Geometric complexity theory III: On deciding nonvanishing of a Littlewood-Richardson coefficient

Abstract

We point out that the positivity of a Littlewood-Richardson coefficient c α,βγ for sl n can be decided in strongly polynomial time. This means that the number of arithmetic operations is polynomial in n and independent of the bit lengths of the specifications of the partitions α,β, and γ, and each operation involves numbers whose bitlength is polynomial in n and the bit lengths α,β, and γ. Secondly, we observe that nonvanishing of a generalized Littlewood-Richardson coefficient of any type can be decided in strongly polynomial time assuming an analogue of the saturation conjecture for these types, and that for weights α,β,γ, the positivity of c α,βγ can (unconditionally) be decided in strongly polynomial time. © 2011 Springer Science+Business Media, LLC.