  
  [1m[4m[31m2. Background[0m
  
  In  this  chapter  we  summarize some of the theoretical concepts with which
  [1mQuaGroup[0m  operates.  Due  to  the rather mathematical nature of this chapter
  everything  has  been  written  in  LaTeX.  Therefore,  it  will  be  almost
  unreadable in the html version.
  
  
  [1m[4m[31m2.1 Gaussian Binomials[0m
  
  Let $v$ be an indeterminate over $\mathbb{Q}$. For a positive integer $n$ we
  set  $$  [n]  =  v^{n-1}+v^{n-3}+\cdots  + v^{-n+3}+v^{-n+1}. $$ We say that
  $[n]$  is  the  [22m[36m  Gaussian  integer  [0m  corresponding  to  $n$. The [22m[36m Gaussian
  factorial  [0m  $[n]!$  is defined by $$ [0]! = 1, ~ [n]! = [n][n-1]\cdots [1],
  \text{ for } n>0.$$ Finally, the [22m[36m Gaussian binomial [0m is $$ \begin{bmatrix} n
  \\ k \end{bmatrix} = \frac{[n]!}{[k]![n-k]!}.$$
  
  
  [1m[4m[31m2.2 Quantized enveloping algebras[0m
  
  Let  $\mathfrak{g}$  be a semisimple Lie algebra with root system $\Phi$. By
  $\Delta=\{\alpha_1,\ldots,  \alpha_l  \}$ we denote a fixed simple system of
  $\Phi$.  Let  $C=(C_{ij})$  be  the Cartan matrix of $\Phi$ (with respect to
  $\Delta$,  i.e., $ C_{ij} = \langle \alpha_i, \alpha_j^{\vee} \rangle$). Let
  $d_1,\ldots,  d_l$ be the unique sequence of positive integers with greatest
  common  divisor  $1$,  such  that  $  d_i  C_{ji}  = d_j C_{ij} $, and set $
  (\alpha_i,\alpha_j)  =  d_j  C_{ij}  $.  (We  note  that  this  implies that
  $(\alpha_i,\alpha_i)$  is  divisible  by  $2$.)  By $P$ we denote the weight
  lattice,  and  we  extend  the  form  $(~,~)$ to $P$ by bilinearity. \par By
  $W(\Phi)$  we denote the Weyl group of $\Phi$. It is generated by the simple
  reflections  $s_i=s_{\alpha_i}$  for  $1\leq i\leq l$ (where $s_{\alpha}$ is
  defined  by  $s_{\alpha}(\beta) = \beta - \langle\beta, \alpha^{\vee}\rangle
  \alpha$).\par  We  work over the field $\mathbb{Q}(q)$. For $\alpha\in\Phi $
  we   set   $$   q_{\alpha}  =  q^{\frac{(\alpha,\alpha)}{2}},$$  and  for  a
  non-negative     integer     $n$,     $[n]_{\alpha}=    [n]_{v=q_{\alpha}}$;
  $[n]_{\alpha}!$  and  $\begin{bmatrix}  n  \\  k \end{bmatrix}_{\alpha}$ are
  defined     analogously.\par     The     quantized     enveloping    algebra
  $U_q(\mathfrak{g})$   is   the   associative   algebra   (with   one)   over
