  
  [1X2 [33X[0;0YBackground[133X[101X
  
  [33X[0;0YIn  this  chapter  we  summarize some of the theoretical concepts with which
  [5XQuaGroup[105X operates.[133X
  
  
  [1X2.1 [33X[0;0YGaussian Binomials[133X[101X
  
  [33X[0;0YLet [23Xv[123X be an indeterminate over [23X\mathbb{Q}[123X. For a positive integer [23Xn[123X we set[133X
  
  
  [24X[33X[0;6Y[n] = v^{n-1}+v^{n-3}+\cdots + v^{-n+3}+v^{-n+1}.[133X
  
  [124X
  
  [33X[0;0YWe  say that [23X[n][123X is the [13X Gaussian integer [113X corresponding to [23Xn[123X. The [13X Gaussian
  factorial [113X [23X[n]![123X is defined by[133X
  
  
  [24X[33X[0;6Y[0]! = 1, ~ [n]! = [n][n-1]\cdots [1], \text{ for } n>0.[133X
  
  [124X
  
  [33X[0;0YFinally, the [13X Gaussian binomial [113X is[133X
  
  
  [24X[33X[0;6Y\begin{bmatrix} n \\ k \end{bmatrix} = \frac{[n]!}{[k]![n-k]!}.[133X
  
  [124X
  
  
  [1X2.2 [33X[0;0YQuantized enveloping algebras[133X[101X
  
  [33X[0;0YLet  [23X\mathfrak{g}[123X  be  a  semisimple  Lie  algebra with root system [23X\Phi[123X. By
  [23X\Delta=\{\alpha_1,\ldots,  \alpha_l  \}[123X  we  denote a fixed simple system of
  [23X\Phi[123X.  Let  [23XC=(C_{ij})[123X be the Cartan matrix of [23X\Phi[123X (with respect to [23X\Delta[123X,
  i.e.,  [23X C_{ij} = \langle \alpha_i, \alpha_j^{\vee} \rangle[123X). Let [23Xd_1,\ldots,
  d_l[123X be the unique sequence of positive integers with greatest common divisor
  [23X1[123X,  such  that [23X d_i C_{ji} = d_j C_{ij} [123X, and set [23X (\alpha_i,\alpha_j) = d_j
  C_{ij} [123X. (We note that this implies that [23X(\alpha_i,\alpha_i)[123X is divisible by
  [23X2[123X.)  By [23XP[123X we denote the weight lattice, and we extend the form [23X(~,~)[123X to [23XP[123X by
  bilinearity.[133X
  
  [33X[0;0YBy  [23XW(\Phi)[123X  we denote the Weyl group of [23X\Phi[123X. It is generated by the simple
  reflections  [23Xs_i=s_{\alpha_i}[123X for [23X1\leq i\leq l[123X (where [23Xs_{\alpha}[123X is defined
  by [23Xs_{\alpha}(\beta) = \beta - \langle\beta, \alpha^{\vee}\rangle \alpha[123X).[133X
  
  [33X[0;0YWe work over the field [23X\mathbb{Q}(q)[123X. For [23X\alpha\in\Phi [123X we set[133X
  
  
  [24X[33X[0;6Yq_{\alpha} = q^{\frac{(\alpha,\alpha)}{2}},[133X
  
  [124X
  
  [33X[0;0Yand   for   a  non-negative  integer  [23Xn[123X,  [23X[n]_{\alpha}=  [n]_{v=q_{\alpha}}[123X;
  [23X[n]_{\alpha}![123X  and [23X\begin{bmatrix} n \\ k \end{bmatrix}_{\alpha}[123X are defined
  analogously.[133X
  
  [33X[0;0YThe  quantized  enveloping  algebra  [23XU_q(\mathfrak{g})[123X  is  the  associative
  algebra  (with  one) over [23X\mathbb{Q}(q)[123X generated by [23XF_{\alpha}[123X, [23XK_{\alpha}[123X,
  [23XK_{\alpha}^{-1}[123X,  [23XE_{\alpha}[123X  for  [23X\alpha\in\Delta[123X, subject to the following
  relations[133X
  
  
  [24X[33X[0;6Y\begin{aligned} 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{aligned}[133X
  
  [124X
  
  [33X[0;0Ytogether with, for [23X\alpha\neq \beta\in\Delta[123X,[133X
  
  
  [24X[33X[0;6Y\begin{aligned}  \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{aligned}[133X
  
