[Paper Review] Upper bounds on Kronecker coefficients with few rows

Igor Pak, Greta Panova|arXiv (Cornell University)|Feb 25, 2020

Limits and Structures in Graph Theory48 references4 citations

TL;DR

This paper establishes new upper bounds for Kronecker coefficients $ g( au, ho, u) $ with few rows using Kostka numbers, contingency tables, and Littlewood–Richardson coefficients. The key result is a clean, general bound: $ g( au, ho, u) \igl(1 + \frac{\ell m r}{n}\bigr)^n \bigl(1 + \frac{n}{\ell m r}\bigr)^{\ell m r} $, which improves upon prior general bounds and is sharp when $ \ell m r \leq n $, yielding $ g \leq 4^n $.

ABSTRACT

We present three different upper bounds for Kronecker coefficients $g(λ,μ,ν)$ in terms of Kostka numbers, contingency tables and Littlewood--Richardson coefficients. We then give various examples, asymptotic applications, and compare them with existing lower bounds.

Motivation & Objective

• To address the long-standing challenge of finding general, effective upper bounds for Kronecker coefficients, which are #P-hard to compute and lack a known combinatorial interpretation.
• To provide general, non-trivial upper bounds that are applicable across a wide range of partitions, especially when the number of rows in the partitions is small.
• To compare the new bounds with existing lower bounds and asymptotic estimates, particularly in relation to binary and 3D contingency tables.
• To explore implications for open problems such as the Saxl conjecture and the asymptotic behavior of Kronecker coefficients.
• To establish a foundation for future explicit constructions and tighter bounds using combinatorial tools like Knutson–Tao puzzles and plane partitions.

Proposed method

• Derives upper bounds using 2D contingency tables (via Kostka numbers) and 3D contingency tables, leveraging known asymptotic estimates for their counts.
• Applies recent results on reduced Kronecker coefficients via a formula by Briand and Rosas, linking them to 3D contingency tables.
• Uses Vallejo’s multi-Littlewood–Richardson coefficients and inverse Kostka numbers to derive a third class of bounds.
• Employs the identity $ g( au, ho, u) \leq \min\{f^\tau, f^\rho, f^\nu\} $ as a baseline for comparison.
• Utilizes asymptotic estimates for partition functions $ p(n) $ and plane partition counts $ p_2(n) $, particularly to analyze lower bounds.
• Applies the hook-length formula and representation-theoretic tools to relate $ f^\lambda $ to Kostka numbers $ K(\lambda, 1^n) $.

Experimental results

Research questions

• RQ1 Can general upper bounds for Kronecker coefficients be derived that are both tight and computationally meaningful, given their #P-hard nature?
• RQ2 How do bounds based on 3D contingency tables compare with those based on Kostka numbers and Littlewood–Richardson coefficients in terms of strength and applicability?
• RQ3 What is the asymptotic behavior of $ g(\lambda, \lambda, \lambda) $ for self-conjugate partitions $ \lambda $, and can this be bounded below by an exponential in $ n^{2/3} $?
• RQ4 To what extent do the bounds derived from contingency tables reflect the true growth rate of Kronecker coefficients?
• RQ5 Are Kronecker coefficients non-vanishing with high probability as $ n \to \infty $, and how does this compare to the vanishing of Kostka numbers?

Key findings

• The paper establishes a clean, general upper bound: $ g(\lambda, \mu, \nu) \leq \left(1 + \frac{\ell m r}{n}\right)^n \left(1 + \frac{n}{\ell m r}\right)^{\ell m r} $, where $ \ell, m, r $ are the number of rows in $ \lambda, \mu, \nu $, respectively.
• When $ \ell m r \leq n $, the bound simplifies to $ g(\lambda, \mu, \nu) \leq 4^n $, which is often tighter than the classical bound $ \min\{f^\lambda, f^\mu, f^\nu\} $.
• A lower bound is established: $ \sum_{\lambda \in \mathcal{L}_n} g(\lambda, \lambda, \lambda) \geq e^{c n^{2/3}} $ for some $ c > 0 $, showing that the sum of Kronecker coefficients over self-conjugate partitions grows faster than any polynomial.
• The paper shows that $ p_2(2100) > p(2100)^3 $, implying the existence of a triple $ (\alpha, \beta, \gamma) $ with $ \text{Pyr}(\alpha, \beta, \gamma) \geq 2 $, thus providing an explicit example where the number of 3D contingency tables exceeds the cube of the number of partitions.
• The authors conjecture that Kronecker coefficients are non-vanishing with probability approaching 1 as $ n \to \infty $, contrasting with the known fact that Kostka numbers vanish a.s.
• The paper provides evidence that the number of triples $ (\lambda, \mu, \nu) $ with non-zero $ g(\lambda, \mu, \nu) $ is asymptotically almost all such triples, supporting the idea that Kronecker coefficients are generically non-zero.