# Background

## Gaussian Binomials

Let $v$ be an indeterminate over ${\mathbb{Q}}$. For a positive integer $n$ we set

$$
[n]_v = v^{n-1}+v^{n-3}+\cdots + v^{-n+3}+v^{-n+1}.
$$

We say that $[n]_v$ is the *Gaussian integer* corresponding to $n$. The *Gaussian factorial* $[n]_v!$ is defined by

$$
[0]_v! = 1, ~ [n]_v! = [n]_v[n-1]_v\cdots [1]_v {\rm ~for~ } n >0.
$$

Finally, the *Gaussian binomial* is

$$
{n \choose k}_v  = {[n]! \over [k]![n-k]!}.
$$

## Quantized Enveloping Algebras

Let $L$ be a semisimple Lie algebra with root system $\Phi$. By $\Delta=\{\alpha_1,\ldots, \alpha_l \}$ we denote a fixed set of simple roots 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.

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 $s_{\alpha}(\beta) = \beta - \langle\beta, \alpha^{\vee}\rangle \alpha$).

We work over the field ${\mathbb{Q}}(q)$. For $\alpha\in\Phi$ we set

$$
q_{\alpha} = q^{{(\alpha,\alpha)\over 2}},
$$

and for a non-negative integer $n$, $[n]_{\alpha}= [n]_{v=q_{\alpha}}$; $[n]_{\alpha}!$ and ${n \choose k }_{\alpha}$ are defined analogously.

The quantized enveloping algebra $U_q(L)$ 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

$$
\begin{aligned}K_{\alpha}K_{\alpha}^{-1} &= K_{\alpha}^{-1}K_{\alpha} = 1,~
K_{\alpha}K_{\beta} = K_{\beta}K_{\alpha}\\
E_{\beta} K_{\alpha} &= q^{-(\alpha,\beta)}K_{\alpha} E_{\beta}\\
K_{\alpha} F_{\beta} &= q^{-(\alpha,\beta)}F_{\beta}K_{\alpha}\\
E_{\alpha} F_{\beta} &= F_{\beta}E_{\alpha} +\delta_{\alpha,\beta}
{K_{\alpha}-K_{\alpha}^{-1} \over q_{\alpha}-q_{\alpha}^{-1}}\end{aligned}
$$

together with, for $\alpha\neq \beta\in\Delta$,

$$
\begin{aligned}\sum_{k=0}^{1-\langle \beta,\alpha^{\vee}\rangle }
(-1)^k {1-\langle \beta,\alpha^{\vee}\rangle \choose k}_{\alpha}
E_{\alpha}^{1-\langle \beta,\alpha^{\vee}\rangle-k}
E_{\beta} E_{\alpha}^k =0 & \\
\sum_{k=0}^{1-\langle \beta,\alpha^{\vee}\rangle } (-1)^k
{1-\langle \beta,\alpha^{\vee}\rangle \choose k}_{\alpha}
F_{\alpha}^{1-\langle \beta,\alpha^{\vee}\rangle-k}
F_{\beta} F_{\alpha}^k =0 &.\end{aligned}
$$

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 $\tau^2=1$.

If the Dynkin diagram of $\Phi$ admits a diagram automorphism $\pi$, then $\pi$ induces an automorphism of $U_q(L)$ in the obvious way ($\pi$ is a permutation of the simple roots; we permute the $F_{\alpha}$, $E_{\alpha}$, $K_{\alpha}^{\pm 1}$ accordingly).

Now we view $U_q(L)$ as an algebra over ${\mathbb{Q}}$, and we let $\overline{\phantom{A}} : U_q(L)\to U_q(L)$ be the automorphism defined by $\overline{F_{\alpha}}=F_{\alpha}$, $\overline{K_{\alpha}}= K_{\alpha}^{-1}$, $\overline{E_{\alpha}}=E_{\alpha}$, $\overline{q}=q^{-1}$. This map is called the *bar-automorphism*.

## Representations of $U_q(L)$

Let $\lambda\in P$ be a dominant weight. Then there is a unique irreducible highest-weight module over $U_q(L)$ with highest weight $\lambda$. We denote it by $V(\lambda)$. It has the same character as the irreducible highest-weight module over $L$ with highest weight $\lambda$. Furthermore, every finite-dimensional $U_q(L)$-module is a direct sum of irreducible highest-weight modules. In [[Graaf, 2003/04](../../references.md#cite-graaf04)] a few algorithms for constructing $V(\lambda)$ are given. In the Magma implementation the algorithm based on Gröbner bases is used.

