数理解析研究所講究録
第 1795 巻 2012 年 163-178

163

KHOVANOV-LAUDA-ROUQUIER ALGEBRAS AND CRYSTAL BASESFOR FINITE CLASSICAL TYPEEUIYONG PARKABSTRACT. We present a crystal basis theoretic construction of irreducible modules over
Khovanov-Lauda-Rouquier algebras and their cyclotomic quotients of finite classical type.

INTRODUCTION
The Khovanov-Lauda-Rouquier algebras (or Hecke quiver algebras) were introduced independently Khovanov-Lauda [11, 12] and Rouquier [16] for providing a categorification ofquantum groups associated with symmetrizable Cartan data. Let
be a quantum groupand letbe the corresponding Khovanov-Lauda-Rouquier algebra of weight $\alpha\in Q+$ . Fora dominant integral weight, the algebra, whichhas a special quotient
gives a categorification of the irreducible highest weight-modulewith highest
weight $\lambda[4,18]$ . The crystal structures ofandalso were interpreted in [14] interms of irreducible modules overand
. The Khovanov-Lauda-Rouquier algebras
were generalized to the quantum generalized Kac-Moody algebras [5, 7] and, when the Cartan
datum is symmetric, geometric realizations of the Khovanov-Lauda-Rouquier algebras were
given in [6, 17] via quiver varieties.
In this paper, we announce the main result of our previous work [1, 8], which is an explicit construction of irreducible modules overandof finite classical type. This
construction differs from the one given in [2, 13] and is based on the theory of crystal bases.
Though this paper is rely on [1], the description of irreducible modules in this paper isdifferent and more combinatorial than the description given in [1].
Let us explain more precisely. Let(resp.
) be the set of all isomorphismclasses of irreducible graded. It is shown-modules) for-modules (resp.
in [14] that there exists an crystal isomorphism
resp.
. We
first define segments to be unordered pairs of comparable elements in the basic crystals$B_{X}(X=A, B, C, D)$ given in Section 3 and give a partial orderto them. Then we set
multisegments of
to be multisets of segments satisfying the conditions (3.1). Proposition$U_{q}(\mathfrak{g})$$R(\alpha)$$\lambda\in p+$$R(\alpha)$$R^{\lambda}(\alpha)$$V(\lambda)$$U_{q}(\mathfrak{g})$$V(\lambda)$$U_{q}^{-}(\mathfrak{g})$$R(\alpha)$$R^{\lambda}(\alpha)$$R(\alpha)$$\mathfrak{B}(\infty)$$R(\alpha)$$R^{\lambda}(\alpha)$$\mathfrak{B}(\lambda)$$\alpha\in Q^{+}$$R^{\lambda}(\alpha)$$\mathfrak{B}(\infty)\simeq B(\infty)$$\mathfrak{B}(\lambda)\simeq B(\lambda))$$($$\epsilon$$\preceq$$\underline{m}$$B_{X}$

This work was supported by JSPS Postdoctoral Fellowships for Foreign Researchers.

164
EUIYONG PARK.
of multisegements is in a 1-1 correspondence to the crystal. Thenout of the crystal
For each segment , we define 1 or 2-dimensional module
gives the outer tensor productofa multisegment
3.1 says that the set$B(\infty)$$\prime r$$\mathcal{M}x$$B_{X}$$\nabla(\epsilon)$$g$$B_{X}$$\underline{m}=\{\epsilon_{1}\preceq\epsilon_{2}\preceq 53\preceq\cdots\}$$\nabla(\underline{m})=\nabla(\epsilon_{1})\otimes\nabla(s_{2})\otimes\nabla(\epsilon_{3})\otimes\cdots$.

It follows from [1] that

hdInd
for $m\in M_{X}$ and the map$\nabla(\underline{m})=\tilde{f}_{\nu_{X}}^{e_{1}(\underline{\mathfrak{m}})}\ldots\tilde{f}_{\nu_{X}}^{e_{n}(\underline{m})}1$: $M_{X}arrow B(\infty)$ defined by$\Psi$$\Psi(\underline{m})=$hdInd$\nabla(\underline{m})$for$\underline{m}\in \mathcal{M}x$is bijective (Theorem 3.2). Hence it can be deduced from Proposition 2.3 that the composition:
is a crystal isomorphism. In the cyclotomic cases, using the, we obtain the same results; i.e., the composition
crystal embedding:
is a crystal isomorphism (Corollary 3.4).
This paper is organized as follows. Section 1 contains a brief review of the crystal basis
theory for quantum generalized Kac-Moody algebras. In Section 2, we give the definition of
Khovanov-Lauda-Rouquier algebras which is the most general version given in [6] associatedwith a Borcherds-Cartan datum (or a quiver positively with loops), and introduce someresults of Khovanov-Lauda-Rouquier algebras on crystal bases. In Section 3, we restrict tothe case of finite classical types and present a crystal basis theoretic construction of irreducible
modules over Khovanov-Lauda-Rouquier algebras and their cyclotomic quotients in terms ofsegments.
$\Psi 0^{\prime r-1}$$B(\infty)arrow \mathfrak{B}(\infty)$$B(\lambda)\mapsto B(\infty)\otimes T_{\lambda}\otimes C$$\Psi^{\lambda}\circ Y_{\lambda}^{-1}$$B(\lambda)arrow \mathfrak{B}(\lambda)$

1. QUANTUM GENERALIZED $KAC$-MOODY ALGEBRAS

Let be an index set. A square matrix $A=(a_{ij})_{i,j\in I}$ is called a symmetrizable Borcherdsfor $i\neq j$ , (iii)for $i\in I$ , (ii)
Cartan matrix if it satisfies (i) $a_{ii}=2$ or$a_{ij}=0$ if $a_{ji}=0$ for $i,j\in I$ , (vi) there is a diagonal matrix $D=$ diag $(d_{i}\in Z_{>0}|i\in I)$ such.
that DA is symmetric. Let $I^{re}=\{i\in I|a_{ii}=2\}$ andA Borcherds-Cartan datum
consists of(1) a symmetrizable Borcherds-Cartan matrix ,(2) a free abelian group , called the weight lattice,$I$$a_{ii}\in 2Z_{\leq 0}$$a_{ij}\in Z_{\leq 0}$$I^{im}=I\backslash I^{re}$$(A, P, \Pi, \Pi^{\vee})$

$A$

$P$

(3) the set $\Pi=\{\alpha_{i}|i\in I\}\subset P$ of simple roots,(4) the set $\Pi^{\vee}=\{h_{i}|i\in I\}\subset p\vee:=Hom(P, Z)$ of simple coroots,

which satisfy the following properties:$:=\alpha_{j}(h_{i})=a_{ij}$ for all $i,j\in I$ ,(i)(ii)
is linearly independent, where り$\langle h_{i},$$\alpha_{j}\rangle$$\Pi\subset \mathfrak{y}*$$:=\mathbb{C}\otimes z^{P^{\vee}}$,165
KLR ALGEBRAS AND CRYSTAL BASES FOR FINITE CLASSICAL TYPE(iii) for each $i\in I$ , there exists$\Lambda_{i}\in P$

such that$\langle h_{j},$$\Lambda_{i}\rangle=\delta_{ij}$for all $j\in I$ .

