant NSF/DMS-1148634. This work benefited from computations
using SageMath [60, 61].

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