It is well-known that $U_q(L)$ is a Hopf algebra. The comultiplication $\Delta : U_q(L)\to U_q(L) \otimes U_q(L)$ is defined by

$$
\begin{aligned}\Delta(E_{\alpha}) &= E_{\alpha}\otimes 1 + K_{\alpha}\otimes E_{\alpha}\\
\Delta(F_{\alpha}) &= F_{\alpha}\otimes K_{\alpha}^{-1} +
1\otimes F_{\alpha}\\
\Delta(K_{\alpha}) &= K_{\alpha}\otimes K_{\alpha}.\end{aligned}
$$

(Note that we use the same symbol ($\Delta$) to denote a set of simple roots of $\Phi$; of course this does not cause confusion.) The counit $\varepsilon : U_q(L) \to {\mathbb{Q}}(q)$ is a homomorphism defined by $\varepsilon(E_{\alpha})=\varepsilon(F_{\alpha})=0$, $\varepsilon(K_{\alpha}) =1$. Finally, the antipode $S: U_q(L)\to U_q(L)$ is an anti-automorphism given by $S(E_{\alpha})=-K_{\alpha}^{-1}E_{\alpha}$, $S(F_{\alpha})=-F_{\alpha} K_{\alpha}$, $S(K_{\alpha})=K_{\alpha}^{-1}$.

Using $\Delta$ we can make the tensor product $V\otimes W$ of two $U_q(L)$-modules $V,W$ into a $U_q(L)$-module. The counit $\varepsilon$ yields a trivial $1$-dimensional $U_q(L)$-module. And with $S$ we can define a $U_q(L)$-module structure on the dual $V^*$ of a $U_q(L)$-module $V$, by $(u\cdot f)(v) = f(S(u)\cdot v)$.

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

$$
S^f = f\circ S\circ f^{-1},
$$

if $f$ is an automorphism, and

$$
S^f = f\circ S^{-1}\circ f^{-1},
$$

if $f$ is an anti-automorphism. The twisted counit is given by $\varepsilon^f = \varepsilon\circ f^{-1}$.

## PBW-type Bases

The first problem one has to deal with when working with $U_q(L)$ 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 $U_q(L)\cong U^-\otimes U^0\otimes U^+$ (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 $U_q(L)$ is formed by all elements $FKE$, where $F$, $K$, $E$ run through bases of $U^-$, $U^0$, $U^+$ respectively.

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 $U^-$ and $U^+$ we use the so-called *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 $T_{\alpha} : U_q(L)\to U_q(L)$ defined by

$$
\begin{aligned}T_{\alpha}(E_{\alpha}) &= -F_{\alpha}K_{\alpha}\\
T_{\alpha}(E_{\beta}) &= \sum_{i=0}^{r_{\beta,\alpha}}
(-1)^i q_{\alpha}^{-i} E_{\alpha}^{(r_{\beta,\alpha}-i)}E_{\beta}
E_{\alpha}^{(i)},\ (\alpha\neq\beta)\\
T_{\alpha}(K_{\beta}) &= K_{\beta}K_{\alpha}^{r_{\beta,\alpha}}\\
T_{\alpha}(F_{\alpha}) &= -K_{\alpha}^{-1} E_{\alpha}\\
T_{\alpha}(F_{\beta}) &= \sum_{i=0}^{r_{\beta,\alpha}}
(-1)^i q_{\alpha}^{i} F_{\alpha}^{(i)}F_{\beta}F_{\alpha}^
{(r_{\beta,\alpha}-i)},\ (\alpha\neq\beta)\end{aligned}
$$

(where $E_{\alpha}^{(k)} = E_{\alpha}^k/[k]_{\alpha}!$, and likewise for $F_{\alpha}^{(k)}$).

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.

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 $\beta=s_{i_1}\cdots s_{i_{k-1}}(\alpha_{i_k})$.

In the following we use the notation $F_k^{(m)} = F_k^m/[m]_{\alpha_{i_k}}!$, and $E_k^{(n)} = E_k^n/[n]_{\alpha_{i_k}}!$.

We refer to [[Graaf, 2001](../../references.md#cite-graaf01)] for an account of algorithms for expressing the product of two elements of a PBW-type basis as a linear combination of such elements. These algorithms are implemented in Magma.