We denote by $p+=\{\lambda\in P|\lambda(h_{i})\in Z_{\geq 0}, i\in I\}$ the set of dominant integral weights. The
free abelian groupis the root lattice, andis the positiveroot lattice. For, the height of is. There is a symmetricbilinear form $(|)$ on
such that$Q=\oplus_{i\in I}Z\alpha_{i}$$Q^{+}=\sum_{i\in I}Z_{\geq 0}\alpha_{i}$$\alpha=\sum_{i\in I}k_{i}\alpha_{i}\in Q+$$\alpha$$|\alpha|$$:= \sum_{i\in I}k_{i}$$\mathfrak{y}*$$(\alpha_{i}|\alpha_{j})=d_{i}a_{ij}$

Let

$q$

for $i,j\in I$ ,be an indeterminate and $m,$$[n]_{i}= \frac{q_{i}^{n}-q_{i}^{-n}}{q_{i}-q_{i}^{-1}}$.

for$\{h_{i}, \lambda\}=\frac{2(\alpha_{i}|\lambda)}{(\alpha_{i}1\alpha_{i})}$$n\in Z_{\geq 0}$. For$i\in I^{re}$$[n]_{i}!= \prod_{k=1}^{n}[k]_{i},$, let$q_{i}=q^{d_{i}}$and $i\in I$ .

and$\{\begin{array}{l}mn\end{array}\}=\frac{\lfloor m]_{i}!}{[m-n]_{i}![n]_{i}!}$Definition 1.1. The quantum generalized Kac-Moody algebm
Cartan datumis the associative algebra over$(A. P, \Pi, \Pi^{\vee})$$(i\in I)$$\lambda\in \mathfrak{h}^{*}$$U_{q}(\mathfrak{g})$$\mathbb{Q}(q)$.

associated with a Borcherds-

with 1 generated by$e_{i},$$f_{l}$

and $q^{h}(h\in P^{\vee})SatiS\mathfrak{h}rlng$ following relations:(1) $q^{0}=1,$ $q^{h}q^{h’}=q^{h+h’}$ for(2)$q^{h}e_{i}q^{-h}=q^{\langle h,\alpha_{i}\rangle}e_{i},$$h,$ $h’\in p\vee$,$q^{h}f_{i}q^{-h}=q^{-(h,\alpha_{i}\rangle}f_{i}$.

(3)$e_{i}f_{j}-f_{j}e_{i}= \delta_{ij}\frac{K_{i}-K_{i}^{-1}}{q_{i}-q_{i}^{-1}}$(4)$\sum_{k=0}^{1-a_{i_{J}}}\{\begin{array}{ll}1- a_{ij}k \end{array}\}e_{i}^{1-a_{ij}-k}e_{j}e_{i}^{k}=0$(5)$\sum_{k=0}^{1-a_{ij}}\{\begin{array}{ll}l- a_{ij}k \end{array}\}f_{i}^{1-a_{ij}-k}f_{j}f_{i}^{k}=0$

where

for $h\in P^{\vee}.i\in I$ ,$K_{i}=q_{i}^{h_{i}}$,if$i\in I^{re}$and $i\neq j$ ,if$i\in I^{re}$and $i\neq j$ ,(6) $e_{i}e_{j}-e_{j}e_{i}=0,$ $f_{i}f_{j}-f_{j}f_{i}=0$ if $a_{ij}=0$ .

Note that, if all diagonal entries of A are 2, then A is a generalized Cartan matrix andis the usual quantum group associated with A.
(resp.
Let) be the subalgebra of(resp.
generated by the elements$i\inI$. For $n\in Z>0$ , setfor$U_{q}(\mathfrak{g})$$U_{q}^{+}(\mathfrak{g})$$f_{i})$$U_{q}(\mathfrak{g})$$U_{q}^{-}(\mathfrak{g})$$e_{i}^{(n)}=\{\begin{array}{ll}\frac{e_{i}^{n}}{[n]_{i}!} if i\in I^{re}.e_{i}^{n} if i\in I^{im},\end{array}$

For an element(1.1)$u\in U_{q}^{-}(\mathfrak{g}),$$u$$f_{i}^{(n)}=\{\begin{array}{ll}\frac{f_{i}^{n}}{[n]_{i}!} if i\in I^{re},f_{i}^{n} if i\in I^{im}.\end{array}$can be expressed uniquely as$u= \sum_{k\geq 0}f_{i}^{(k)}u_{k}$,$e_{i}$

166
EUIYONG PARK

and $u_{k}=0$ for $k\gg 0$ . Here, is the endomorphism
wherein [3, Section 6] and [9, Section 3.3]. The Kashiwam opemtors
defined by.
.
$u_{k}\in kere_{i}’$$U_{q}^{-}(\mathfrak{g})arrow U_{q}^{-}(\mathfrak{g})$$e_{i}’$$\tilde{e}_{i},\tilde{f_{i}}(i\in I)$

of

given

$U_{q}^{-}(\mathfrak{g})$are$\tilde{f_{i}}u=\sum_{k\geq 0}f_{i}^{(k+1)}uk$$\tilde{e}_{i}u=\sum_{k\geq 1}f_{i}^{(k-1)}u_{k}$Let $A_{0}=\{f/g\in \mathbb{Q}(q)|f,g\in \mathbb{Q}[q],g(O)\neq 0\}$ .
is a pair $(L, B)$ satisfying the following conditions:Definition 1.2. A crystal basis of,andsuch that(1)
is a free -module of,where$:=B\cap(L_{-\alpha}/qL_{-\alpha})$ ,, where(2)
is a Q-basis of $L/qL$ such thatfor all $i\in I$ ,(3).
if and only if(4) For $b’\in B$ and $i\in I,$$U_{q}^{-}(g)$$L$$A_{0}$$L_{-\alpha}$$L=\oplus_{\alpha\in Q+}L_{-\alpha}$$U_{q}^{-}(\mathfrak{g})=\mathbb{Q}(q)\otimes_{A_{0}}L$$U_{q}^{-}(\mathfrak{g})$$:=L\cap U_{q}^{-}(\mathfrak{g})_{-\alpha}$$B$$B_{-\alpha}$$B=\lfloor\rfloor_{\alpha\in Q+}B_{-\alpha}$$\tilde{e}_{i}B\subset Bu\{0\},\tilde{f_{i}}B\subset B$$b’=\tilde{f_{i}}b$

$b,$

Let

$L(\infty)$

be the free$\mathbb{A}_{0}$-module of$b=\tilde{e}_{i}b’$generated by$U_{q}^{-}(\mathfrak{g})$$\{\tilde{f_{i_{1}}}\cdots\tilde{f_{i_{r}}}1|r\geq 0, i_{k}\in I\}$and let.

$B(\infty)=\{\tilde{f_{i_{1}}}\cdots\tilde{f_{i_{r}}}1+qL(\infty)|r\geq 0, i_{k}\in I\}\backslash \{0\}$.
Then, it is proved in [3, 9] that the pair $(L(\infty), B(\infty))$ is a unique crystal basis of
$i\in
I$ ,
$M$
defined in [3. Definition 3.1]. For anyLetbe a-module in the category
any element $u\in M_{\mu}$ can be expressed uniquely as$U_{q}^{-}(\mathfrak{g})$$\mathcal{O}_{int}$$U_{q}(\mathfrak{g})$$u= \sum_{k\geq 0}f_{i}^{(k)}u_{k}$where$u_{k}\in M_{\mu+k\alpha_{i}}\cap kere_{i}$,. The Kashiwara opemtors$\tilde{e}_{i}u=\sum_{k\geq l}f_{i}^{(k-1)}u_{k}$Definition 1.3. A crystal basis of$U_{q}(\mathfrak{g})$.

