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<div class="ChapSects"><a href="chap2.html#X84AF2F1D7D4E7284">2 Background</a>
<div class="ContSect"> <a href="chap2.html#X7AAC838B7CEB5E54">2.1 Gaussian Binomials</a>

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<div class="ContSect"> <a href="chap2.html#X81394E207F6AA6CF">2.2 Quantized enveloping algebras</a>

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<div class="ContSect"> <a href="chap2.html#X7DCC28DE7C626D33">2.3 Representations of U_q(\mathfrak{g}) </a>

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<div class="ContSect"> <a href="chap2.html#X83E7F39F7D16793B">2.4 PBW-type bases </a>

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<div class="ContSect"> <a href="chap2.html#X798D979F7A53E05D">2.5 The {\mathbb Z}-form of U_q(\mathfrak{g}) </a>

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<div class="ContSect"> <a href="chap2.html#X78BE3EB980F0A295">2.6 The canonical basis </a>

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<div class="ContSect"> <a href="chap2.html#X83BBB99685FB9FB7">2.7 The path model </a>

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<div class="ContSect"> <a href="chap2.html#X7BB6B9777D54FA07">2.8 Notes</a>

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<h3>2 Background</h3>

In this chapter we summarize some of the theoretical concepts with which QuaGroup operates.

<a id="X7AAC838B7CEB5E54" name="X7AAC838B7CEB5E54"></a>

<h4>2.1 Gaussian Binomials</h4>

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 Gaussian integer corresponding to n. The Gaussian factorial [n]! is defined by

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

Finally, the Gaussian binomial is

 \begin{bmatrix} n \\ k \end{bmatrix} = \frac{[n]!}{[k]![n-k]!}.

<a id="X81394E207F6AA6CF" name="X81394E207F6AA6CF"></a>

<h4>2.2 Quantized enveloping algebras</h4>

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.

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^{\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.

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{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}


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


\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}


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(\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).

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}.

<a id="X7DCC28DE7C626D33" name="X7DCC28DE7C626D33"></a>

<h4>2.3 Representations of U_q(\mathfrak{g}) </h4>

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.

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{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 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}.

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 ).

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\text{ is an
automorphism}\\ f\circ S^{-1}\circ f^{-1}
& \text{ if }f\text{ is an anti-automorphism.}\end{cases}

And the twisted counit by \varepsilon^f = \varepsilon\circ f^{-1} (see <a href="chapBib.html#biBJ96">[Jan96]</a>, 3.8).

<a id="X83E7F39F7D16793B" name="X83E7F39F7D16793B"></a>

<h4>2.4 PBW-type bases </h4>

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.

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 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{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} 

(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 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}}!.

<a id="X798D979F7A53E05D" name="X798D979F7A53E05D"></a>

<h4>2.5 The {\mathbb Z}-form of U_q(\mathfrak{g}) </h4>

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">[Lus90]</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.

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

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.

<a id="X78BE3EB980F0A295" name="X78BE3EB980F0A295"></a>

<h4>2.6 The canonical basis </h4>

As in Section <a href="chap2.html#X83E7F39F7D16793B">2.4</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">[Lus0a]</a> Lusztig introduced a basis of U^- with very nice properties, called the 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">[Com06]</a>.)

Let w_0=s_{i_1}\cdots s_{i_t}, and the elements F_k be as in Section <a href="chap2.html#X83E7F39F7D16793B">2.4</a>. 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.

Now we let \overline{\phantom{a}} be the automorphism of U^- defined in Section <a href="chap2.html#X81394E207F6AA6CF">2.2</a>. Elements that are invariant under \overline{\phantom{a}} are said to be bar-invariant.

By results of Lusztig (<a href="chapBib.html#biBL93">[Lus93]</a> Theorem 42.1.10, <a href="chapBib.html#biBL96">[Lus96]</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 principal monomial of X.

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}.

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 n_1 if we write \widetilde{x}=F_1^{(n_1)}\cdots F_t^{(n_t)}). 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}.

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 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}.

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 canonical basis, or 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">[Jan96]</a>, Chapter 9 a different definition is given; however, by <a href="chapBib.html#biBJ96">[Jan96]</a>, Proposition 10.9, Lemma 10.13, the two definitions agree).

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 crystal graph of V(\lambda).

<a id="X83BBB99685FB9FB7" name="X83BBB99685FB9FB7"></a>

<h4>2.7 The path model </h4>

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

From Section <a href="chap2.html#X81394E207F6AA6CF">2.2</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">[Lit95]</a>, 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 <a href="chapBib.html#biBL94">[Lit94]</a>, <a href="chapBib.html#biBL95">[Lit95]</a>). We note that the path \xi_{\lambda} = ( (\lambda), (0,1) ) joining the origin and \lambda by a straight line is an LS-path.

Now from <a href="chapBib.html#biBL94">[Lit94]</a>, <a href="chapBib.html#biBL95">[Lit95]</a> we transcribe the following:

<ol>
<li>Let \pi be an LS-path. Then f_{\alpha}\pi is an LS-path or 0; and the same holds for e_{\alpha}\pi.

</li>
<li>The action of f_{\alpha},e_{\alpha} can easily be described combinatorially (see <a href="chapBib.html#biBL94">[Lit94]</a>).

</li>
<li>The endpoint of an LS-path is an integral weight.

</li>
<li>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.

</li>
</ol>
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">[Kas96]</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#X78BE3EB980F0A295">2.6</a>).

<a id="X7BB6B9777D54FA07" name="X7BB6B9777D54FA07"></a>

<h4>2.8 Notes</h4>

I refer to <a href="chapBib.html#biBH90">[Hum90]</a> for more information on Weyl groups, and to <a href="chapBib.html#biBS01">[Ste01]</a> for an overview of algorithms for computing with weights, Weyl groups and their elements.

For general introductions into the theory of quantized enveloping algebras I refer to <a href="chapBib.html#biBC98">[Car98]</a>, <a href="chapBib.html#biBJ96">[Jan96]</a> (from where most of the material of this chapter is taken), <a href="chapBib.html#biBL92">[Lus92]</a>, <a href="chapBib.html#biBL93">[Lus93]</a>, <a href="chapBib.html#biBR91">[Ros91]</a>. I refer to the papers by Littelmann (<a href="chapBib.html#biBL94">[Lit94]</a>, <a href="chapBib.html#biBL95">[Lit95]</a>, <a href="chapBib.html#biBL98">[Lit98]</a>) for more information on the path model. The paper by Kashiwara (<a href="chapBib.html#biBK96">[Kas96]</a>) contains a proof of the connection between path operators and Kashiwara operators.

Finally, I refer to <a href="chapBib.html#biBG01">[Gra01]</a> (on computing with PBW-type bases), <a href="chapBib.html#biBG02">[Gra02]</a> (computation of elements of the canonical basis) for an account of some of the algorithms used in QuaGroup.

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