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<title>GAP (quagroup) - Chapter 2: Background</title>
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<h3>2. Background</h3>
<p>In this chapter we summarize some of the theoretical concepts with which <strong class="pkg">QuaGroup</strong> 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.</p>
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<h4>2.1 Gaussian Binomials</h4>
<p>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 <em> Gaussian integer </em> corresponding to $n$. The <em> Gaussian factorial </em> $[n]!$ is defined by $$ [0]! = 1, ~ [n]! = [n][n-1]\cdots [1], \text{ for } n>0.$$ Finally, the <em> Gaussian binomial </em> is $$ \begin{bmatrix} n \\ k \end{bmatrix} = \frac{[n]!}{[k]![n-k]!}.$$</p>
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<h4>2.2 Quantized enveloping algebras</h4>
<p>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 $\overline{F_{\alpha}}=F_{\alpha}$, $\overline{K_{\alpha}}= K_{\alpha}^{-1}$, $\overline{E_{\alpha}}=E_{\alpha}$, $\overline{q}=q^{-1}$.</p>
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<h4>2.3 Representations of $U_q(\mathfrak{g})$ </h4>
<p>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 <a href="chapBib.html#biBJ96">[J96]</a>, 3.8).</p>
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<h4>2.4 PBW-type bases </h4>
<p>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</p>
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<h4>2.5 The ${\mathbb Z}$-form of $U_q(\mathfrak{g})$ </h4>
<p>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 <a href="chapBib.html#biBL90">[L90]</a>, 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.</p>
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<h4>2.6 The canonical basis </h4>
<p>As in Section <a href="chap2.html#s4ss0"><b>2.4</b></a> we let $U^-$ be the subalgebra of $U_q(\mathfrak{g})$ generated by the $F_{\alpha}$ for $\alpha\in\Delta$. In <a href="chapBib.html#biBL0a">[L0a]</a> 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 <a href="chapBib.html#biBC06">[C06]</a>.) \par Let $w_0=s_{i_1}\cdots s_{i_t}$, and the elements $F_k$ be as in Section <a href="chap2.html#s4ss0"><b>2.4</b></a>. 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 <a href="chap2.html#s2ss0"><b>2.2</b></a>. Elements that are invariant under $\overline{\phantom{a}}$ are said to be bar-invariant. \par By results of Lusztig (<a href="chapBib.html#biBL93">[L93]</a> Theorem 42.1.10, <a href="chapBib.html#biBL96">[L96]</a>, 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 <a href="chapBib.html#biBJ96">[J96]</a>, Chapter 9 a different definition is given; however, by <a href="chapBib.html#biBJ96">[J96]</a>, 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 $V(\lambda)$.</p>
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<h4>2.7 The path model </h4>
<p>In this section we recall some basic facts on Littelmann's path model. \par From Section <a href="chap2.html#s2ss0"><b>2.2</b></a> 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 $V(\lambda)$ (<a href="chapBib.html#biBL95">[L95]</a>, 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 <a href="chapBib.html#biBL94">[L94]</a>, <a href="chapBib.html#biBL95">[L95]</a>). 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 <a href="chapBib.html#biBL94">[L94]</a>, <a href="chapBib.html#biBL95">[L95]</a> 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 <a href="chapBib.html#biBL94">[L94]</a>). \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 <a href="chapBib.html#biBK96">[K96]</a> 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 <a href="chap2.html#s6ss0"><b>2.6</b></a>).</p>
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<h4>2.8 Notes</h4>
<p>I refer to <a href="chapBib.html#biBH90">[H90]</a> for more information on Weyl groups, and to <a href="chapBib.html#biBS01">[S01]</a> 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 <a href="chapBib.html#biBC98">[C98]</a>, <a href="chapBib.html#biBJ96">[J96]</a> (from where most of the material of this chapter is taken), <a href="chapBib.html#biBL92">[L92]</a>, <a href="chapBib.html#biBL93">[L93]</a>, <a href="chapBib.html#biBR91">[R91]</a>. I refer to the papers by Littelmann (<a href="chapBib.html#biBL94">[L94]</a>, <a href="chapBib.html#biBL95">[L95]</a>, <a href="chapBib.html#biBL98">[L98]</a>) for more information on the path model. The paper by Kashiwara (<a href="chapBib.html#biBK96">[K96]</a>) contains a proof of the connection between path operators and Kashiwara operators.\par Finally, I refer to <a href="chapBib.html#biBG01">[G01]</a> (on computing with PBW-type bases), <a href="chapBib.html#biBG02">[G02]</a> (computation of elements of the canonical basis) for an account of some of the algorithms used in <strong class="pkg">QuaGroup</strong>.</p>
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