  $\mathbb{Q}(q)$  generated by $F_{\alpha}$, $K_{\alpha}$, $K_{\alpha}^{-1}$,
  $E_{\alpha}$  for  $\alpha\in\Delta$,  subject  to  the  following relations
  \begin{align*}  K_{\alpha}K_{\alpha}^{-1} &= K_{\alpha}^{-1}K_{\alpha} = 1,~
  K_{\alpha}K_{\beta}   =   K_{\beta}K_{\alpha}\\   E_{\beta}   K_{\alpha}  &=
  q^{-(\alpha,\beta)}K_{\alpha}    E_{\beta}\\    K_{\alpha}    F_{\beta}   &=
  q^{-(\alpha,\beta)}F_{\beta}K_{\alpha}\\     E_{\alpha}     F_{\beta}     &=
  F_{\beta}E_{\alpha}                                   +\delta_{\alpha,\beta}
  \frac{K_{\alpha}-K_{\alpha}^{-1}}{q_{\alpha}-q_{\alpha}^{-1}}   \end{align*}
  together    with,    for    $\alpha\neq    \beta\in\Delta$,   \begin{align*}
  \sum_{k=0}^{1-\langle  \beta,\alpha^{\vee}\rangle  }  (-1)^k \begin{bmatrix}
  1-\langle    \beta,\alpha^{\vee}\rangle    \\    k    \end{bmatrix}_{\alpha}
  E_{\alpha}^{1-\langle  \beta,\alpha^{\vee}\rangle-k}  E_{\beta} E_{\alpha}^k
  =0   &   \\   \sum_{k=0}^{1-\langle   \beta,\alpha^{\vee}\rangle   }  (-1)^k
  \begin{bmatrix}      1-\langle      \beta,\alpha^{\vee}\rangle      \\     k
  \end{bmatrix}_{\alpha}  F_{\alpha}^{1-\langle  \beta,\alpha^{\vee}\rangle-k}
  F_{\beta}  F_{\alpha}^k  =0 &. \end{align*} The quantized enveloping algebra
  has an automorphism $\omega$ defined by $\omega( F_{\alpha} ) = E_{\alpha}$,
  $\omega(E_{\alpha})=  F_{\alpha}$  and $\omega(K_{\alpha})=K_{\alpha}^{-1}$.
  Also     there     is     an    anti-automorphism    $\tau$    defined    by
  $\tau(F_{\alpha})=F_{\alpha}$,     $\tau(E_{\alpha})=     E_{\alpha}$    and
  $\tau(K_{\alpha})=K_{\alpha}^{-1}$. We have $\omega^2=1$ and $\tau^2=1$.\par
  If  the  Dynkin  diagram of $\Phi$ admits a diagram automorphism $\pi$, then
  $\pi$  induces  an  automorphism  of  $U_q(\mathfrak{g})$ in the obvious way
  ($\pi$  is  a  permutation of the simple roots; we permute the $F_{\alpha}$,
  $E_{\alpha}$,   $K_{\alpha}^{\pm   1}$   accordingly).\par   Now   we   view
  $U_q(\mathfrak{g})$   as   an   algebra   over   $\mathbb{Q}$,  and  we  let
  $\overline{\phantom{A}}  :  U_q(\mathfrak{g})\to  U_q(\mathfrak{g})$  be the
  automorphism       defined       by      $\overline{F_{\alpha}}=F_{\alpha}$,
  $\overline{K_{\alpha}}=                                    K_{\alpha}^{-1}$,
  $\overline{E_{\alpha}}=E_{\alpha}$, $\overline{q}=q^{-1}$.
  