$\tilde{e}_{i},\tilde{f}_{i}(i\in I)$$\tilde{f_{i}}u=\sum_{k\geq 0}f_{i}^{(k+1)}u_{k}$are defined by.

-module $M$ is a pair $(L. B)$ satisfying the following

conditions:
(1)

$L$

is a free$\mathbb{A}_{\Phi}$-module of $M$ such that,
(2) $B$ is (Q-basis of $L/qL$ such that

and

$M=\mathbb{Q}(q)\otimes_{A_{0}}L$$L=\oplus_{\lambda\in P}L_{\lambda}$, where$L_{\lambda}:=L\cap M_{\lambda}$(3)(4) For$\tilde{e}_{i}B\subset Bu\{0\},\tilde{f_{i}}B\subset BU\{0\}$$b,$$b’\in B$

and$i\in I,$$b’=\tilde{f_{i}}b$$B=u_{\lambda\in P}B_{\lambda}$,
for all
if and only if, where$B_{\lambda}$$:=B\cap L_{\lambda}/qL_{\lambda}$,$i\in I$

$b=\tilde{e}_{i}b’$.

, we denote by
the irreducible highest weight
For a dominant integral weight. Let
be theis contained in-module with highest weight . Note thatand letgenerated byfree -module of$\lambda\in p+$$\lambda$$U_{q}(\mathfrak{g})$$A_{0}$$V(\lambda)$$V(\lambda)$$V(\lambda)$$\mathcal{O}_{int}$$\{\tilde{f_{i_{1}}}\cdots\tilde{f_{i_{r}}}v_{\lambda}|r\geq 0, i_{k}\in I\}$.

$B(\lambda)=\{\tilde{f_{i_{1}}}\cdots\tilde{f_{i_{r}}}v_{\lambda}+qL(\lambda)|r\geq 0, i_{k}\in I\}\backslash \{0\}$$L(\lambda)$

167
KLR ALGEBRAS AND CRYSTAL BASES FOR FINITE CLASSICAL TYPE

Then the pair$(L(\lambda), B(\lambda))$is a unique crystal basis of $V(\lambda)[3,9]$ .

2. KHOVANOV-LAUDA-ROUQUIER ALGEBRAS
Let

be a commutative graded ring such thatwith $|\alpha|=m$ , let$k=\oplus_{n\in \mathbb{Z}}k_{n}$$n<0$ . For$\alpha\in Q^{+}$$k_{0}$

is a field and $k_{n}=0$ for.

$I^{\alpha}=\{\nu=(\nu_{1}\ldots., \nu_{m})\in I^{m}|\alpha_{\nu_{1}}+\cdots+\alpha_{\nu_{m}}=\alpha\}$Note that the symmetric group $S_{m}=\langle s_{k}|k=1,$ $\ldots m-1\}$ acts naturally on$t=1,$$m-1$ , we define the operator
on $k[x_{1}, \ldots, x_{m}]$ by$I^{\alpha}$. For$\partial_{t}$$\ldots,$$\partial_{t}(f)=\frac{s_{t}f-f}{x_{t}-x_{t+1}}$,

where $wf(x_{1,}\ldots. , x_{m})=f(x_{w(1)}, \ldots, x_{w(m)})$ for $w\in S_{m}$ and $f(x_{1}, \ldots , x_{m})\in k[x_{1}, \ldots.x_{m}]$ .
For each $i\in I$ , we choose a polynomialof the form$\mathcal{P}_{i}(u, v)\in k[u.v]$$\mathcal{P}_{i}(u, v)=\sum_{k,l\geq 0}p_{i;k,l}u^{k}v^{l}$where $p_{i;k,l}\in k_{d_{i}(2-a)-2d_{i}k-2d_{i}l}ii$ andin $k[u, v]$ such that

$p_{i10}-,’ p_{i;0,1-};_{2^{1}2}\in k_{0}^{\cross}$$\mathcal{Q}_{ij}(u, v)=\mathcal{Q}_{ji}(v, u)$$(\mathcal{Q}_{ij}(u, v))_{i,j\in I}$

,

and

.

We also take a matrix
has the form$\mathcal{Q}_{ij}(u, v)$$\mathcal{Q}_{ij}(u, v)=\{\begin{array}{ll}0 if i=j,\sum_{k,l\geq 0}q_{i,j;k,l}u^{k}v^{l} if i\neq j,\end{array}$where$q_{i,j;k,l}\in k_{-2(\alpha_{i}|\alpha_{j})-2d_{i}k-2d_{j}l}$and$q_{i,j;-a_{ij}.0}\in k_{0}^{\cross}$.

Definition 2.1. Letwith height $m$ .
be a Borcherds-Cartan datum and,
The Khovanov-Lauda-Rouquier algebm
of weight associated with the dataandis the associative graded k-algebra generated by $e(\nu)(\nu=$$x_{k}(1\leq k\leq m),$ $\tau_{t}(1\leq t\leq m-1)$ satisfying the following defining relations:$\alpha\in Q+$$(A, P, \Pi, \Pi^{\vee})$$R(\alpha)$$(\mathcal{P}_{i})_{i\in I}$$(A, P, \Pi, \Pi^{\vee})$$\alpha$$(\nu_{1}, \ldots , \nu_{m})\in$$(\mathcal{Q}_{ij})_{i,j\in I}$$I^{\alpha}),$$e( \nu)e(\nu’)=\delta_{\nu,\nu’}e(\nu),\sum_{\nu\in I^{\alpha}}e(\nu)=1,$$\tau_{t}e(\nu)=e(s_{t}(\nu))\tau_{t},$$\tau_{t}\tau_{s}=\tau_{s}\tau_{t}$$x_{k}e(\nu)=e(\nu)xk,$$x_{k}x_{l}=x_{l}x_{k}$,if $|t-s|>1$ ,$\tau_{t}^{2}e(\nu)=\{\begin{array}{ll}\partial_{t}\mathcal{P}_{\nu_{t}}(x_{t}, x_{t+1})\tau_{t}e(\nu) if \nu_{t}=\nu_{t+1},\mathcal{Q}_{\nu_{t}\nu_{t+1}}(x_{t}, x_{t+1})e(\nu) if \nu_{t}\neq\nu_{t+1},\end{array}$$(\tau_{t}x_{k}-x_{s\iota(k)}\tau_{t})e(\nu)=\{\begin{array}{ll}-\mathcal{P}_{\nu_{t}}(x_{t}, x_{t+1})e(\nu) if k=t and \nu_{t}=\nu_{t+1},\mathcal{P}_{\nu_{l}}(x_{t}, x_{t+1})e(\nu) if k=t+1 and \nu_{t}=\nu_{t+1},0 otherwise,\end{array}$