  [124X
  
  [33X[0;0YThe  quantized  enveloping  algebra  has  an  automorphism [23X\omega[123X defined by
  [23X\omega(  F_{\alpha}  )  =  E_{\alpha}[123X,  [23X\omega(E_{\alpha})=  F_{\alpha}[123X  and
  [23X\omega(K_{\alpha})=K_{\alpha}^{-1}[123X.  Also there is an anti-automorphism [23X\tau[123X
  defined  by  [23X\tau(F_{\alpha})=F_{\alpha}[123X,  [23X\tau(E_{\alpha})=  E_{\alpha}[123X and
  [23X\tau(K_{\alpha})=K_{\alpha}^{-1}[123X. We have [23X\omega^2=1[123X and [23X\tau^2=1[123X.[133X
  
  [33X[0;0YIf  the  Dynkin  diagram of [23X\Phi[123X admits a diagram automorphism [23X\pi[123X, then [23X\pi[123X
  induces  an  automorphism  of [23XU_q(\mathfrak{g})[123X in the obvious way ([23X\pi[123X is a
  permutation  of  the  simple  roots;  we permute the [23XF_{\alpha}[123X, [23XE_{\alpha}[123X,
  [23XK_{\alpha}^{\pm 1}[123X accordingly).[133X
  
  [33X[0;0YNow  we  view  [23XU_q(\mathfrak{g})[123X  as  an algebra over [23X\mathbb{Q}[123X, and we let
  [23X\overline{\phantom{A}}   :  U_q(\mathfrak{g})\to  U_q(\mathfrak{g})[123X  be  the
  automorphism        defined       by       [23X\overline{F_{\alpha}}=F_{\alpha}[123X,
  [23X\overline{K_{\alpha}}=   K_{\alpha}^{-1}[123X,  [23X\overline{E_{\alpha}}=E_{\alpha}[123X,
  [23X\overline{q}=q^{-1}[123X.[133X
  
  
  [1X2.3 [33X[0;0YRepresentations of [23XU_q(\mathfrak{g})[123X[101X[1X[133X[101X
  
  [33X[0;0YLet  [23X\lambda\in  P[123X  be a dominant weight. Then there is a unique irreducible
  highest-weight module over [23XU_q(\mathfrak{g})[123X with highest weight [23X\lambda[123X. We
  denote  it  by  [23XV(\lambda)[123X.  It  has  the  same character as the irreducible
  highest-weight   module  over  [23X\mathfrak{g}[123X  with  highest  weight  [23X\lambda[123X.
  Furthermore,  every  finite-dimensional [23XU_q(\mathfrak{g})[123X-module is a direct
  sum of irreducible highest-weight modules.[133X
  
  [33X[0;0YIt   is   well-known   that   [23XU_q(\mathfrak{g})[123X   is  a  Hopf  algebra.  The
  comultiplication  [23X\Delta  :  U_q(\mathfrak{g})\to  U_q(\mathfrak{g}) \otimes
  U_q(\mathfrak{g})[123X is defined by[133X
  
  
  [24X[33X[0;6Y\begin{aligned}     \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{aligned}[133X
  
  [124X
  
  [33X[0;0Y(Note  that  we  use  the  same symbol to denote a simple system of [23X\Phi[123X; of
  course   this   does   not   cause  confusion.)  The  counit  [23X\varepsilon  :
  U_q(\mathfrak{g})   \to   \mathbb{Q}(q)[123X   is   a   homomorphism  defined  by
  [23X\varepsilon(E_{\alpha})=\varepsilon(F_{\alpha})=0[123X,  [23X\varepsilon( K_{\alpha})
  =1[123X.  Finally,  the  antipode [23XS: U_q(\mathfrak{g})\to U_q(\mathfrak{g})[123X is an
  anti-automorphism    given    by   [23XS(E_{\alpha})=-K_{\alpha}^{-1}E_{\alpha}[123X,
  [23XS(F_{\alpha})=-F_{\alpha} K_{\alpha}[123X, [23XS(K_{\alpha})=K_{\alpha}^{-1}[123X.[133X
  
  [33X[0;0YUsing   [23X\Delta[123X   we   can   make  the  tensor  product  [23XV\otimes  W[123X  of  two
  [23XU_q(\mathfrak{g})[123X-modules  [23XV,W[123X  into  a [23XU_q(\mathfrak{g})[123X-module. The counit
  [23X\varepsilon[123X  yields  a  trivial  [23X1[123X-dimensional [23XU_q(\mathfrak{g})[123X-module. And
  with [23XS[123X we can define a [23XU_q(\mathfrak{g})[123X-module structure on the dual [23XV^*[123X of
  a [23XU_q(\mathfrak{g})[123X-module [23XV[123X, by [23X(u\cdot f)(v) = f(S(u)\cdot v )[123X.[133X
  
