In this chapter we summarize some of the theoretical concepts with which QuaGroup operates.
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
\[ \begin{bmatrix} n \\ k \end{bmatrix} = \frac{[n]!}{[k]![n-k]!}.\]
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 \(\overline{F_{\alpha}}=F_{\alpha}\), \(\overline{K_{\alpha}}= K_{\alpha}^{-1}\), \(\overline{E_{\alpha}}=E_{\alpha}\), \(\overline{q}=q^{-1}\).
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 [Jan96], 3.8).
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 \(E_k^{(n)} = E_k^n/[n]_{\alpha_{i_k}}!\).
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 [Lus90], 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.
As in Section 2.4 we let \(U^-\) be the subalgebra of \(U_q(\mathfrak{g})\) generated by the \(F_{\alpha}\) for \(\alpha\in\Delta\). In [Lus0a] 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 [Com06].)
Let \(w_0=s_{i_1}\cdots s_{i_t}\), and the elements \(F_k\) be as in Section 2.4. 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 2.2. Elements that are invariant under \(\overline{\phantom{a}}\) are said to be bar-invariant.
By results of Lusztig ([Lus93] Theorem 42.1.10, [Lus96], 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 [Jan96], Chapter 9 a different definition is given; however, by [Jan96], 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 \(V(\lambda)\).
In this section we recall some basic facts on Littelmann's path model.
From Section 2.2 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 \(V(\lambda)\) ([Lit95], 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 [Lit94], [Lit95]). 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 [Lit94], [Lit95] we transcribe the following:
Let \(\pi\) be an LS-path. Then \(f_{\alpha}\pi\) is an LS-path or \(0\); and the same holds for \(e_{\alpha}\pi\).
The action of \(f_{\alpha},e_{\alpha}\) can easily be described combinatorially (see [Lit94]).
The endpoint of an LS-path is an integral weight.
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 [Kas96] 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 2.6).
I 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.
For 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.
Finally, 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 QuaGroup.
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