Computing Clebsch–Gordan coefficients with Hive
Polytopes and BZ-polytopes
The Maple notebooks below generate the defining inequalities for
Hive polytopes () or Berenstein–Zelevinsky (BZ) polytopes
() in a file-format suitable for input into LattE
macchiato. LattE macchiato is a program that enumerates the
integer lattice points in a rational polytope via the algorithm of
Barvinok (). Applying Barvinok's algorithm to the hive and
BZ-polytopes yields an algorithm for computing Clebsch–Gordan
coefficients that runs in polynomial time for fixed rank. In
particular, enumerating the lattice points in hive polytopes yields
the Littlewood–Richardson coefficients (which are the
Clebsch–Gordan coefficients in type Ar).
The strength of the polyhedral algorithm is that it works for weights
with very large entry sizes (at least into the millions) with little
effect on the running time. On the other hand, the algorithm requires
lower ranks than the standard methods, such as those based on Klimyk's
formula. In this sense, polyhedral algorithm complements the standard
methods. Generally speaking, the polyhedral algorithm is an effective
means for computing Littlewood–Richardson coefficients for ranks
r ≤ 6. For the other classical roots systems, the
algorithm is effective in ranks r ≤ 4, roughly speaking.
See  for a study of the practical effectiveness of this
Instructions for using the Maple notebooks are in the introductory
comments of each file. These files have been tested on Maple 9.
They may not run on earlier versions of Maple.
Hive Polytopes (Type Ar)
BZ-polytopes for Type Br
BZ-polytopes for Type Cr
BZ-polytopes for Type Dr
 Alexander I. Barvinok, A polynomial time algorithm for counting
integral points in polyhedra when the dimension is fixed, Math. Oper. Res.
19 (1994), no. 4, 769–779.
 Arkady Berenstein and Andrei Zelevinsky, Tensor product multiplicities,
canonical bases and totally positive varieties, Invent. Math. 143 (2001),
no. 1, 77–128, arXiv:math.RT/9912012.
 Anders Skovsted Buch,
The saturation conjecture (after A. Knutson and T. Tao), Enseign. Math.
(2) 46 (2000), no. 1-2, 43–60, arXiv:math.CO/9810180, With an appendix
by William Fulton.
 Jesús A. De Loera and Tyrrell B. McAllister, On the computation
of Clebsch-Gordan coefficients and the dilation effect,
Experiment. Math. 15, (2006), no. 1, 7–20, arXiv:math.CO/0505094.