The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in [this paper](http://arxiv.org/abs/math/9912012). It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^\*$. Let ${\bf i}=(i\_1, \ldots, i\_l)$. Then $\pi=(\gamma = \gamma\_0, \gamma\_1, \ldots, \gamma\_l=\delta)$ is called an ${\bf i}$-trail if for $k=1,\ldots, l$, $\gamma\_{k-1}-\gamma\_{k} = c\_k\alpha\_{i\_k}$ for some $c\_k \in \mathbb{Z}\_{\geq 0}$ and $e\_{i\_1}^{c\_1} \cdots e\_{i\_l}^{c\_l}$ is a non-zero linear map from $V(\delta)$ to $V(\gamma)$. 


One main result in [the paper](http://arxiv.org/abs/math/9912012) is a polyhedral formula for tensor product multiplicities (Theorems 2.2 and 2.3 in the above paper). 


My question is: let $\mathfrak{g}$ be the simple complex Lie algebra of type $G\_2$. Let $\lambda=k\_1 \omega\_1 + l\_1 \omega\_2$ and $\mu=k\_2 \omega\_1 + l\_2 \omega\_2$, where $\omega\_1, \omega\_2$ are fundamental weights. Are there some polyhedral formula which describe the decomposition of $V(\lambda) \otimes V(\mu)$ into irreducible $\mathfrak{g}$-modules using only $k\_1, k\_2, l\_1, l\_2$? The formula will be of the form 
$$
V(\lambda) \otimes V(\mu) = \oplus\_{(r\_1, r\_2) \in A} V(r\_1 \omega\_1 + r\_2 \omega\_2), \quad (1)
$$
where
$$A = \left\{(r\_1, r\_2) \in \mathbb{Z}\_{\geq 0}^2 \middle\vert 
\begin{aligned} r\_1 &=g\_1(k\_1,k\_2,l\_1,l\_2),\\
 r\_2 &=g\_1(k\_1,k\_2,l\_1,l\_2),\\
 0\ &\le f\_1(k\_1,k\_2,l\_1,l\_2),\\
 &\quad\ldots,\\
 0\ &\le f\_m(k\_1,k\_2,l\_1,l\_2)
\end{aligned} \right\},$$
and where $f\_1,\ldots, f\_m,g\_1,g\_2$ are some (linear) polynomials. Maybe use the definition of ${\bf i}$-trail, we can translate Theorem 2.2 in [the paper](http://arxiv.org/abs/math/9912012) to the form $(1)$. But I don't know how to verify that $e\_{i\_1}^{c\_1} e\_{i\_2}^{c\_2}$ is a non-zero linear map from $V(s\_i \omega\_i^{\vee})$ to $V(w\_0 \omega\_i^{\vee})$ (in the case of type $G\_2$, $w\_0 = s\_1s\_2s\_1s\_2s\_1s\_2$). Are there some references which have some formula similar to $(1)$? Thank you very much.



Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky
There is a bit of a mistake in the question. If a formula like (1) held, then tensor product multiplicities for $G\_2$ could only be at most 1. 


On the other hand, Theorem 2.2 of the paper does give a polyhedral formula of the form:
$$ V(\lambda)\otimes V(\mu) = \oplus\_{(t\_1, t\_2, t\_3, t\_4, t\_5, t\_6) \in A} V(\lambda + \mu - \sum\_{k=1}^6 t\_k \beta\_k)$$ 
where $ \beta\_1, \dots, \beta\_6 $ are the positive roots of $G\_2$ (ordered using a reduced word for $ w\_0$) and $ A $ is a polyhedron.


In Theorem 2.2, $ A$ is described using $\mathbb i$-trails. I agree that this description of $ A $ is a bit confusing. However, there is a simpler description given in Corollary 3.4 of the same paper.
$$
A = \{ (t\_1, \dots, t\_6) : t\_k \ge 0, t\_1 \le \langle \lambda, \alpha\_1^\vee \rangle, p\_1 \le \langle \lambda, \alpha\_2^\vee \rangle, t\_6 \le \langle \mu, \alpha\_2^\vee \rangle, p\_6 \le \langle \mu, \alpha\_1^\vee \rangle \} $$
where $p\_1, p\_6 $ are computed from $ t\_1, \dots, t\_6 $ using the tropicalization of the formula in Proposition 7.1.(4). Since $p\_1, p\_6 $ are tropical expressions in $ t\_1, \dots, t\_6 $, the above description of $ A $ consists only of inequalities. Unfortunately, the formulas for $ p\_1, p\_6 $ are pretty complicated, but this is an explicit description of the polyhedron. 