  
  [1m[4m[31m2.3 Representations of $U_q(\mathfrak{g})$[0m
  
  Let  $\lambda\in P$ be a dominant weight. Then there is a unique irreducible
  highest-weight   module   over   $U_q(\mathfrak{g})$   with  highest  weight
  $\lambda$.  We  denote  it by $V(\lambda)$. It has the same character as the
  irreducible  highest-weight  module  over $\mathfrak{g}$ with highest weight
  $\lambda$.  Furthermore, every finite-dimensional $U_q(\mathfrak{g})$-module
  is  a direct sum of irreducible highest-weight modules.\par It is well-known
  that  $U_q(\mathfrak{g})$  is a Hopf algebra. The comultiplication $\Delta :
  U_q(\mathfrak{g})\to U_q(\mathfrak{g}) \otimes U_q(\mathfrak{g})$ is defined
  by    \begin{align*}    \Delta(E_{\alpha})    &=   E_{\alpha}\otimes   1   +
  K_{\alpha}\otimes   E_{\alpha}\\   \Delta(F_{\alpha})  &=  F_{\alpha}\otimes
  K_{\alpha}^{-1}    +    1\otimes    F_{\alpha}\\    \Delta(K_{\alpha})    &=
  K_{\alpha}\otimes K_{\alpha}. \end{align*} (Note that we use the same symbol
  to  denote  a  simple  system  of  $\Phi$;  of  course  this  does not cause
  confusion.)  The  counit $\varepsilon : U_q(\mathfrak{g}) \to \mathbb{Q}(q)$
  is             a             homomorphism             defined             by
  $\varepsilon(E_{\alpha})=\varepsilon(F_{\alpha})=0$,           $\varepsilon(
  K_{\alpha})    =1$.   Finally,   the   antipode   $S:   U_q(\mathfrak{g})\to
  U_q(\mathfrak{g})$      is      an      anti-automorphism      given      by
  $S(E_{\alpha})=-K_{\alpha}^{-1}E_{\alpha}$,       $S(F_{\alpha})=-F_{\alpha}
  K_{\alpha}$, $S(K_{\alpha})=K_{\alpha}^{-1}$.\par Using $\Delta$ we can make
  the  tensor  product  $V\otimes  W$ of two $U_q(\mathfrak{g})$-modules $V,W$
  into a $U_q(\mathfrak{g})$-module. The counit $\varepsilon$ yields a trivial
  $1$-dimensional  $U_q(\mathfrak{g})$-module.  And  with  $S$ we can define a
  $U_q(\mathfrak{g})$-module    structure    on    the   dual   $V^*$   of   a
  $U_q(\mathfrak{g})$-module  $V$,  by  $(u\cdot f)(v) = f(S(u)\cdot v )$.\par
  The  Hopf  algebra  structure  given above is not the only one possible. For
  example,  we  can  twist  $\Delta,\varepsilon,S$  by  an automorphism, or an
  anti-automorphism $f$. The twisted comultiplication is given by $$\Delta^f =
  f\otimes  f  \circ\Delta\circ  f^{-1}.$$  The  twisted  antipode by $$ S^f =
  \begin{cases}  f\circ  S\circ f^{-1} \text{ ~~~~if $f$ is an automorphism}\\
  f\circ      S^{-1}\circ      f^{-1}     \text{     ~if     $f$     is     an
  anti-automorphism.}\end{cases}$$  And the twisted counit by $\varepsilon^f =
  \varepsilon\circ f^{-1}$ (see [J96], 3.8).
  