  [33X[0;0YThe  Hopf  algebra  structure  given above is not the only one possible. For
  example,  we  can  twist  [23X\Delta,\varepsilon,S[123X  by  an  automorphism,  or an
  anti-automorphism [23Xf[123X. The twisted comultiplication is given by[133X
  
  
  [24X[33X[0;6Y\Delta^f = f\otimes f \circ\Delta\circ f^{-1}.[133X
  
  [124X
  
  [33X[0;0YThe twisted antipode by[133X
  
  
  [24X[33X[0;6YS^f  =  \begin{cases}  f\circ  S\circ  f^{-1}  &  \text{  if  }f\text{ is an
  automorphism}\\  f\circ  S^{-1}\circ  f^{-1}  &  \text{  if  }f\text{  is an
  anti-automorphism.}\end{cases}[133X
  
  [124X
  
  [33X[0;0YAnd  the  twisted  counit  by  [23X\varepsilon^f  = \varepsilon\circ f^{-1}[123X (see
  [Jan96], 3.8).[133X
  
  
  [1X2.4 [33X[0;0YPBW-type bases[133X[101X
  
  [33X[0;0YThe  first  problem one has to deal with when working with [23XU_q(\mathfrak{g})[123X
  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 [23XU_q(\mathfrak{g})\cong U^-\otimes U^0\otimes U^+[123X (as vector
  spaces), where [23XU^-[123X is the subalgebra generated by the [23XF_{\alpha}[123X, [23XU^0[123X is the
  subalgebra  generated  by  the  [23XK_{\alpha}[123X,  and  [23XU^+[123X  is  generated  by the
  [23XE_{\alpha}[123X.  So  a basis of [23XU_q(\mathfrak{g})[123X is formed by all elements [23XFKE[123X,
  where [23XF[123X, [23XK[123X, [23XE[123X run through bases of [23XU^-[123X, [23XU^0[123X, [23XU^+[123X respectively.[133X
  
  [33X[0;0YFinding  a  basis  of  [23XU^0[123X  is easy: it is spanned by all [23XK_{\alpha_1}^{r_1}
  \cdots  K_{\alpha_l}^{r_l}[123X,  where [23Xr_i\in\mathbb{Z}[123X. For [23XU^-[123X, [23XU^+[123X we use the
  so-called    [13XPBW-type[113X    bases.   They   are   defined   as   follows.   For
  [23X\alpha,\beta\in\Delta[123X    we   set   [23Xr_{\beta,\alpha}   =   -\langle   \beta,
  \alpha^{\vee}\rangle[123X.  Then  for  [23X\alpha\in\Delta[123X  we  have the automorphism
  [23XT_{\alpha} : U_q(\mathfrak{g})\to U_q(\mathfrak{g})[123X defined by[133X
  
  
  [24X[33X[0;6Y\begin{aligned}     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{  if  }  \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{     if
  }\alpha\neq\beta,\\ \end{aligned}[133X
  
  [124X
  
  [33X[0;0Y(where  [23XE_{\alpha}^{(k)}  =  E_{\alpha}^k/[k]_{\alpha}![123X,  and  likewise  for
  [23XF_{\alpha}^{(k)}[123X).[133X
  
  [33X[0;0YLet  [23Xw_0=s_{i_1}\cdots  s_{i_t}[123X  be  a  reduced  expression  for the longest
  element   in   the  Weyl  group  [23XW(\Phi)[123X.  For  [23X1\leq  k\leq  t[123X  set  [23XF_k  =
  T_{\alpha_{i_1}}\cdots  T_{\alpha_{i_{k-1}}}(F_{\alpha_{i_k}})[123X,  and  [23XE_k  =
  T_{\alpha_{i_1}}\cdots  T_{\alpha_{i_{k-1}}}(E_{\alpha_{i_k}})[123X.  Then [23XF_k\in
  U^-[123X,  and  [23XE_k\in U^+[123X. Furthermore, the elements [23XF_1^{m_1} \cdots F_t^{m_t}[123X,
  [23XE_1^{n_1}\cdots  E_t^{n_t}[123X  (where  the  [23Xm_i[123X, [23Xn_i[123X are non-negative integers)
  form bases of [23XU^-[123X and [23XU^+[123X respectively.[133X
  
