Geometric Complexity III: on deciding positivity of Littlewood-Richardson coefficients

We point out that the remarkable Knutson and Tao Saturation Theorem [9] and polynomial time algorithms for linear programming [14] have together an important, immediate consequence in geometric complexity theory [15, 16]: The problem of deciding positivity of Littlewood-Richardson coefficients belongs to P ; cf.[10]. Specifically, for GLn(C), positivity of a Littlewood-Richardson coefficient cα,β,γ can be decided in time that is polynomial in n and the bit lengths of the specifications of the partitions α, β and γ. Furthermore, the algorithm is strongly polynomial in the sense of [14]. The main goal of this article is to explain the significance of this result in the context of geometric complexity theory. Furthermore, it is also conjectured that an analogous result holds for arbitrary symmetrizable Kac-Moody algebras. The fundamental Littlewood-Richardson rule in the representation theory of GLn(C) [4] states that the tensor product of two irreducible representations (Weyl modules) Vα and Vβ of GLn(C) decomposes as follows: Vα ⊗ Vβ = ⊕γcα,β,γVγ , (1) ∗Visiting faculty member