  
  [1m[4m[31m2.4 PBW-type bases[0m
  
  The first problem one has to deal with when working with $U_q(\mathfrak{g})$
  is finding a basis of it, along with an algorithm for expressing the product
  of  two  basis  elements as a linear combination of basis elements. First of
  all  we  have  that  $U_q(\mathfrak{g})\cong  U^-\otimes U^0\otimes U^+$ (as
  vector spaces), where $U^-$ is the subalgebra generated by the $F_{\alpha}$,
  $U^0$  is  the  subalgebra  generated  by  the  $K_{\alpha}$,  and  $U^+$ is
  generated  by  the $E_{\alpha}$. So a basis of $U_q(\mathfrak{g})$ is formed
  by  all  elements  $FKE$,  where  $F$,  $K$, $E$ run through bases of $U^-$,
  $U^0$,  $U^+$  respectively.\par  Finding  a  basis  of $U^0$ is easy: it is
  spanned   by   all  $K_{\alpha_1}^{r_1}  \cdots  K_{\alpha_l}^{r_l}$,  where
  $r_i\in\mathbb{Z}$.  For  $U^-$,  $U^+$  we use the so-called {\em PBW-type}
  bases.  They  are  defined  as  follows.  For $\alpha,\beta\in\Delta$ we set
  $r_{\beta,\alpha}   =   -\langle   \beta,  \alpha^{\vee}\rangle$.  Then  for
  $\alpha\in\Delta$     we     have    the    automorphism    $T_{\alpha}    :
  U_q(\mathfrak{g})\to    U_q(\mathfrak{g})$    defined    by   \begin{align*}
  T_{\alpha}(E_{\alpha})  &=  -F_{\alpha}K_{\alpha}\\ T_{\alpha}(E_{\beta}) &=
  \sum_{i=0}^{r_{\beta,\alpha}}             (-1)^i             q_{\alpha}^{-i}
  E_{\alpha}^{(r_{\beta,\alpha}-i)}E_{\beta}   E_{\alpha}^{(i)}   \text{  (for
  $\alpha\neq\beta$)}\\                T_{\alpha}(K_{\beta})                &=
  K_{\beta}K_{\alpha}^{r_{\beta,\alpha}}\\      T_{\alpha}(F_{\alpha})      &=
  -K_{\alpha}^{-1}         E_{\alpha}\\        T_{\alpha}(F_{\beta})        &=
  \sum_{i=0}^{r_{\beta,\alpha}}              (-1)^i             q_{\alpha}^{i}
  F_{\alpha}^{(i)}F_{\beta}F_{\alpha}^    {(r_{\beta,\alpha}-i)}\text{    (for
  $\alpha\neq\beta$),}     \end{align*}     (where     $E_{\alpha}^{(k)}     =
  E_{\alpha}^k/[k]_{\alpha}!$,  and likewise for $F_{\alpha}^{(k)}$). \par Let
  $w_0=s_{i_1}\cdots  s_{i_t}$ be a reduced expression for the longest element
  in   the   Weyl   group   $W(\Phi)$.   For   $1\leq  k\leq  t$  set  $F_k  =
  T_{\alpha_{i_1}}\cdots  T_{\alpha_{i_{k-1}}}(F_{\alpha_{i_k}})$,  and $E_k =
  T_{\alpha_{i_1}}\cdots T_{\alpha_{i_{k-1}}}(E_{\alpha_{i_k}})$. Then $F_k\in
  U^-$,   and  $E_k\in  U^+$.  Furthermore,  the  elements  $F_1^{m_1}  \cdots
  F_t^{m_t}$,   $E_1^{n_1}\cdots   E_t^{n_t}$  (where  the  $m_i$,  $n_i$  are
  non-negative  integers) form bases of $U^-$ and $U^+$ respectively. \par The
