Computing Multiplicities of Lie Group Representations

Abstract

Abstract-For fixed compact connected Lie groups H ⊆ G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok's algorithm for counting integral points in polytopes.The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be #P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Nonasymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.

On the complexity of computing Kronecker coefficients

“…With Valiant's theory of determinant computations as its starting point, their approach relies, among other things, on the (conjectural) ability to decide in polynomial time the positivity of Kronecker coefficients and their plethystic generalizations. Envisioned as a far reaching mathematical program requiring over 100 years to complete [F2], this approach led to a flurry of activity in an attempt to understand and establish some critical combinatorial and computational properties of Kronecker coefficients (see [BOR1,BI1,CDW,Ike,M1]). This paper is a new advance in this direction.…”

Computation of dilated Kronecker coefficients

“…When the stabilizer of λ in the Weyl group of G is large, we explain how to take advantage of this situation. Similarly as in [11,10], our method for computing g(kλ 1 , kλ 2 , . .…”

Untitled

“…Algorithms for computing Kronecker coefficients have been implemented in for example Schur [26], Sage [20] and the Python package Kronecker [6].…”

“…We will apply the method described above to the tensor 2, 2, 2 q , see Theorem 2. [5,14,8,2,12,0,1,15,6,11,18,13,4,3,9,17,7,10,16,19], [14,5,9,0,6,13,16,15,4,11,3,10,12,8,2,17,7,19,18,1]), 18,2,12,10,5,1,17,19,9,3,4,7,6,…”