168
EUIYONG PARK$(\tau_{t+1}\tau_{t}\tau_{t+1}-\tau_{t}\tau_{t+1}\tau_{t})e(\nu)$$=\{\begin{array}{ll}\mathcal{P}_{\nu_{t}}(x_{t}, x_{t+2})\overline{\mathcal{Q}}_{\nu_{t},\nu_{t+1}}(x_{t}, x_{t+1}, x_{t+2})e(\nu) if \nu_{t}=\nu_{t+2}\neq v_{t+1},\overline{\mathcal{P}}_{\nu_{t}}’(x_{t}, x_{t+1}, x_{t+2})\tau_{t}e(\nu)+\overline{\mathcal{P}}_{\nu_{t}}’’(x_{t}, x_{t+1}, x_{t+2})\tau_{t+1}e(\nu) if\nu_{t}=\nu_{t+1}=\nu_{t+2},0 otherwise.\end{array}$where$\overline{\mathcal{P}}_{i}’(u.v, w):=\frac{\mathcal{P}_{i}(v,u)\mathcal{P}_{i}(u,w)}{(u-v)(u-w)}+\frac{\mathcal{P}_{i}(u,w)P_{i}(v,w)}{(u-w)(v-w)}-\frac{\mathcal{P}_{i}(u,v)\mathcal{P}_{i}(v,w)}{(u-v)(v-w)}$,$\overline{\mathcal{P}}_{i}’’(u, v, w):=-\frac{\mathcal{P}_{i}(u.v)\mathcal{P}_{i}(u,w)}{(u-v)(u-w)}-\frac{\mathcal{P}_{i}(u.w)\mathcal{P}_{i}(w,v)}{(u-w)(v-w)}+\frac{\mathcal{P}_{i}(u,v)P_{i}(v,w)}{(u-v)(v-w)}$$\overline{\mathcal{Q}}_{i,j}(u.v, w):=\frac{\mathcal{Q}_{i,j}(u,v)-Q_{i,j}(w,v)}{u-w}$

The algebra$R(\alpha)$,.

has the Z-grading given by$\deg(e(\nu))=0$ ,$\deg(x_{k}(\nu))=2d_{\nu_{k}}$,$\deg(\tau_{t}(\nu))=-(\alpha_{\nu_{t}}|\alpha_{\nu_{t+1}})$,. A diagrammatic presentation offorwhere $x_{k}(\nu)=x_{k}e(\nu)$ andusing planar diagrams with dots and strands is given in [7, 11, 12].
of the form
For
and $i\in I$ , let us fix a polynomial$\nu\in I^{\alpha}$$\tau_{t}(\nu)=\tau_{t}e(\nu)$$\lambda\in p+$$R(\alpha)$$a_{i}^{\lambda}(u)$$a_{i}^{\lambda}(u)= \sum_{k=0}^{\lambda(h_{i})}c_{i;k}^{\lambda}u^{\lambda(h_{i})-k}$

where
and
Lauda-Rouquier algebm$c_{i;k}^{\lambda}\in k_{2d_{:}k}$$c_{i;0}^{\lambda}=1$. Set$R^{\lambda}(\alpha)$,$a^{\lambda}(x)= \sum_{\nu\in I}a_{\nu_{1}}^{\lambda}(x)e(\nu)$. Then the cyclotomic Khovanov-is defined to be the quotient algebra$R^{\lambda}(\alpha)=R(\alpha)/R(\alpha)a^{\lambda}(x)R(\alpha)$.

-mod) be the category of finite-dimensional graded left
Let $R(\alpha)-mod$ (resp.
-modules). For a Z-graded module $M=$
modules (resp. finite-dimensional graded left
$t\in
Z$
, let
be the Z-graded module obtained from $M$ by
and$:=M_{t+k}$ . The q-chamcter $ch_{q}(M)$ and chamcter ch $(M)$ of $M$ are defined by
setting$R(\alpha)-$$R^{\lambda}(\alpha)$$R^{\lambda}(\alpha)$$\oplus_{k\in Z}M_{k}$$M\langle t\rangle=\oplus_{k\in Z}M\langle t\}_{k}$$M\langle t\rangle_{k}$$ch_{q}(M)=\sum_{\nu\in I^{\alpha}}\dim_{q}(e(\nu)M)\nu$,

ch $(M)= \sum_{\nu\in I^{a}}\dim(e(\nu)M)\nu$,where $\dim_{q}(N)$ $:= \sum_{i\in Z}(\dim N_{i})q^{i}$ for any graded module $N=\oplus_{i\in Z}N_{i}$ .
For $M,$ $N\in R(\alpha)$ -mod, let $Hom(M, N)$ be the set of homogeneous homomorphisms of, we define
degree , and let HOM $(M, N)=\oplus_{k\in Z}Hom(M, N\langle k\rangle)$ . For$0$$\beta_{1},$$\ldots,$,$e( \beta_{1}, \ldots, \beta_{m})=\sum_{\nu_{j}\in I^{\beta_{j}}}e(\nu_{1}*\cdots*\nu_{m})$$\beta_{m}\in Q^{+}$169
KLR ALGEBRAS AND CRYSTAL BASES FOR FINITE CLASSICAL TYPE

where$\nu_{1}*\cdots*\nu_{m}$$R(\beta_{1}+\cdots+\beta_{m})$is the concatenation ofgives the functors$\nu_{k}’ s$. The natural embedding$Ind_{\beta_{1},\ldots,\beta_{m}}:R(\beta_{1})\otimes\cdots\otimes R(\beta_{m})-$${\rm Res}_{\beta_{1},\ldots,\beta_{m}}:R(\beta_{1}+\cdots+\beta_{m})-$$R(\beta_{1})\otimes\cdots\otimes R(\beta_{m})\epsilonarrow$

mod $arrow R(\beta_{1}+\cdots+\beta_{m})$ -mod,

mod

-mod

$arrow R(\beta_{1})\otimes\cdots\otimes R(\beta_{m})$$:=e(\beta_{1}, \ldots, \beta_{m})M$
defined byandfor-mod and $M\in R(\beta_{1}+\cdots+\beta_{m})-mod$ .
(resp.
Let) be the set of all isomorphism classes of irreducible graded
modules (resp.
-modules) for all. Set 1 to be the l-dimensional trivial $R(O)-$$M\in
R(\beta)$-mod,module. For
we define$Ind_{\beta_{1},\ldots,\beta_{m}}L$$:=R(\beta_{1}+\cdots+\beta_{m})\otimes_{R(\beta_{1})\otimes\cdots\otimes R(\beta_{m})}L$${\rm Res}_{\beta_{1},\ldots,\beta_{m}}M$$L\in R(\beta_{1})\otimes\cdots\otimes R(\beta_{m})$$\mathfrak{B}(\infty)$$R(\beta)-$$\mathfrak{B}(\lambda)$$\beta\in Q^{+}$$R^{\lambda}(\beta)$$\tilde{e}_{i}(M)=$soc$\tilde{f_{i}}(M)=$hd$({\rm Res}_{\alpha_{i},\beta-\alpha_{i}}M)\in R(\beta-\alpha_{i})$-mod$Ind_{\alpha_{i},\beta}(L(i)\otimes M)\in R(\beta+\alpha_{i})$-mod,$wt(M)=\{\begin{array}{ll}-\beta if M\in R(\beta)- mod,\lambda-\beta if M\in R^{\lambda}(\beta)- mod,\end{array}$$\epsilon_{i}(M)=\max\{k\geq 0|\tilde{e}_{i}^{k}M\neq 0\}$,$\varphi_{i}(M)=\epsilon_{i}(M)+$wt $(M)(h_{i})$ .

where $L(i)$ is the l-dimensional-module with $\dim_{q}L(i)=1$ . Here, for an-module$N$ , soc $(N)$ (resp.
$(N)$) is the maximal completely reducible submodule (resp. the maximal
hd
completely reducible quotient) of $N$ . Then, when
, the sets
for allandwith the above maps have crystal structures [7, 14].
$R(\alpha)$$R(\alpha_{i})$$\iota\in I$$a_{ii}\neq 0$$\mathfrak{B}(\infty)$$\mathfrak{B}(\lambda)$

Theorem 2.2 ([7, 14]). If
for all $i\in I$ , then the crystal
is isomorphic to the crystal $B(\infty)$ (resp.
$a_{ii}\neq 0$$(\mathfrak{B}(\lambda).