  elements $F_{\alpha}$ and $E_{\alpha}$ are said to have weight $-\alpha$ and
  $\alpha$  respectively,  where  $\alpha$  is a simple root. Furthermore, the
  weight  of  a  product  $ab$  is  the sum of the weights of $a$ and $b$. Now
  elements  of  $U^-$,  $U^+$  that are linear combinations of elements of the
  same  weight  are  said to be homogeneous. It can be shown that the elements
  $F_k$,   and   $E_k$   are   homogeneous  of  weight  $-\beta$  and  $\beta$
  respectively, where $\beta=s_{i_1}\cdots s_{i_{k-1}}(\alpha_{i_k})$. \par In
  the  sequel we use the notation $F_k^{(m)} = F_k^m/[m]_{\alpha_{i_k}}!$, and
  $E_k^{(n)} = E_k^n/[n]_{\alpha_{i_k}}!$. \par
  
  
  [1m[4m[31m2.5 The ${\mathbb Z}$-form of $U_q(\mathfrak{g})$[0m
  
  For  $\alpha\in\Delta$ set $$\begin{bmatrix} K_{\alpha} \\ n \end{bmatrix} =
  \prod_{i=1}^n     \frac{q_{\alpha}^{-i+1}K_{\alpha}    -    q_{\alpha}^{i-1}
  K_{\alpha}^{-1}}  {q_{\alpha}^i-q_{\alpha}^{-i}}.$$ Then according to [L90],
  Theorem     6.7     the     elements     $$F_1^{(k_1)}\cdots     F_t^{(k_t)}
  K_{\alpha_1}^{\delta_1}  \begin{bmatrix}  K_{\alpha_1}  \\ m_1 \end{bmatrix}
  \cdots   K_{\alpha_l}^{\delta_l}   \begin{bmatrix}   K_{\alpha_l}   \\   m_l
  \end{bmatrix}  E_1^{(n_1)}\cdots  E_t^{(n_t)},$$ (where $k_i,m_i,n_i\geq 0$,
  $\delta_i=0,1$)  form  a basis of $U_q(\mathfrak{g})$, such that the product
  of  any  two  basis  elements is a linear combination of basis elements with
  coefficients  in  $\mathbb{Z}[q,q^{-1}]$.  The  quantized enveloping algebra
  over  $\mathbb{Z}[q,q^{-1}]$ with this basis is called the $\mathbb{Z}$-form
  of    $U_q(\mathfrak{g})$,    and   denoted   by   $U_{\mathbb{Z}}$.   Since
  $U_{\mathbb{Z}}$  is  defined  over $\mathbb{Z}[q,q^{-1}]$ we can specialize
  $q$  to any nonzero element $\epsilon$ of a field $F$, and obtain an algebra
  $U_{\epsilon}$  over  $F$.  \par We call $q\in \mathbb{Q}(q)$, and $\epsilon
  \in  F$  the  quantum  parameter  of  $U_q(\mathfrak{g})$ and $U_{\epsilon}$
  respectively.  \par Let $\lambda$ be a dominant weight, and $V(\lambda)$ the
  irreducible   highest   weight  module  of  highest  weight  $\lambda$  over
  $U_q(\mathfrak{g})$.  Let  $v_{\lambda}\in  V(\lambda)$  be  a fixed highest
  weight    vector.    Then    $U_{\mathbb{Z}}\cdot    v_{\lambda}$    is    a
  $U_{\mathbb{Z}}$-module.  So by specializing $q$ to an element $\epsilon$ of
  a  field  $F$, we get a $U_{\epsilon}$-module. We call it the Weyl module of
  highest  weight  $\lambda$  over  $U_{\epsilon}$.  We  note  that  it is not
  necessarily irreducible.
  