  [33X[0;0YThe  elements  [23XF_{\alpha}[123X and [23XE_{\alpha}[123X are said to have weight [23X-\alpha[123X and
  [23X\alpha[123X  respectively, where [23X\alpha[123X is a simple root. Furthermore, the weight
  of  a  product [23Xab[123X is the sum of the weights of [23Xa[123X and [23Xb[123X. Now elements of [23XU^-[123X,
  [23XU^+[123X  that are linear combinations of elements of the same weight are said to
  be  homogeneous.  It  can  be  shown  that  the  elements  [23XF_k[123X,  and [23XE_k[123X are
  homogeneous    of    weight    [23X-\beta[123X    and   [23X\beta[123X   respectively,   where
  [23X\beta=s_{i_1}\cdots s_{i_{k-1}}(\alpha_{i_k})[123X.[133X
  
  [33X[0;0YIn the sequel we use the notation [23XF_k^{(m)} = F_k^m/[m]_{\alpha_{i_k}}![123X, and
  [23XE_k^{(n)} = E_k^n/[n]_{\alpha_{i_k}}![123X.[133X
  
  
  [1X2.5 [33X[0;0YThe [23X{\mathbb Z}[123X[101X[1X-form of [23XU_q(\mathfrak{g})[123X[101X[1X[133X[101X
  
  [33X[0;0YFor [23X\alpha\in\Delta[123X set[133X
  
  
  [24X[33X[0;6Y\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}}.[133X
  
  [124X
  
  [33X[0;0YThen according to [Lus90], Theorem 6.7 the elements[133X
  
  
  [24X[33X[0;6YF_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)},[133X
  
  [124X
  
  [33X[0;0Y(where  [23Xk_i,m_i,n_i\geq  0[123X, [23X\delta_i=0,1[123X) form a basis of [23XU_q(\mathfrak{g})[123X,
  such  that  the product of any two basis elements is a linear combination of
  basis  elements  with  coefficients  in  [23X\mathbb{Z}[q,q^{-1}][123X. The quantized
  enveloping  algebra  over [23X\mathbb{Z}[q,q^{-1}][123X with this basis is called the
  [23X\mathbb{Z}[123X-form  of  [23XU_q(\mathfrak{g})[123X, and denoted by [23XU_{\mathbb{Z}}[123X. Since
  [23XU_{\mathbb{Z}}[123X  is  defined over [23X\mathbb{Z}[q,q^{-1}][123X we can specialize [23Xq[123X to
  any   nonzero  element  [23X\epsilon[123X  of  a  field  [23XF[123X,  and  obtain  an  algebra
  [23XU_{\epsilon}[123X over [23XF[123X.[133X
  
  [33X[0;0YWe  call  [23Xq\in  \mathbb{Q}(q)[123X,  and  [23X\epsilon \in F[123X the quantum parameter of
  [23XU_q(\mathfrak{g})[123X and [23XU_{\epsilon}[123X respectively.[133X
  
  [33X[0;0YLet  [23X\lambda[123X  be  a  dominant weight, and [23XV(\lambda)[123X the irreducible highest
  weight   module  of  highest  weight  [23X\lambda[123X  over  [23XU_q(\mathfrak{g})[123X.  Let
  [23Xv_{\lambda}\in   V(\lambda)[123X   be   a   fixed  highest  weight  vector.  Then
  [23XU_{\mathbb{Z}}\cdot   v_{\lambda}[123X   is   a   [23XU_{\mathbb{Z}}[123X-module.   So  by
  specializing   [23Xq[123X   to   an   element  [23X\epsilon[123X  of  a  field  [23XF[123X,  we  get  a
  [23XU_{\epsilon}[123X-module.  We  call  it the Weyl module of highest weight [23X\lambda[123X
  over [23XU_{\epsilon}[123X. We note that it is not necessarily irreducible.[133X
  
  
  [1X2.6 [33X[0;0YThe canonical basis[133X[101X
  
  [33X[0;0YAs  in  Section  [14X2.4[114X  we  let  [23XU^-[123X  be  the  subalgebra of [23XU_q(\mathfrak{g})[123X
  generated   by  the  [23XF_{\alpha}[123X  for  [23X\alpha\in\Delta[123X.  In  [Lus0a]  Lusztig
  introduced  a  basis  of [23XU^-[123X with very nice properties, called the [13Xcanonical
  basis[113X.  (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 [Com06].)[133X
  
  [33X[0;0YLet  [23Xw_0=s_{i_1}\cdots  s_{i_t}[123X,  and the elements [23XF_k[123X be as in Section [14X2.4[114X.
  Then, in order to stress the dependency of the monomial[133X
  
