For a complex semisimple Lie algebra, a simple example of a "model" for its finite dimensional representations are the characters: A representation is completely determined by its character, and the tensor product of representations translates into the language of characters as the product of characters. A problem of this "character model" is that formulas like the ones of Steinberg and Klimyk to decompose tensor products are involved alternating sums through it is a priori clear that only nonnegative results can occur.
In the following we consider weights with "tails", or more precisely, piecewise linear paths that connect a weight with the origin in the space of rational weights. A special case of such a construction is well known for the group ${\rm GL}_n({\bf C})$: The Young tableaux can be naturally considered as "piecewise linear paths". The advantage of using paths is that this setting is independent of the type of the Lie algebras, and it enables us for example to give a straightforward generalization of the classical Littlewood-Richardson rule to arbitrary semisimple Lie algebras and even to arbitrary symmetrizable Kac-Moody algebras.
To describe the construction [3,4] of our path model, let ${\bf Z}\Pi$ be the free ${\bf Z}$-module with basis the set of all piecewise linear paths that connect a weight with the origin in the space of rational weights. We associate to a simple root $\alpha$ linear operators $e_\alpha$ and $f_\alpha$ on ${\bf Z}\Pi$. Such an operator either replaces parts of a path by the image with respect to the reflection $s_\alpha$ (see Fig. 1), or it maps the path to zero.
For a path $\pi$ let $M_\pi\subset{\bf Z}\Pi$ be the module obtained by applying the $e_\alpha$'s and $f_\alpha$'s to $\pi$. If the image of $\pi$ is contained in the dominant Weyl chamber ${\scr C}$ (as in Fig.\ 2a), then $M_\pi$ is a "model" for the irreducible representation $V_\lambda$ of highest weight $\lambda\coloneq \pi(1)$: The sum $\sum e^{\eta(1)}$ over the endpoints of all paths $\eta\in M_\pi$ is the character of $V_\lambda$ [3,4]. Figure 2 shows such a model for the adjoint representation of ${\rm sl}_3({\bf C})$: One path for each root (Fig. 2a) and two paths ending in the zero weight (Fig. 2b).
Denote by $M_\pi *M_\eta$ the ${\bf Z}$-span of the concatenations of all paths in $M_\pi$ with the paths in $M_\eta$. If the images of $\pi$ and $\eta$ are contained in the dominant Weyl chamber ${\scr C}$, then $M_\pi *M_\eta$ is a direct sum of modules of the same type. In fact, $M_\pi *M_\eta=\bigoplus M_{\pi *\tau}$, where the sum runs over all paths $\tau$ in $M_\eta$ such that the iamge of $\pi *\tau$ is contained in ${\scr C}$ [3,4]. Figure 3 shows for the Lie algebra ${\rm sl}_3({\bf C})$ the concatenation of a model for the standard representation on ${\bf C}^3$ with a model for its dual representation, and the decomposition into a model for the adjoint representation and a model for the trivial representation.
If the images of $\pi$ and $\eta$ are contained in ${\scr C}$, then the "characters" of $M_\pi * M_\eta$ and $V_{\pi(1)}\otimes V_{\eta(1)}$ are equal. So the concatenation of paths gives a "model" for the tensor product of representations, and the decomposition formula for $M_\pi *M_\eta$ leads to decomposition formulas for tensor products of representations which can be naturally viewed as generalizations of the classical Littelwood-Richardson formula for tensor products of representations of the group ${\rm GL}_n({\bf C})$. In a similar way one can also use our path model to get decomposition rules for the restriction of an irreducible highest weight representation to a Levi subalgebra of the Kac-Moody algebra ${\germ g}$ [3,4].
Recently the following nice connection between the path model and the crystal base has been found by Joseph and Kashiwara: For two paths $\pi_1,\pi_2$ put an arrow $\pi_1\overset\alpha\to \pi_2$ with color $\alpha$ if and only if $f_\alpha(\pi_1)=\pi_2$. They proved [1,2] that this graph associated to $M_\pi$ is isomorphic to the crystal graph of the corresponding representation $V_|pi(1)|$ of the $q$-analogue $U_q({\germ g})$ of the enveloping algebra of the Kac-Moody algebra ${\germ g}$.
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