  
  [1m[4m[31m2.6 The canonical basis[0m
  
  As  in  Section  [1m2.4[0m  we  let $U^-$ be the subalgebra of $U_q(\mathfrak{g})$
  generated  by  the  $F_{\alpha}$  for  $\alpha\in\Delta$.  In  [L0a] Lusztig
  introduced  a  basis  of  $U^-$  with  very nice properties, called the {\em
  canonical basis}. (Later this basis was also constructed by Kashiwara, using
  a  different  method. For a brief overview on the history of canonical bases
  we  refer  to [C06].) \par Let $w_0=s_{i_1}\cdots s_{i_t}$, and the elements
  $F_k$  be  as in Section [1m2.4[0m. Then, in order to stress the dependency of the
  monomial     \begin{equation}\label{eq0}    F_1^{(n_1)}\cdots    F_t^{(n_t)}
  \end{equation}  on  the choice of reduced expression for the longest element
  in   $W(\Phi)$   we  say  that  it  is  a  $w_0$-monomial.\par  Now  we  let
  $\overline{\phantom{a}}$  be  the  automorphism  of $U^-$ defined in Section
  [1m2.2[0m.  Elements that are invariant under $\overline{\phantom{a}}$ are said to
  be  bar-invariant. \par By results of Lusztig ([L93] Theorem 42.1.10, [L96],
  Proposition  8.2),  there  is  a  unique  basis  ${\bf B}$ of $U^-$ with the
  following  properties. Firstly, all elements of ${\bf B}$ are bar-invariant.
  Secondly, for any choice of reduced expression $w_0$ for the longest element
  in  the  Weyl group, and any element $X\in{\bf B}$ we have that $X = x +\sum
  \zeta_i  x_i$,  where  $x,x_i$ are $w_0$-monomials, $x\neq x_i$ for all $i$,
  and $\zeta_i\in q\mathbb{Z}[q]$. The basis ${\bf B}$ is called the canonical
  basis. If we work with a fixed reduced expression for the longest element in
  $W(\Phi)$,  and  write  $X\in{\bf  B}$ as above, then we say that $x$ is the
  {\em   principal   monomial}   of   $X$.\par   Let   $\mathcal{L}$   be  the
  $\mathbb{Z}[q]$-lattice  in  $U^-$ spanned by {\bf B}. Then $\mathcal{L}$ is
  also  spanned  by  all  $w_0$-monomials  (where  $w_0$  is  a  fixed reduced
  expression  for the longest element in $W(\Phi)$). Now let $\widetilde{w}_0$
  be a second reduced expression for the longest element in $W(\Phi)$. Let $x$
  be  a  $w_0$-monomial,  and let $X$ be the element of {\bf B} with principal
  monomial     $x$.     Write    $X$    as    a    linear    combination    of
  $\widetilde{w}_0$-monomials,   and  let  $\widetilde{x}$  be  the  principal
  monomial    of   that   expression.   Then   we   write   $\widetilde{x}   =
  R_{w_0}^{\tilde{w}_0}(x)$. Note that $x = \widetilde{x} \bmod q\mathcal{L}$.
  \par  Now  let  $\mathcal{B}$  be  the  set  of  all  $w_0$-monomials $\bmod
  q\mathcal{L}$.  Then  $\mathcal{B}$  is  a  basis of the $\mathbb{Z}$-module
  $\mathcal{L}/q\mathcal{L}$.  Moreover,  $\mathcal{B}$  is independent of the
  choice  of  $w_0$.  Let  $\alpha\in\Delta$,  and  let $\widetilde{w}_0$ be a
  reduced  expression  for  the  longest  element  in $W(\Phi)$, starting with
  $s_{\alpha}$.   The   Kashiwara   operators   $\widetilde{F}_{   \alpha}   :
  \mathcal{B}\to  \mathcal{B}$  and  $\widetilde{E}_{\alpha}  : \mathcal{B}\to
  \mathcal{B}\cup\{0\}$  are defined as follows. Let $b\in\mathcal{B}$ and let
  $x$  be  the  $w_0$-monomial  such  that  $b  =  x  \bmod q\mathcal{L}$. Set
  $\widetilde{x}  =  R_{w_0}^  {\tilde{w}_0}(x)$. Then $\widetilde{x}'$ is the
  $\widetilde{w}_0$-monomial  constructed  from  $\widetilde{x}$ by increasing
  its  first exponent by $1$ (the first exponent is the $n_1$ in (\ref{eq0})).
  Then  $\widetilde{F}_{  \alpha}(b)  =  R_{\tilde{w}_0}^{w_0}(\widetilde{x}')