  
  [24X[33X[0;6YF_1^{(n_1)}\cdots F_t^{(n_t)}[133X
  
  [124X
  
  [33X[0;0Yon  the  choice  of reduced expression for the longest element in [23XW(\Phi)[123X we
  say that it is a [23Xw_0[123X-monomial.[133X
  
  [33X[0;0YNow  we  let  [23X\overline{\phantom{a}}[123X  be  the automorphism of [23XU^-[123X defined in
  Section  [14X2.2[114X.  Elements  that are invariant under [23X\overline{\phantom{a}}[123X are
  said to be bar-invariant.[133X
  
  [33X[0;0YBy  results  of Lusztig ([Lus93] Theorem 42.1.10, [Lus96], Proposition 8.2),
  there  is  a  unique  basis  [23X{\bf  B}[123X  of [23XU^-[123X with the following properties.
  Firstly, all elements of [23X{\bf B}[123X are bar-invariant. Secondly, for any choice
  of reduced expression [23Xw_0[123X for the longest element in the Weyl group, and any
  element  [23XX\in{\bf  B}[123X  we have that [23XX = x +\sum \zeta_i x_i[123X, where [23Xx,x_i[123X are
  [23Xw_0[123X-monomials, [23Xx\neq x_i[123X for all [23Xi[123X, and [23X\zeta_i\in q\mathbb{Z}[q][123X. The basis
  [23X{\bf  B}[123X  is  called  the  canonical  basis. If we work with a fixed reduced
  expression  for  the  longest  element  in [23XW(\Phi)[123X, and write [23XX\in{\bf B}[123X as
  above, then we say that [23Xx[123X is the [13Xprincipal monomial[113X of [23XX[123X.[133X
  
  [33X[0;0YLet [23X\mathcal{L}[123X be the [23X\mathbb{Z}[q][123X-lattice in [23XU^-[123X spanned by [23X{\bf B}[123X. Then
  [23X\mathcal{L}[123X  is  also  spanned  by  all  [23Xw_0[123X-monomials (where [23Xw_0[123X is a fixed
  reduced   expression   for   the   longest  element  in  [23XW(\Phi)[123X).  Now  let
  [23X\widetilde{w}_0[123X  be  a  second reduced expression for the longest element in
  [23XW(\Phi)[123X.  Let  [23Xx[123X be a [23Xw_0[123X-monomial, and let [23XX[123X be the element of [23X{\bf B}[123X with
  principal    monomial    [23Xx[123X.   Write   [23XX[123X   as   a   linear   combination   of
  [23X\widetilde{w}_0[123X-monomials,  and  let [23X\widetilde{x}[123X be the principal monomial
  of  that expression. Then we write [23X\widetilde{x} = R_{w_0}^{\tilde{w}_0}(x)[123X.
  Note that [23Xx = \widetilde{x} \bmod q\mathcal{L}[123X.[133X
  