$(\mathfrak{B}(\infty), wt, \tilde{e}_{i},\tilde{f_{i}}, \epsilon_{i}, \varphi_{i})$wt, \tilde{e}_{i},\tilde{f_{i}}, \epsilon_{i}, \varphi_{i}))$$B(\lambda)$(resp.

).

We now assume that $a_{ii}\neq 0$ for all $i\in I$ . Let $n=|I|$ be the rank of, and let$(k=1, \ldots , n)$ be subsets of $I=I_{(n+1)}$ such thatand $|I_{(k)}|=k$ for all .
$f_{i}(i\in I_{(k)})$ and $q^{h}(h\in P^{\vee})$ and letLetdenote the subalgebra ofgenerated by
be the crystal obtained from. Thenby forgetting the i-arrows forcanbe understood as a -crystal and every connected component of
has a unique highest
weight vector [1, Lemma 1.9].
Take an element $v\in B(\infty)$ . Let $u_{0}=v$ and let
be the highest weight vector of theconnected component, . By construction, there is aofcontaining for $k=1,$
chain of injective maps
$U_{q}(\mathfrak{g})$$I_{(k)}\subset I$$U_{q}(\mathfrak{g})$$e_{i},$$B(\infty)$$\mathcal{B}_{k}$$i\not\in I_{(k)}$$U_{k}$$\mathcal{B}_{k}$$u_{k}$$C_{k}$$\mathcal{B}_{k}$$v$$\ldots$$n$$C_{1}\mapsto C_{2}arrow\cdots=\rangle C_{n-1}arrow B(\infty)$

For $k=1,$
(2.1)

$k$

$I_{(k)}\subset I_{(k+1)}$$U_{k}$$\ldots$

,

$n$

, let
$\mathcal{N}_{k}(v)=\tilde{f}_{\nu_{k,1}}\cdots\tilde{f}_{\nu_{k,t_{k}}}1\in \mathfrak{B}$

(oo),

.

$\mathcal{B}_{k}$170
EUIYONG PARK.
is a sequence of such that
Hence, for each $v\in B(\infty)$ , we obtain the corresponding n-tuple. Then, using the same argument as in [1, Proposition 1.10],
of irreducible modules in
one can obtain the following proposition.
where$\nu_{k}=$$(\nu_{k,1}\ldots.

$I$, \nu_{k,t_{k}})$$u_{k-1}=\tilde{f}_{\nu_{k,1}}\cdots\tilde{f}_{\nu_{k.t_{k}}}u_{k}$$(\mathcal{N}_{1}(v),\mathcal{N}_{2}(v)\ldots.,\mathcal{N}_{n}(v))$$\mathfrak{B}(\infty)$

Proposition 2.3. [1, Proposition 1.10](1) For $v\in B(\infty)$ , hd
(2) The map :$B(\infty)arrow \mathfrak{B}(\infty)$$\Phi$

is irreducible.
defined by$Ind(H_{k=1}^{n}\mathcal{N}_{k}(v))$$\Phi(v)=$hd$Ind(\mathbb{E}_{k=1}^{n}\mathcal{N}_{k}(v))$

for$v\in B(\infty)$

is a crystal isomo rphism.

3. CRYSTAL BASES AND IRREDUCIBLE REPRESENTATIONSIn this section, we give a crystal basis theoretic construction of irreducible modules over
Khovanov-Lauda-Rouquier algebras and their cyclotomic quotients for finite classical types.
Though this section is based on our previous work [1], the description of irreducible modules inthis section is different and more combinatorial than the description given in [1]. Throughout
this section, we assume thatand A is a generalized Cartan matrix of finite classical
type
.
and
Let $I=\{1,2, \ldots , n\}$ and let be the following Dynkin diagram:$k=\mathbb{C}$$A_{n},$$B_{n},$$D_{n}$$C_{n}$$\Gamma$$(A_{n})$$(B_{n})$$(C_{n})$$\overline{12}$.....

–......

$\overline{n-1n}$$\overline{12}\overline{n-1n}-\cdots\cdots\cdots$$\overline{12}$.....

$-\overline{n-1n}$$(D_{n})$We set$B_{X}$$(X=A, B. C, D)$ to be the crystal defined by$\overline{1}\underline{1}\overline{2}\underline{2}$$(A_{n})$$(B_{n})$$(C_{n})$$\overline{1}\underline{1}$$\overline{1}\underline{1}$

...

...

$\underline{n-1}\overline{n}\underline{n}\overline{n+1}$. $..arrow^{n-1}$ fi $\underline{n}0\underline{n}n\underline{n-1}$ .. . $\underline{1}1$$\underline{n-2}\overline{n-1}\underline{n-1\underline{n}}\underline{\prod\overline{n}}n\underline{n-1}n-1\underline{n-2}\ldots\underline{1}1$

171
KLR ALGEBRAS AND CRYSTAL BASES FOR FINITE CLASSICAL TYPE
$n$$\nearrow^{n}$$\overline{1}\underline{1}$$(D_{n})$...

$\backslash ^{n-1}$$\underline{n-2}\overline{n-1}$$n-1\underline{n-2}\ldots\underline{1}\underline{\prod 1}$$\nearrow^{n}$$n\backslash _{-1}$$\overline{n}$with the entries ordered by,$(A_{n})$$\overline{1}\succ\overline{2}\succ\cdots\succ\overline{n+1}$$(B_{n})$$\overline{1}\succ\overline{2}\succ\cdots\succ\overline{n}\succ 0\succ n\succ\cdots\succ 2\succ 1$$(C_{n})$$\overline{1}\succ\overline{2}\succ\cdots\succ\overline{n-1}\succ\overline{n}\succ n\succ n-1\succ\cdots\succ 1$$(D_{n})$$\overline{1}\succ\overline{2}\succ\cdots\succ\overline{n-1}\succ\overline{n},$For $\nu=$.

$n\succ n-1\succ\cdots\succ 1$,.

(resp.
and $k=(k_{1}\ldots., k_{m})\in(Z_{\geq 0})^{m}$ , let. If $k=(1, \ldots, 1)$ , then we write(resp. ).
(resp. ) for
A segment of the crystal
is a subset $\epsilon=\{a.b\}\subset B_{X}$ such that the two elements
and of are distinct and comparable. Note that any distinct two elements $\{a, b\}$ ofexcept the case $\{a, b\}=\{n, \overline{n}\}(X=D)$ can be viewed as a segment of. For a segment$s=\{a, b\}$ with $a\succ b$ , set $h(s)=a$ and $t(s)=b$ , respectively. We define a partial orderonthe set of segments of,as follows: for two segments and of$(\nu_{1}, \ldots , \nu_{m})\in I^{m}$$\tilde{f}_{\nu}^{k}=\tilde{f}_{\nu_{1}}^{k_{1}}\cdots\tilde{f}_{\nu_{m}^{m}}^{k}$$\tilde{f}_{\nu}$$\tilde{e}_{\nu}^{k}=\tilde{e}_{\nu_{1}}^{k_{1}}\cdots\tilde{e}_{\nu_{m}}^{k_{m}})$$\tilde{e}_{\nu}$$\tilde{e}_{\nu}^{k}$$B_{X}$$\epsilon$$b$$\tilde{f}_{\nu}^{k}$$a$$B_{X}$$\epsilon$$B_{X}$$\succeq$$B_{X}$$\epsilon\succeq\epsilon^{l}$$\epsilon’$$\epsilon$

if and only if$(h(s)\succ h(z^{l}))$$B_{X}$or $(h(z)=h(s’)$ and$(t(s’)\succ t(\epsilon))$

For $b\in B_{X}$ and $X=B$ , C. $D$ , we set
to be a unique element insuch that $wt(b^{\vee})=$$-$ wt $(b)$ .
Let $b_{X}=\overline{n}(X=A, B, C),$ $b_{X}=\overline{n-1}(X=D)$ . A multisegment ofis amultiset of segments of the crystal
such that$b^{\vee}$$B_{X}$$\underline{m}$$B_{X}$$B_{X}$(i) $h(s)\succeq b_{X}$ for

(ii)

(3.1)

$t(\epsilon)\succeq h(\epsilon)^{\vee}$$\epsilon\in\underline{m}$,if $X=B,$ $C,$ $D$ ,(iii) any two segments in$\underline{m}$are comparable.

