On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients

Kostka numbers and Littlewood-Richardson coefficients appear in combinatorics and representation theory. Interest in their computation stems from the fact that they are present in quantum mechanical computations since Wigner [15]. In recent times, there have been a number of algorithms proposed to perform this task [1–3, 11, 12]. The issue of their computational complexity has received at-tention in the past, and was raised recently by E. Rassart in [11]. We prove that the problem of computing either quantity is #P-complete. Thus, unless P = NP, which is widely disbelieved, there do not exist efficient algorithms that compute these numbers.