  \bmod q\mathcal{L}$. For $\widetilde{E}_{\alpha}$ we let $\widetilde{x}'$ be
  the   $\widetilde{w}_0$-monomial   constructed   from   $\widetilde{x}$   by
  decreasing  its  first  exponent  by $1$, if this exponent is $\geq 1$. Then
  $\widetilde{E}_{\alpha}(b)    =   R_{\tilde{w}_0}^{w_0}(\widetilde{x}')\bmod
  q\mathcal{L}$.  Furthermore,  $\widetilde{E}_{\alpha}(b)  =0$  if  the first
  exponent  of  $\widetilde{x}$  is  $0$. It can be shown that this definition
  does  not  depend  on the choice of $w_0$, $\widetilde{w}_0$. Furthermore we
  have         $\widetilde{F}_{\alpha}\widetilde{E}_{\alpha}(b)=b$,         if
  $\widetilde{E}_{\alpha}(b)\neq      0$,      and     $\widetilde{E}_{\alpha}
  \widetilde{F}_   {\alpha}(b)=b$   for   all   $b\in   \mathcal{B}$.\par  Let
  $w_0=s_{i_1}\cdots  s_{i_t}$  be  a fixed reduced expression for the longest
  element in $W(\Phi)$. For $b\in\mathcal{B}$ we define a sequence of elements
  $b_k\in\mathcal{B}$  for  $0\leq  k\leq t$, and a sequence of integers $n_k$
  for  $1\leq k\leq t$ as follows. We set $b_0=b$, and if $b_{k-1}$ is defined
  we   let   $n_k$   be   maximal   such  that  $\widetilde{E}_{\alpha_{i_k}}^
  {n_k}(b_{k-1})\neq 0$. Also we set $b_k = \widetilde{E}_{\alpha_{i_k}}^{n_k}
  (b_{k-1})$.  Then the sequence $(n_1,\ldots,n_t)$ is called the {\em string}
  of  $b\in\mathcal{B}$  (relative  to  $w_0$). We note that $b=\widetilde{F}_
  {\alpha_{i_1}}^{n_1}\cdots  \widetilde{F}_{\alpha_{i_t}}^ {n_t}(1)$. The set
  of  all  strings  parametrizes  the  elements of $\mathcal{B}$, and hence of
  ${\bf  B}$.\par  Now  let  $V(\lambda)$  be  a  highest-weight  module  over
  $U_q(\mathfrak{g})$,  with  highest weight $\lambda$. Let $v_{\lambda}$ be a
  fixed   highest   weight   vector.  Then  ${\bf  B}_{\lambda}  =  \{  X\cdot
  v_{\lambda}\mid  X\in {\bf B}\} \setminus \{0\}$ is a basis of $V(\lambda)$,
  called  the  {\em  canonical basis}, or {\em crystal basis} of $V(\lambda)$.
  Let  $\mathcal{L}(\lambda)$  be  the $\mathbb{Z}[q]$-lattice in $V(\lambda)$
  spanned  by  ${\bf B}_{\lambda}$. We let $\mathcal{B}({\lambda})$ be the set
  of  all  $x\cdot  v_{\lambda}\bmod  q\mathcal{L}(\lambda)$,  where  $x$ runs
  through  all  $w_0$-monomials,  such that $X\cdot v_{\lambda} \neq 0$, where
  $X\in  {\bf  B}$  is  the  element  with  principal  monomial  $x$. Then the
  Kashiwara   operators  are  also  viewed  as  maps  $\mathcal{B}(\lambda)\to
  \mathcal{B}(\lambda)\cup\{0\}$,   in   the   following  way.  Let  $b=x\cdot
  v_{\lambda}\bmod     q\mathcal{L}(\lambda)$     be     an     element     of
  $\mathcal{B}(\lambda)$,    and   let   $b'=x\bmod   q\mathcal{L}$   be   the
  corresponding  element  of $\mathcal{B}$. Let $y$ be the $w_0$-monomial such
  that $\widetilde{F}_{\alpha}(b')=y\bmod q\mathcal{L}$. Then $\widetilde{F}_{
  \alpha}(b)   =   y\cdot   v_{\lambda}   \bmod   q\mathcal{L}(\lambda)$.  The
  description of $\widetilde{E}_{\alpha}$ is analogous. (In [J96], Chapter 9 a
  different  definition  is  given; however, by [J96], Proposition 10.9, Lemma
  10.13,  the  two  definitions agree).\par The set $\mathcal{B}(\lambda)$ has
  $\dim  V(\lambda)$  elements. We let $\Gamma$ be the coloured directed graph
  defined   as   follows.   The   points  of  $\Gamma$  are  the  elements  of
  $\mathcal{B}(\lambda)$,  and there is an arrow with colour $\alpha\in\Delta$
  connecting  $b,b'\in  \mathcal{B}$,  if  $\widetilde{F}_{\alpha}(b)=b'$. The
  graph $\Gamma$ is called the {\em crystal graph} of $V(\lambda)$.
  