When no confusion can arise, we write$\underline{m}=\{z_{1}\preceq z_{2}\preceq 53\preceq\cdots\}$$\underline{m}(i)=\{z\in\underline{m}|\tilde{e}_{i}(h(\epsilon))\neq 0\}$. For $i=1,$$\ldots$, $n$ , we set.

Note that.
Let
be the set of all multisegments ofparameterizes the. We will show thatcrystal (oo). Let $p_{A}=n,$ $\ell_{B}=\ell_{C}=2n-1,$ $\ell_{D}=2n-2$ and let
be the sequence of the$\underline{m}=\bigcup_{i\in I}\underline{m}(i)$$\mathcal{M}_{X}$$B$$B_{X}$$\mathcal{M}x$$\nu_{X}$172
EUIYONG PARK

colors over the arrows of$B_{X}$, i.e.,$\nu_{A}=(1.2, \ldots, n-1.n)$ ,$v_{B}=\nu_{C}=(1,2, \ldots, n-1, n, n-1, \ldots, 2,1)$ ,$\nu_{D}=$For a segment , let$\epsilon$$(1. 2, \ldots , n-1.n,n-2, \ldots , 2.

$e(5)=(e_{1}, \ldots, e_{\ell_{X}})\in Z^{\ell_{X}}$$h(\epsilon)=\tilde{f}_{\nu_{X}}^{e(\epsilon)}t(\epsilon)$

Here we consider$Z^{\ell_{X}}$,be the$e_{j}\neq 0$

as an abelian group. Let$p_{X}$

1)$.

-tuple of$Z$

defined byfor some $1\leq j\leq n$ .
$(0, \ldots , 0.1, 0ith\downarrow, \ldots , 0)$$e_{i}=$

for $i=1,$$\ldots,$$\ell_{X}$.

Proposition 3.1. Let $T:M_{X}arrow B(\infty)$ be the map defined by$\prime r(\underline{m})=\tilde{f}_{\nu_{X}}^{e_{1}(\underline{m})}\ldots\tilde{f}_{\nu_{X}}^{e_{n}(\underline{m})}1$

where

,

$q(\underline{m})=\{\begin{array}{ll}\sum_{s\in\underline{\mathfrak{m}}(n+1-i)}e(\mathfrak{s}) if either X=A, B, C or i\neq 1,2(X=D),\#\underline{m}(n-1)e_{n-1} if i=2(X=D),\#\underline{m}(n)e_{n} if i=1(X=D).\end{array}$

Then the map$\prime r$

is bijective.

Proof. We focus on the case $X=D$ since the remaining cases can be proved in a similarmanner. Let$S$

be the set of $t=$$(t_{1}, \ldots , t_{n})\in Z_{\geq 0}^{n(2n-2)}$

such that(a) $t_{j}=(t_{j,1}, \ldots.t_{j,2n-2})\in Z_{\geq 0}^{(2n-2)}$ for $j=1,$,(b) $t_{j,n+1-j}\geq t_{j,n+2-j}\geq\cdots\geq t_{j,n-1},$ $t_{j,n}\geq t_{j,n+1}\geq\cdots\geq t_{j,n-2+j}$ for $j=3,$ $\ldots.n$ ,, then $t_{j,k}=0$ for either $k<n+1-j$ or $k>n-2+j$ ,(c) if $j=3,$(d) $t_{2,k}=0$ for $k\neq n-1$ and $t_{1,k}=0$ for $k\neq n$ .
$\ldots,$$\ldots,$

$n$

$n$

It follows from [15, Section 7] that the map $Sarrow B(\infty)$ mapping $t=(t_{1}, \ldots , t_{n})\in S$ tois bijective.
. By (ii) and (iii) of (3.1),
Let
for $j=1,$
and write
satisfy the above condition (b), (c) and (d). Hence the map $\phi:M_{X}arrow S$
the sequences
given by$\tilde{f}_{\nu_{D}^{1}}^{t}\ldots\tilde{f}_{\nu_{D}}^{t_{\mathfrak{n}}}1\in B(\infty)$$\underline{m}\in M_{X}$$e_{j}(\underline{m})=(e_{j,1}, \ldots, e_{j,2n-2})$$\ldots,$$e_{j}(\underline{m})$$\phi(\underline{m})=(e_{1}(\underline{m}), \ldots , e_{n}(\underline{m}))$$(\underline{m}\in M_{X})$is well-defined.
On the other hand, let
へ$i=\{\begin{array}{ll}\overline{i} if 1\leq i\leq n,2n+1-i if n+1\leq i\leq 2n.\end{array}$

$n$

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KLR ALGEBRAS AND CRYSTAL BASES FOR FINITE CLASSICAL TYPE

and, for a segmentof in . Let :such that, if $i=1,$$s$$\psi$$\underline{m}$$\mathfrak{s}$, let us denote by $\sigma(m,s)$ the multiplicity
and a multisegment
be the map mapping $t\in S$ to the multisegment $\psi(t)\in Mx$$n-2$ , then$\underline{m}\in \mathcal{M}_{X}$$Sarrow \mathcal{M}_{X}$$\ldots,$$\sigma(\psi(t), \{\hat{i},\overline{j+1}\})=\{\begin{array}{ll}t_{n+1-i,j}-t_{n+1-\iota,j+1} if j\leq n-3,t_{n+1-i,n-2}-\max\{t_{n+1-i,n-1}, t_{n+1-i,n}\} if j=n-2,ma \cross\{0, t_{n+1-i,n-1}-t_{n+1-i.n}\} if j=n-1,\max\{0, t_{n+1-i,n}-t_{n+1-i,n-1}\} if j=n,m|n\{t_{n+1-i,n-1}, t_{n+1-i,n}\}-t_{n+1-i,n+1} if j=n+1,t_{n+1-i,j-1}-t_{n+1-i,j} if j\geq n+2,\end{array}$and, if $i=n-1$ , then$\sigma(\psi(t).

\{\overline{n-1}.\overline{j+1}\})=\{\begin{array}{ll}\max\{0, t_{2,n-1}-t_{1,n}\} if j=n-1,\max\{0, t_{1.n}-t_{2,n-1}\} if j=n,\min\{t_{1,n}.t_{2,n-1}\} if j=n+1.\end{array}$

Then it is straightforward to verify that $\phi 0\psi=id_{S}$ and$\psi 0\phi=id_{\Lambda t_{X}}$.

$\square$

We now retum to the Khovanov-Lauda-Rouquier algebras. Let 1 be the trivial $R(O)-$module. For a segment . we define$\epsilon$$\nabla(s)=\tilde{f}_{\nu_{X}}^{e(\epsilon)}1\in \mathfrak{B}(\infty)$.

