  We prove the GGS conjecture (1993), due to Gerstenhaber, Giaquinto, and
Schack, which gives a particularly simple explicit quantization of classical
r-matrices for Lie algebras gl(n) in terms of an element R satisfying the
quantum Yang-Baxter equation and the Hecke condition. The r-matrices were
classified by Belavin and Drinfeld in the 1980s in terms of combinatorial
objects known as Belavin-Drinfeld triples. We prove this conjecture by showing
that the GGS matrix coincides with another quantization due to Etingof,
Schiffmann, and the author, which is a more general construction. We do this by
explicitly expanding the product from the aforementioned paper using detailed
combinatorial analysis in terms of Belavin-Drinfeld triples.