  
  [1m[4m[31m2.7 The path model[0m
  
  In  this section we recall some basic facts on Littelmann's path model. \par
  From  Section  [1m2.2[0m  we  recall  that  $P$  denotes  the  weight lattice. Let
  $P_{\mathbb{R}}$  be  the vector space over $\mathbb{R}$ spanned by $P$. Let
  $\Pi$ be the set of all piecewise linear paths $\xi : [0,1]\to P_{\mathbb{R}
  $,  such that $\xi(0)=0$. For $\alpha\in\Delta$ Littelmann defined operators
  $f_{\alpha},  e_{\alpha}  :  \Pi  \to  \Pi\cup  \{0\}$.  Let  $\lambda$ be a
  dominant  weight  and  let $\xi_{\lambda}$ be the path joining $\lambda$ and
  the origin by a straight line. Let $\Pi_{\lambda}$ be the set of all nonzero
  $f_{\alpha_{i_1}}\cdots f_{\alpha_{i_m}}(\xi_{\lambda})$ for $m\geq 0$. Then
  $\xi(1)\in  P$  for  all $\xi\in \Pi_{\lambda}$. Let $\mu\in P$ be a weight,
  and  let  $V(\lambda)$ be the highest-weight module over $U_q(\mathfrak{g})$
  of  highest weight $\lambda$. A theorem of Littelmann states that the number
  of  paths  $\xi\in  \Pi_{\lambda}$  such  that  $\xi(1)=\mu$ is equal to the
  dimension  of  the  weight  space  of  weight  $\mu$ in $V(\lambda)$ ([L95],
  Theorem  9.1).\par  All  paths  appearing  in  $\Pi_{\lambda}$ are so-called
  Lakshmibai-Seshadri paths (LS-paths for short). They are defined as follows.
  Let   $\leq$   denote   the  Bruhat  order  on  $W(\Phi)$.  For  $\mu,\nu\in
  W(\Phi)\cdot   \lambda$   (the  orbit  of  $\lambda$  under  the  action  of
  $W(\Phi)$),  write  $\mu\leq \nu$ if $\tau\leq\sigma$, where $\tau,\sigma\in
  W(\Phi)$   are   the   unique   elements   of   minimal   length  such  that
  $\tau(\lambda)=\mu$,  $\sigma(\lambda)=  \nu$.  Now a rational path of shape
  $\lambda$  is  a  pair $\pi=(\nu,a)$, where $\nu=(\nu_1,\ldots, \nu_s)$ is a
  sequence of elements of $W(\Phi)\cdot \lambda$, such that $\nu_i> \nu_{i+1}$
  and  $a=(a_0=0,  a_1,  \cdots  ,a_s=1)$ is a sequence of rationals such that
  $a_i  <a_{i+1}$. The path $\pi$ corresponding to these sequences is given by
  $$  \pi(t)  =\sum_{j=1}^{r-1}  (a_j-a_{j-1})\nu_j  +  \nu_r(t-a_{r-1})$$ for
  $a_{r-1}\leq  t\leq  a_r$.  Now  an LS-path of shape $\lambda$ is a rational
  path  satisfying a certain integrality condition (see [L94], [L95]). We note
  that  the path $\xi_{\lambda} = ( (\lambda), (0,1) )$ joining the origin and
  $\lambda$  by  a  straight  line is an LS-path.\par Now from [L94], [L95] we
  transcribe  the  following:  \begin{itemize}  \item Let $\pi$ be an LS-path.
  Then  $f_{\alpha}\pi$  is  an  LS-path  or  $0$;  and  the  same  holds  for
  $e_{\alpha}\pi$.  \item  The action of $f_{\alpha},e_{\alpha}$ can easily be
  described  combinatorially  (see [L94]). \item The endpoint of an LS-path is
  an  integral  weight.  \item  Let  $\pi=(\nu,a)$  be  an  LS-path.  Then  by
  $\phi(\pi)$  we  denote the unique element $\sigma$ of $W(\Phi)$ of shortest
  length  such  that $\sigma(\lambda)=\nu_1$. \end{itemize} Let $\lambda$ be a
  dominant  weight.  Then  we  define  a  labeled  directed  graph $\Gamma$ as
  follows.  The  points of $\Gamma$ are the paths in $\Pi_{\lambda}$. There is
  an   edge   with   label   $\alpha\in\Delta$  from  $\pi_1$  to  $\pi_2$  if
  $f_{\alpha}\pi_1  =\pi_2$. Now by [K96] this graph $\Gamma$ is isomorphic to
  the   crystal  graph  of  the  highest-weight  module  with  highest  weight
  $\lambda$.  So  the  path  model  provides an efficient way of computing the
  crystal  graph  of  a highest-weight module, without constructing the module
  first.       Also       we       see       that      $f_{\alpha_{i_1}}\cdots
  f_{\alpha_{i_r}}\xi_{\lambda}        =0$        is       equivalent       to
  $\widetilde{F}_{\alpha_{i_1}}\cdots                           \widetilde{F}_
  {\alpha_{i_r}}v_{\lambda}=0$, where $v_{\lambda}\in V(\lambda)$ is a highest
  weight   vector  (or  rather  the  image  of  it  in  $\mathcal{L}(\lambda)/
  q\mathcal{L}   (\lambda)$),   and  the  $\widetilde{F}_{\alpha_k}$  are  the
  Kashiwara operators on $\mathcal{B}(\lambda)$ (see Section [1m2.6[0m).
  
  
  [1m[4m[31m2.8 Notes[0m
  
  I  refer  to  [H90] for more information on Weyl groups, and to [S01] for an
  overview  of  algorithms  for  computing with weights, Weyl groups and their
  elements.\par  For  general  introductions  into  the  theory  of  quantized
  enveloping algebras I refer to [C98], [J96] (from where most of the material
  of  this  chapter  is  taken), [L92], [L93], [R91]. I refer to the papers by
  Littelmann ([L94], [L95], [L98]) for more information on the path model. The
  paper  by  Kashiwara ([K96]) contains a proof of the connection between path
  operators  and  Kashiwara  operators.\par  Finally,  I  refer  to  [G01] (on
  computing  with  PBW-type  bases),  [G02]  (computation  of  elements of the
  canonical basis) for an account of some of the algorithms used in [1mQuaGroup[0m.
  