  [33X[0;0YNow let [23X\mathcal{B}[123X be the set of all [23Xw_0[123X-monomials [23X\bmod q\mathcal{L}[123X. Then
  [23X\mathcal{B}[123X  is  a  basis of the [23X\mathbb{Z}[123X-module [23X\mathcal{L}/q\mathcal{L}[123X.
  Moreover,   [23X\mathcal{B}[123X   is   independent   of   the  choice  of  [23Xw_0[123X.  Let
  [23X\alpha\in\Delta[123X,  and  let  [23X\widetilde{w}_0[123X  be a reduced expression for the
  longest   element  in  [23XW(\Phi)[123X,  starting  with  [23Xs_{\alpha}[123X.  The  Kashiwara
  operators   [23X\widetilde{F}_{   \alpha}   :   \mathcal{B}\to  \mathcal{B}[123X  and
  [23X\widetilde{E}_{\alpha}  : \mathcal{B}\to \mathcal{B}\cup\{0\}[123X are defined as
  follows.  Let [23Xb\in\mathcal{B}[123X and let [23Xx=[123X be the [23Xw_0[123X-monomial such that [23Xb = x
  \bmod  q\mathcal{L}[123X.  Set  [23X\widetilde{x}  =  R_{w_0}^ {\tilde{w}_0}(x)[123X. Then
  [23X\widetilde{x}'[123X    is    the    [23X\widetilde{w}_0[123X-monomial   constructed   from
  [23X\widetilde{x}[123X  by  increasing its first exponent by [23X1[123X (the first exponent is
  [23Xn_1[123X   if   we   write   [23X\widetilde{x}=F_1^{(n_1)}\cdots  F_t^{(n_t)}[123X).  Then
  [23X\widetilde{F}_{  \alpha}(b)  =  R_{\tilde{w}_0}^{w_0}(\widetilde{x}')  \bmod
  q\mathcal{L}[123X.  For  [23X\widetilde{E}_{\alpha}[123X  we  let  [23X\widetilde{x}'[123X  be  the
  [23X\widetilde{w}_0[123X-monomial  constructed  from  [23X\widetilde{x}[123X by decreasing its
  first    exponent    by    [23X1[123X,   if   this   exponent   is   [23X\geq   1[123X.   Then
  [23X\widetilde{E}_{\alpha}(b)    =    R_{\tilde{w}_0}^{w_0}(\widetilde{x}')\bmod
  q\mathcal{L}[123X.   Furthermore,   [23X\widetilde{E}_{\alpha}(b)  =0[123X  if  the  first
  exponent  of  [23X\widetilde{x}[123X  is [23X0[123X. It can be shown that this definition does
  not  depend  on  the  choice  of  [23Xw_0[123X,  [23X\widetilde{w}_0[123X. Furthermore we have
  [23X\widetilde{F}_{\alpha}\widetilde{E}_{\alpha}(b)=b[123X,                        if
  [23X\widetilde{E}_{\alpha}(b)\neq  0[123X,  and [23X\widetilde{E}_{\alpha} \widetilde{F}_
  {\alpha}(b)=b[123X for all [23Xb\in \mathcal{B}[123X.[133X
  
  [33X[0;0YLet  [23Xw_0=s_{i_1}\cdots s_{i_t}[123X be a fixed reduced expression for the longest
  element  in  [23XW(\Phi)[123X.  For  [23Xb\in\mathcal{B}[123X we define a sequence of elements
  [23Xb_k\in\mathcal{B}[123X  for  [23X0\leq  k\leq  t[123X,  and a sequence of integers [23Xn_k[123X for
  [23X1\leq k\leq t[123X as follows. We set [23Xb_0=b[123X, and if [23Xb_{k-1}[123X is defined we let [23Xn_k[123X
  be  maximal  such  that  [23X\widetilde{E}_{\alpha_{i_k}}^ {n_k}(b_{k-1})\neq 0[123X.
  Also  we  set  [23Xb_k  = \widetilde{E}_{\alpha_{i_k}}^{n_k} (b_{k-1})[123X. Then the
  sequence  [23X(n_1,\ldots,n_t)[123X is called the [13Xstring[113X of [23Xb\in\mathcal{B}[123X (relative
  to   [23Xw_0[123X).   We   note   that   [23Xb=\widetilde{F}_  {\alpha_{i_1}}^{n_1}\cdots
  \widetilde{F}_{\alpha_{i_t}}^  {n_t}(1)[123X. The set of all strings parametrizes
  the elements of [23X\mathcal{B}[123X, and hence of [23X{\bf B}[123X.[133X
  
  [33X[0;0YNow  let  [23XV(\lambda)[123X be a highest-weight module over [23XU_q(\mathfrak{g})[123X, with
  highest  weight  [23X\lambda[123X.  Let [23Xv_{\lambda}[123X be a fixed highest weight vector.
  Then  [23X{\bf B}_{\lambda} = \{ X\cdot v_{\lambda}\mid X\in {\bf B}\} \setminus
  \{0\}[123X is a basis of [23XV(\lambda)[123X, called the [13Xcanonical basis[113X, or [13Xcrystal basis[113X
  of  [23XV(\lambda)[123X.  Let  [23X\mathcal{L}(\lambda)[123X  be  the [23X\mathbb{Z}[q][123X-lattice in
  [23XV(\lambda)[123X  spanned  by  [23X{\bf B}_{\lambda}[123X. We let [23X\mathcal{B}({\lambda})[123X be
  the  set  of all [23Xx\cdot v_{\lambda}\bmod q\mathcal{L}(\lambda)[123X, where [23Xx[123X runs
  through  all  [23Xw_0[123X-monomials, such that [23XX\cdot v_{\lambda} \neq 0[123X, where [23XX\in
  {\bf  B}[123X  is  the  element  with  principal  monomial  [23Xx[123X. Then the Kashiwara
  operators     are    also    viewed    as    maps    [23X\mathcal{B}(\lambda)\to
  \mathcal{B}(\lambda)\cup\{0\}[123X,   in   the   following   way.   Let  [23Xb=x\cdot
  v_{\lambda}\bmod      q\mathcal{L}(\lambda)[123X     be     an     element     of
  [23X\mathcal{B}(\lambda)[123X,  and  let  [23Xb'=x\bmod q\mathcal{L}[123X be the corresponding
  element   of   [23X\mathcal{B}[123X.   Let   [23Xy[123X   be   the   [23Xw_0[123X-monomial   such  that
  [23X\widetilde{F}_{\alpha}(b')=y\bmod    q\mathcal{L}[123X.    Then   [23X\widetilde{F}_{
  \alpha}(b) = y\cdot v_{\lambda} \bmod q\mathcal{L}(\lambda)[123X. The description
  of  [23X\widetilde{E}_{\alpha}[123X  is analogous. (In [Jan96], Chapter 9 a different
  definition is given; however, by [Jan96], Proposition 10.9, Lemma 10.13, the
  two definitions agree).[133X
  