## The ${\mathbb{Z}}$-form of $U_q(L)$

For $\alpha\in\Delta$ set

$$
{K_{\alpha} \choose n } = \prod_{i=1}^n
{q_{\alpha}^{-i+1}K_{\alpha} - q_{\alpha}^{i-1} K_{\alpha}^{-1}\over
q_{\alpha}^i-q_{\alpha}^{-i}}.
$$

Then according to [[Lusztig, 1990](../../references.md#cite-lusztig90)], Theorem 6.7 the elements

$$
F_1^{(k_1)}\cdots F_t^{(k_t)} K_{\alpha_1}^{\delta_1}
{K_{\alpha_1} \choose m_1 }
\cdots K_{\alpha_l}^{\delta_l}
{K_{\alpha_l} \choose m_l }
E_1^{(n_1)}\cdots E_t^{(n_t)},
$$

(where $k_i,m_i,n_i\geq 0$, $\delta_i=0,1$) form a basis of $U_q(L)$, 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 $U_q(L)$, 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 $F$. In particular, if we take $\epsilon = 1$, then we obtain an algebra $U_1$ over ${\mathbb{Q}}$. Let $I$ be the ideal of $U_1$ generated by $K_{\alpha_1}-1,\ldots, K_{\alpha_l}-1$. Then $U_1/I$ is isomorphic to the universal enveloping algebra $U(L)$ of $L$. Also, the homomorphism $U_q(L) \to U(L)$ maps the basis above onto an integral basis of $U(L)$ ([[Lusztig, 1990](../../references.md#cite-lusztig90)]).

We call $q\in {\mathbb{Q}}(q)$, and $\epsilon \in F$ the quantum parameter of $U_q(L)$ and $U_{\epsilon}$ respectively.

## The Canonical Basis

As in Section [PBW-type Bases](#pbwbases) we let $U^-$ be the subalgebra of $U_q(L)$ generated by the $F_{\alpha}$ for $\alpha\in\Delta$. Kashiwara and Lusztig have (independently) given constructions of a basis of $U^-$ with very nice properties, called the *canonical basis*.

Let $w_0=s_{i_1}\cdots s_{i_t}$, and the elements $F_k$ be as in Section [PBW-type Bases](#pbwbases). Then, in order to stress the dependency of the monomial

$$
F_1^{(n_1)}\cdots F_t^{(n_t)}
$$

on the choice of reduced expression for the longest element in $W(\Phi)$ we say that it is a $w_0$-monomial. The integer $n_1$ is called its first exponent.

Now we let $\overline{\phantom{a}}$ be the automorphism of $U^-$ defined in Section [Quantized Enveloping Algebras](#quantizeduea). Elements that are invariant under $\overline{\phantom{a}}$ are said to be bar-invariant.

By results of Lusztig ([[Lusztig, 1993](../../references.md#cite-lusztig93)], Theorem 42.1.10, [[Lusztig, 1996](../../references.md#cite-lusztig96)], 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 $X = x +\sum_i \zeta_i x_i$, 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 *principal monomial* of $X$.

Let ${\cal L}$ be the ${\mathbb{Z}}[q]$-lattice in $U^-$ spanned by **B**. Then ${\cal L}$ 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 **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 $x = \widetilde{x} \bmod q{\cal L}$.

Now let ${\cal B}$ be the set of all $x \bmod q{\cal L}$, where $x$ runs through the set of $w_0$-monomials. Then ${\cal B}$ is a basis of the ${\mathbb{Z}}$-module ${\cal L}/q{\cal L}$. Moreover, ${\cal B}$ 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 $\widetilde{F}_{ \alpha} : {\cal B}\to {\cal B}$ and $\widetilde{E}_{\alpha} : {\cal B}\to {\cal B}\cup\{0\}$ are defined as follows. Let $b\in{\cal B}$ and let $x$ be the $w_0$-monomial such that $b = x \bmod q{\cal L}$. Set $\widetilde{x} = R_{w_0}^ {\tilde{w}_0}(x)$. Let $\widetilde{x}'$ be the $\widetilde{w}_0$-monomial constructed from $\widetilde{x}$ by increasing its first exponent by $1$. Then $\widetilde{F}_{ \alpha}(b) = R_{\tilde{w}_0}^{w_0}(\widetilde{x}') \bmod q{\cal L}$. 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 $\widetilde{E}_{\alpha}(b) = R_{\tilde{w}_0}^{w_0}(\widetilde{x}')\bmod q{\cal L}$. 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 $b\in {\cal B}$.

