Computing Multiplicities of Lie Group Representations

For fixed compact connected Lie groups H \subseteq 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. Non-asymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.

💡 Research Summary

The paper addresses a fundamental computational problem in representation theory: given compact connected Lie groups H ⊂ G and an irreducible representation V_λ of G, compute the multiplicity m_{λ}^{μ}=dim Hom_H(V_μ, Res^G_H V_λ) of an irreducible H‑module V_μ in the restriction of V_λ. While the general problem is known to be #P‑hard, the authors develop a polynomial‑time algorithm for a broad class of instances by exploiting a finite‑difference formula that translates multiplicities into lattice‑point counting problems inside rational polytopes.

The core technical contribution is the derivation of a finite‑difference identity based on Weyl’s character formula. By applying a sequence of Weyl reflections (difference operators) to the weight multiplicity function, the authors show that each multiplicity equals the number of integer points in a polytope whose dimension is determined solely by the number of rows (or, more abstractly, by the rank of a certain auxiliary lattice). This reduction is crucial because Barvinok’s algorithm can count integer points in a fixed‑dimension polytope in time polynomial in the input size. Consequently, whenever the dimension of the polytope is bounded—most notably when the number of rows of the Young diagrams involved is bounded—the multiplicity can be computed in polynomial time.

A particularly important special case is the computation of Kronecker coefficients of the symmetric group S_n. These coefficients appear as multiplicities in the tensor product V_α ⊗ V_β decomposed into irreducibles V_γ, and they are central to the geometric complexity theory (GCT) program, which seeks representation‑theoretic obstructions to separate complexity classes such as P and NP. The authors prove that, for Young diagrams with a bounded number of rows, the Kronecker coefficient can be obtained by their algorithm in time polynomial in the size of the diagrams. This contrasts sharply with the known #P‑hardness when the number of rows is unrestricted, and it provides a concrete computational tool for experimental work in GCT.

Beyond algorithmic results, the paper investigates the theoretical implications for GCT. In the GCT framework, one studies orbit closures of group actions on tensors and examines the coordinate rings of these closures. The asymptotic growth rates of multiplicities in these coordinate rings give rise to moment polytopes, and it has been conjectured that the geometry of these polytopes could yield new complexity‑theoretic obstructions. The authors demonstrate that knowledge of asymptotic growth alone does not produce stronger obstructions than those already obtainable from the moment polytope itself. In other words, the moment polytope captures precisely the “asymptotic” information, and any additional obstruction must rely on non‑asymptotic, exact multiplicity data.

This insight has two major consequences. First, it justifies the focus on exact multiplicity computation rather than merely on asymptotic invariants when searching for GCT obstructions. Second, it underscores the practical relevance of the presented algorithm: by delivering exact multiplicities in polynomial time for a wide class of cases, it equips researchers with the precise data that may be essential for constructing new representation‑theoretic barriers.

In summary, the paper makes the following contributions:

1. Finite‑difference reduction: A novel formula that rewrites Lie‑group multiplicities as integer‑point counts in bounded‑dimension polytopes.
2. Polynomial‑time algorithm: An implementation based on Barvinok’s counting technique that runs in time polynomial in the input size whenever the polytope dimension (e.g., the number of rows of Young diagrams) is fixed.
3. Application to Kronecker coefficients: A concrete, efficient method for computing Kronecker coefficients with a bounded number of rows, bridging a gap between theoretical hardness and practical computation in GCT.
4. Complexity‑theoretic analysis: A proof that asymptotic multiplicity growth does not yield new GCT obstructions beyond those derived from moment polytopes, highlighting the necessity of exact multiplicity data.

Overall, the work provides both a powerful computational tool for representation theorists and a clarified theoretical perspective for researchers pursuing geometric complexity‑theoretic approaches to fundamental problems such as P vs NP.