  [33X[0;0YThe  set [23X\mathcal{B}(\lambda)[123X has [23X\dim V(\lambda)[123X elements. We let [23X\Gamma[123X be
  the coloured directed graph defined as follows. The points of [23X\Gamma[123X are the
  elements  of  [23X\mathcal{B}(\lambda)[123X,  and  there  is  an  arrow  with  colour
  [23X\alpha\in\Delta[123X        connecting        [23Xb,b'\in       \mathcal{B}[123X,       if
  [23X\widetilde{F}_{\alpha}(b)=b'[123X.  The  graph [23X\Gamma[123X is called the [13Xcrystal graph[113X
  of [23XV(\lambda)[123X.[133X
  
  
  [1X2.7 [33X[0;0YThe path model[133X[101X
  
  [33X[0;0YIn this section we recall some basic facts on Littelmann's path model.[133X
  
  [33X[0;0YFrom  Section  [14X2.2[114X  we  recall  that  [23XP[123X  denotes  the  weight  lattice.  Let
  [23XP_{\mathbb{R}}[123X  be the vector space over [23X\mathbb{R}[123X spanned by [23XP[123X. Let [23X\Pi[123X be
  the  set  of all piecewise linear paths [23X\xi : [0,1]\to P_{\mathbb{R}} [123X, such
  that  [23X\xi(0)=0[123X. For [23X\alpha\in\Delta[123X Littelmann defined operators [23Xf_{\alpha},
  e_{\alpha} : \Pi \to \Pi\cup \{0\}[123X. Let [23X\lambda[123X be a dominant weight and let
  [23X\xi_{\lambda}[123X be the path joining [23X\lambda[123X and the origin by a straight line.
  Let   [23X\Pi_{\lambda}[123X   be  the  set  of  all  nonzero  [23Xf_{\alpha_{i_1}}\cdots
  f_{\alpha_{i_m}}(\xi_{\lambda})[123X for [23Xm\geq 0[123X. Then [23X\xi(1)\in P[123X for all [23X\xi\in
  \Pi_{\lambda}[123X.  Let  [23X\mu\in  P[123X  be  a  weight,  and  let  [23XV(\lambda)[123X  be the
  highest-weight  module  over  [23XU_q(\mathfrak{g})[123X of highest weight [23X\lambda[123X. A
  theorem  of  Littelmann states that the number of paths [23X\xi\in \Pi_{\lambda}[123X
  such that [23X\xi(1)=\mu[123X is equal to the dimension of the weight space of weight
  [23X\mu[123X in [23XV(\lambda)[123X ([Lit95], Theorem 9.1).[133X
  
  [33X[0;0YAll paths appearing in [23X\Pi_{\lambda}[123X are so-called Lakshmibai-Seshadri paths
  (LS-paths  for  short).  They  are  defined  as follows. Let [23X\leq[123X denote the
  Bruhat  order  on [23XW(\Phi)[123X. For [23X\mu,\nu\in W(\Phi)\cdot \lambda[123X (the orbit of
  [23X\lambda[123X  under  the action of [23XW(\Phi)[123X), write [23X\mu\leq \nu[123X if [23X\tau\leq\sigma[123X,
  where  [23X\tau,\sigma\in W(\Phi)[123X are the unique elements of minimal length such
  that  [23X\tau(\lambda)=\mu[123X,  [23X\sigma(\lambda)= \nu[123X. Now a rational path of shape
  [23X\lambda[123X is a pair [23X\pi=(\nu,a)[123X, where [23X\nu=(\nu_1,\ldots, \nu_s)[123X is a sequence
  of  elements  of  [23XW(\Phi)\cdot  \lambda[123X,  such  that  [23X\nu_i>  \nu_{i+1}[123X  and
  [23Xa=(a_0=0,  a_1,  \cdots  ,a_s=1)[123X  is  a  sequence of rationals such that [23Xa_i
  <a_{i+1}[123X. The path [23X\pi[123X corresponding to these sequences is given by[133X
  