Now let $V(\lambda)$ be a highest-weight module over $U_q(L)$, 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 *canonical basis* of $V(\lambda)$. Let ${\cal L}(\lambda)$ be the ${\mathbb{Z}}[q]$-lattice in $V(\lambda)$ spanned by ${\bf B}_{\lambda}$. We let ${\cal B}({\lambda})$ be the set of all $x\cdot v_{\lambda}\bmod q{\cal L}(\lambda)$, 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 ${\cal B}(\lambda)\to {\cal B}(\lambda)\cup\{0\}$, in the following way. Let $b=x\cdot v_{\lambda}\bmod q{\cal L}(\lambda)$ be an element of ${\cal B}(\lambda)$, and let $b'=x\bmod q{\cal L}$ be the corresponding element of ${\cal B}$. Let $y$ be the $w_0$-monomial such that $\widetilde{F}_{\alpha}(b')=y\bmod q{\cal L}$. Then $\widetilde{F}_{ \alpha}(b) = y\cdot v_{\lambda} \bmod q{\cal L}(\lambda)$. The description of $\widetilde{E}_{\alpha}$ is analogous. (In [[Jantzen, 1996](../../references.md#cite-jantzen96)], Chapter 9 a different definition is given; however, by [[Jantzen, 1996](../../references.md#cite-jantzen96)], Proposition 10.9, Lemma 10.13, the two definitions agree).

The set ${\cal B}(\lambda)$ has $\dim V(\lambda)$ elements. We let $\Gamma$ be the coloured directed graph defined as follows. The points of $\Gamma$ are the elements of ${\cal B}(\lambda)$, and there is an arrow with colour $\alpha\in\Delta$ connecting $b,b'\in {\cal B}$, if $\widetilde{F}_{\alpha}(b)=b'$. The graph $\Gamma$ is called the *crystal graph* of $V(\lambda)$.

In [[Graaf, 2002](../../references.md#cite-graaf02)] algorithms are given for computing the action of the Kashiwara operators on ${\cal B}$ (without computing **B** first), and for computing elements of **B**.

## The Path Model

In this section we recall some basic facts on Littelmann’s path model.

From Section [Quantized Enveloping Algebras](#quantizeduea) we recall that $P$ denotes the weight lattice. Let $P_{R}$ be the vector space over $R$ spanned by $P$. Let $\Pi$ be the set of all piecewise linear paths $\xi : [0,1]\to P_R$, such that $\xi(0)=0$. For $\alpha\in\Delta$ Littelmann defined *path 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 $U_q(L)$ 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 $V(\lambda)$ ([[Littelmann, 1995](../../references.md#cite-littelmann95)], Theorem 9.1).

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 [[Littelmann, 1994](../../references.md#cite-littelmann94)], [[Littelmann, 1995](../../references.md#cite-littelmann95)]). We note that the path $\xi_{\lambda} = ((\lambda), (0,1))$ joining the origin and $\lambda$ by a straight line is an LS-path of shape $\lambda$. Furthermore, all paths obtained from $\xi_{\lambda}$ by applying the path operators are LS-paths of shape $\lambda$.

From [[Littelmann, 1994](../../references.md#cite-littelmann94)], [[Littelmann, 1995](../../references.md#cite-littelmann95)]) we transcribe the following:

**(a)**
Let $\pi$ be an LS-path. Then $f_{\alpha}\pi$ is an LS-path or $0$; and the same holds for $e_{\alpha}\pi$.

**(b)**
The action of $f_{\alpha},e_{\alpha}$ can easily be described combinatorially (see [[Littelmann, 1994](../../references.md#cite-littelmann94)]).

**(c)**
The endpoint of an LS-path is an integral weight.

**(d)**
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 $\sigma(\lambda)=\nu_1$.

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 [[Kashiwara, 1996](../../references.md#cite-kashiwara96)] 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 ${\cal L}(\lambda)/ q{\cal L} (\lambda)$), and the $\widetilde{F}_{\alpha_k}$ are the Kashiwara operators on ${\cal B}(\lambda)$ (see Section [The Canonical Basis](#canbas)).