We give a explicit description of the module structure of
as follows.
$(A_{n}, C_{n})$ , either$h(s)\succ t(s)$$l=$$(h(s))-$
followingholds:
wt
If
one
of
the
Let
wt
.
$(D_{n})$ , then the module$t(s)\succeq O$ or $0\succeq h(s)$, either $t(s)\succ n-1$ oris the l-dimensional module
given by$\nabla(\epsilon)$$(t(\epsilon))|$$(B_{n})$$\overline{n-1}\succ h(\epsilon)$$\nabla(\epsilon)$$\mathbb{C}v$$x_{i}v=0$ ,wheresuch that$h(s)\succ
0\succt(s)$ for typeIf$\nu(\epsilon)\in I^{l}$$x_{i}u=0$ ,$\tau_{j}v=0$,$h(\epsilon)=\tilde{f}_{\nu(\mathfrak{s})}t(z)$$B_{n}$, then$e(\nu)v=\{\begin{array}{ll}v if \nu=\nu(\epsilon),0 otherwise,\end{array}$.

$\nabla(\epsilon)$is the 2-dimensional module

where $v(5)\in I^{l}$ such that$h(\epsilon)=\tilde{f}_{\nu(s)}t(\epsilon)$$\tau_{j}v=0$

, and

with

$e(\nu)u=\{\begin{array}{ll}u if \nu=\nu(\epsilon),0 otherwise,\end{array}$$\tau_{j}u=\{\begin{array}{ll}v if j=d,0 otherwise,\end{array}$$x_{i}v=\{\begin{array}{ll}u if i=d,-u if i=d+1,0 otherwise\end{array}$$\mathbb{C}u\oplus \mathbb{C}v$$d$,$e(\nu)u=\{\begin{array}{ll}u if \nu=\nu(\epsilon),0 otherwise,\end{array}$is an integer such that$s_{d}(\nu(\epsilon))=\nu(\mathfrak{s})$.

174
EUIYONG PARK

If
module$h(5)\succeq\overline{n-1}$$\mathbb{C}u\oplus \mathbb{C}?f$and $n-1\succeq t(5)$ for typedefined by$D_{n}$, then the module$\nabla(\epsilon)$is the 2-dimensional$x_{i}u=0$ ,$\tau_{j}u=\{\begin{array}{ll}v if j=d,0 otherwise,\end{array}$$e(\nu)u=\{\begin{array}{ll}\mathcal{Q}_{n-1,n}(x_{n-1}, x_{n})u if \nu=\nu(\epsilon)^{+},0 otherwise,\end{array}$$x_{i}v=0$ ,$\tau_{j}v=\{\begin{array}{ll}u if j=d,0 otherwise,\end{array}$$e(\nu)v=\{\begin{array}{ll}v if\nu=\nu(\epsilon)^{-}0 otherwise,\end{array}$where
such thatis an integer such that
description above, the character ch$\nu(\epsilon)^{-}\in I^{l}$$\nu(\epsilon)^{+},$

$d$

and
Note thatis given as follows$\nu(5)^{+}\neq\nu(\mathfrak{s})^{-}$$h(\epsilon)=\tilde{f}_{\nu(s)}+t(\epsilon)=\tilde{f}_{\nu(\epsilon)^{-}}t(\mathfrak{s})$$s_{d}(\nu(\epsilon)^{+})=\nu(\epsilon)^{-}$$\nabla(\epsilon)$

(3.2)

ch

$\nabla(\epsilon)=\{\begin{array}{ll}\nu(5)^{+}+\nu(5)^{-} if h(\mathfrak{s})\succeq\overline{n-1}.

For a multisegment$\mathcal{Q}_{n-1,n}(x_{n-1}, x_{n})\in \mathbb{C}^{*}$, and. For then-1\succeq t(\epsilon)(D_{n}),2v(s) if h(s)\succ O\succ t(s)(B_{n}),\nu(\mathfrak{s}) otherwise.\end{array}$$\underline{m}=\{\epsilon_{1}\preceq\epsilon_{2}\preceq 53\preceq\cdots\}$. we define$\nabla(\underline{m})=\nabla(\epsilon_{1})\otimes\nabla(\mathfrak{s}_{2})\otimes\nabla(\epsilon_{3})\otimes\cdots$.

Then we have the following theorem.
Theorem 3.2. [1, Theorem 3.2]
(1) For a multisegment$\underline{m}$, we have

hdInd
where
(2) Let$\Psi$$\nabla(\underline{m})=\tilde{f}_{\nu_{X}}^{e_{1}(\underline{m})}\ldots\tilde{f}_{\nu_{X}}^{e_{n}(\underline{m})}1$,$e_{i}(\underline{m})=\{\begin{array}{ll}\sum_{5\in\underline{m}(n+1-i)}e(5) if either X=A.B, C or i\neq 1,2(X=D),\#\underline{m}(n-1)\iota_{n-1} if i=2(X=D),\#m(n)e_{n} if i=1(X=D).\end{array}$:$\mathcal{M}_{X}arrow \mathfrak{B}(\infty)$

be the map defined by$\Psi(\underline{m})=$Then the map$\Psi$

hdInd$\nabla(\underline{m})$for.

$\underline{m}\in M_{X}$is bijective.

Pmof. We give a sketch of the proof of [1, Theorem 3.2]. When $X=D$ , without loss ofgenerality, we may assume that$\sigma(\underline{m}, \{\overline{n-1}, \overline{n}\})\geq\sigma(\underline{m}, \{\overline{n-1}, n\})$,is the multiplicity of in . Note that$\#m(n)e_{n}$ if $X=D$ . It follows from [1, Lemma 4.3] and the definition of

where

(a)

$\sigma(\underline{m}, \epsilon)$$\epsilon_{i}(Ind\nabla(\underline{m}(k)))=0$$\epsilon$$\underline{m}$for $i=n+1-k,$ $n+2-k,$$\sum_{s\in\underline{m}(n-1)}e(\epsilon)=\neq\underline{m}(n-1)e_{n-1}+$$\ldots$, $n$ ,$\underline{m}(k)$

that

175
KLR ALGEBRAS AND CRYSTAL BASES FOR FINITE CLASSICAL TYPE(b) hdInd$\nabla(m(k))=\{\begin{array}{l}\tilde{f}_{\nu_{X}^{n+1-k(\underline{m})}}^{e}1 if k=1, \ldots, n(X=A, B, C),k=1, \ldots , n-2(X=D),\tilde{f}_{\nu_{X}^{1}}^{e(\underline{m})}\tilde{f}_{\nu_{X}^{2}}^{e(\underline{m})}1 if k=n-1(X=D),\end{array}$

Let $I_{(k)}=\{n+1-k, \ldots , n\}$ and let
description given in [15], we have

hdInd

be the module given in (2.1). Then, by the crystal$\mathcal{N}_{k}$$\nabla(m(k))\simeq\{\begin{array}{l}\mathcal{N}_{n+1-k} if k=1, \ldots, n(X=A, B, C),k=1, \ldots , n-2(X=D),Ind(\mathcal{N}_{1}\otimes \mathcal{N}_{2}) if k=n-1(X=D).\end{array}$

