  The purpose of this paper is to present an interpretation for the
decomposition of the tensor product of two or more irreducible representations
of GL(N) in terms of a system of quantum particles. Our approach is based on a
certain scattering matrix that satisfies a Yang-Baxter type equation. The
corresponding piecewise-linear transformations of parameters give a solution to
the tetrahedron equation. These transformation maps are naturally related to
the dual canonical bases for modules over the quantum enveloping algebra
$U_q(sl_n)$. A byproduct of our construction is an explicit description for the
cone of Kashiwara's parametrizations of dual canonical bases. This solves a
problem posed by Berenstein and Zelevinsky. We present a graphical
interpretation of the scattering matrices in terms of web functions, which are
related to honeycombs of Knutson and Tao.