  
  [24X[33X[0;6Y\pi(t) =\sum_{j=1}^{r-1} (a_j-a_{j-1})\nu_j + \nu_r(t-a_{r-1})[133X
  
  [124X
  
  [33X[0;0Yfor  [23Xa_{r-1}\leq  t\leq  a_r[123X.  Now an LS-path of shape [23X\lambda[123X is a rational
  path  satisfying  a certain integrality condition (see [Lit94], [Lit95]). We
  note  that  the path [23X\xi_{\lambda} = ( (\lambda), (0,1) )[123X joining the origin
  and [23X\lambda[123X by a straight line is an LS-path.[133X
  
  [33X[0;0YNow from [Lit94], [Lit95] we transcribe the following:[133X
  
  [31X1[131X   [33X[0;6YLet  [23X\pi[123X be an LS-path. Then [23Xf_{\alpha}\pi[123X is an LS-path or [23X0[123X; and the
        same holds for [23Xe_{\alpha}\pi[123X.[133X
  
  [31X2[131X   [33X[0;6YThe   action   of   [23Xf_{\alpha},e_{\alpha}[123X   can  easily  be  described
        combinatorially (see [Lit94]).[133X
  
  [31X3[131X   [33X[0;6YThe endpoint of an LS-path is an integral weight.[133X
  
  [31X4[131X   [33X[0;6YLet  [23X\pi=(\nu,a)[123X be an LS-path. Then by [23X\phi(\pi)[123X we denote the unique
        element   [23X\sigma[123X   of   [23XW(\Phi)[123X   of   shortest   length   such   that
        [23X\sigma(\lambda)=\nu_1[123X.[133X
  
  [33X[0;0YLet  [23X\lambda[123X  be  a dominant weight. Then we define a labeled directed graph
  [23X\Gamma[123X  as  follows.  The  points  of [23X\Gamma[123X are the paths in [23X\Pi_{\lambda}[123X.
  There  is  an  edge  with  label  [23X\alpha\in\Delta[123X  from  [23X\pi_1[123X  to  [23X\pi_2[123X if
  [23Xf_{\alpha}\pi_1  =\pi_2[123X.  Now  by [Kas96] this graph [23X\Gamma[123X is isomorphic to
  the  crystal graph of the highest-weight module with highest weight [23X\lambda[123X.
  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   [23Xf_{\alpha_{i_1}}\cdots   f_{\alpha_{i_r}}\xi_{\lambda}   =0[123X  is
  equivalent      to     [23X\widetilde{F}_{\alpha_{i_1}}\cdots     \widetilde{F}_
  {\alpha_{i_r}}v_{\lambda}=0[123X,  where  [23Xv_{\lambda}\in  V(\lambda)[123X is a highest
  weight   vector   (or  rather  the  image  of  it  in  [23X\mathcal{L}(\lambda)/
  q\mathcal{L}  (\lambda)[123X), and the [23X\widetilde{F}_{\alpha_k}[123X are the Kashiwara
  operators on [23X\mathcal{B}(\lambda)[123X (see Section [14X2.6[114X).[133X
  
  
  [1X2.8 [33X[0;0YNotes[133X[101X
  
  [33X[0;0YI  refer  to [Hum90] for more information on Weyl groups, and to [Ste01] for
  an  overview of algorithms for computing with weights, Weyl groups and their
  elements.[133X
  
  [33X[0;0YFor general introductions into the theory of quantized enveloping algebras I
  refer  to  [Car98], [Jan96] (from where most of the material of this chapter
  is  taken),  [Lus92],  [Lus93], [Ros91]. I refer to the papers by Littelmann
  ([Lit94],  [Lit95],  [Lit98])  for  more  information on the path model. The
  paper by Kashiwara ([Kas96]) contains a proof of the connection between path
  operators and Kashiwara operators.[133X
  
  [33X[0;0YFinally,  I  refer  to  [Gra01]  (on computing with PBW-type bases), [Gra02]
  (computation  of  elements of the canonical basis) for an account of some of
  the algorithms used in [5XQuaGroup[105X.[133X
  