Combining Proposition 2.3 and [1, Lemma 1.8] with the above conditions (a) and (b), we
obtainhd$Ind(\otimes_{k=1}^{n}\mathcal{N}_{k}(v))\simeq$

hd Ind$\simeq$hd$\simeq$hdInd$(\mathbb{B}_{k=n}^{1},Ind\nabla(m(k)))$$Ind(\mathbb{B}_{k=n’}^{1}\nabla(m(k)))$$\nabla(m)$,where $n’=n(X=A, B, C)$ and $n’=n-1(X=D)$ . Therefore, the assertion follows from
Proposition 2.3.
$\square$From Proposition 3.1 and Theorem 3.2, we have the following corollary.
Corollary 3.3. The composition$\Psi 0^{\prime r^{-1}}:B(\infty)arrow \mathfrak{B}(\infty)$

is a crystal isomorphism.

be the crystal of the irreducibleLetbe the dominant integral weight and let
highest weight module. It was shown in [10] that there is a unique strict crystal$\lambda\in p+$$B(\lambda)$$V(\lambda)$

embedding$B(\lambda)arrow B(\infty)\otimes T_{\lambda}\otimes C$,$v_{\lambda}\mapsto 1\otimes t_{\lambda}\otimes c$(resp. $C=\{c\}$ ) is a crystal. Here,whereis the highest weight vector ofoo (resp. wt
with wt. We denote by the composition of the strict embedding and the naturalprojection:$B(\lambda)$$v_{\lambda}$$(t_{\lambda})=\lambda,$$T_{\lambda}=\{t_{\lambda}\}$$(t_{\lambda})=0,$$\epsilon_{i}(t_{\lambda})=\varphi_{i}(t_{\lambda})=0.\tilde{e}_{i}t_{\lambda}=\tilde{f_{i}}t_{\lambda}=-$$\epsilon_{i}(t_{\lambda})=\varphi_{i}(t_{\lambda})=$$0,\tilde{e}_{i}t_{\lambda}=\tilde{f_{i}}t_{\lambda}=0)$$\iota_{\lambda}$$B(\lambda)\mapsto B(oo)\otimes T_{\lambda}\otimes Carrow B(oo)$Letto. By Proposition 3.1, the set:. Then the map$\mathcal{M}_{X}(\lambda)=lr^{-1}\circ\iota_{\lambda}(B(\lambda))$$\mathcal{M}_{X}(\lambda)$

via$l^{-1}\circ\iota_{\lambda}$$\wedge r_{\lambda}$$:=(’r^{-1}\circ\iota_{\lambda})^{-1}$$Y_{\lambda}(m)=\tilde{f}_{\nu_{X}}^{e_{1}(\underline{m})}\ldots\tilde{f}_{\nu_{X}}^{e_{n}(\underline{m})}v_{\lambda}$

where

for

.

$B(\lambda)$

is in 1-1 correspondence
is give by$\mathcal{M}_{X}(\lambda)arrow B(\lambda)$$m\in \mathcal{M}x(\lambda)$,$e_{i}(m)=\{\begin{array}{ll}\sum_{\mathfrak{s}\in\underline{\mathfrak{m}}(n+1-i)}e(\epsilon) if either X=A. B. C or i\neq 1,2(X=D),\#m(n-1)\iota_{n-1} if i=2(X=D),\#m(n)e_{n} if i=1(X=D).\end{array}$

176
EUIYONG PARK

We remark that the set
can be described explicitly from the string parametrizationofgiven in [15]. By Theorem 3.2 and Corollary 3.3, we have the following Corollary.
$\mathcal{M}_{X}(\lambda)$$B(\lambda)$

Corollary 3.4. Let(1) Let$\Psi^{\lambda}$:$\lambda\in p+$

be a dominant integml weight.

$\mathcal{M}_{X}(\lambda)arrow \mathfrak{B}(\lambda)$$\Psi^{\lambda}(m)=$be the map defined by

hdInd

for

$\nabla(m)$Then the map
is bijective.
(2) The composition$m\in \mathcal{M}_{X}(\lambda)$.

$\Psi^{\lambda}$

is a crystal isomorphism.

$\Psi^{\lambda_{O}\oint}r_{\lambda}^{-1}:B(\lambda)arrow \mathfrak{B}(\lambda)$

Example 3.5. Let$U_{q}(\mathfrak{g})$be of type

$D_{5}$. Then the crystal

5$B_{D}$

is given as follows:5$\prime^{5}$$\backslash ^{4}$

4$\overline{1}\underline{1}\overline{2}\underline{2}\overline{3}\underline{3}\overline{4}$$\underline{3}2\underline{2}2arrow^{1}1$$\swarrow 5/$$\backslash _{4}$$\overline{5}$Note that $\nu_{D}=(1,2,3,4,5,3,2,1)$ . We choose the following segments $5_{k}(k=1, \ldots, 6)$ of:$B_{X}$$s_{1}$$S$$:=\{4,5\}$ ,4 $:=\{2,4\}$ ,and letNote that

52 $:=\{3,5\}$ ,$S$

5$m=\{51\preceq s_{2}\preceq s_{3}\preceq s_{4}\preceq s_{5}\preceq s_{6}\}$

ch

Since

$\nabla(\mathfrak{s}_{1})=(4)$.

$;=\{\overline{1},$$\overline{4}\}$$s_{3}$.

$;=\{\overline{2},$ $\overline{4}\}$,$\epsilon_{6}:=\{\overline{1},4\}$be the multisegment consisting $ofs_{k}$ $(k=1, \ldots , 6)$ .

ch $\nabla(5_{2})=(3,5)$ ,ch $\nabla(s_{3})=(2,3)$ ,ch $\nabla(s_{4})=(2,3,4,5)+(2,3,5,4)$ ,ch $\nabla(s_{5})=(1,2,3)$ ,ch $\nabla(s_{6})=(1,2,3,4,5)+(1,2,3.5,4)$ .

$m(1)=\{\epsilon_{5},\epsilon_{6}\},$$\underline{m}(2)=\{\epsilon_{3},\mathfrak{s}_{4}\},$$m(3)=\{\epsilon_{2}\},$$\underline{m}(4)=\{\epsilon_{1}\}$and $m(5)=\emptyset,$ we have$e_{1}(m)=(2,2,2,1,1,0,0,0)$ ,$e_{2}(m)=(0,2,2,1,1,0,0,0)$ ,$e_{3}(m)=(0,0,1,0,1.0,0,0)$ ,$e_{4}(m)=(0,0,0,1,0.0,0,0)$ ,$e_{5}(m)=(0,0,0,0.0,0,0,0)$ .

177
KLR ALGEBRAS AND CRYSTAL BASES FOR FINITE CLASSICAL TYPE

It follows from Theorem 3.2 that

hdInd$\nabla(m)\simeq$

hdInd$(\nabla(\epsilon_{1})\otimes\nabla(g_{2})\otimes\nabla(s_{3})\otimes\nabla(\epsilon_{4})\otimes\nabla(5_{5})\otimes\nabla(\epsilon_{6}))$霧$\simeq\tilde{f}_{4}\tilde{f}_{3}f_{5}\tilde{f}_{2}^{2}\tilde{f}_{3}^{2}f_{4}\tilde{f}_{5}\tilde{f}_{1}^{2}$$\tilde{f}_{3}^{2}\tilde{f}_{4}\tilde{f}_{5}1$.

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178
EUIYONG PARKDEPARTMENT OF PURE AND APPLIED MATHEMATICS, GRADUATE SCHOOL OF INFORMATION SCIENCE ANDTECHNOLOGY, OSAKA UNIVERSITY, TOYONAKA, OSAKA 560-0043, JAPANE-mail address: pwy@cr.math.sci.osaka-u.ac.jp