On the computation of Kronecker coefficients I: column-row polytopes

TL;DR

The paper develops a novel, geometry-based framework for Kronecker coefficients by introducing column-row polytopes ${\rm CR}(\lambda,\mu;\gamma)$, which count as LR-associated lattice points via a Robinson–Taulbee style expansion. It proves ${\rm lr}(\lambda,\mu;\tau) = \# {\rm CR}(\lambda,\mu;\tau)$ and expresses ${\rm g}(\lambda,\mu,\nu)$ as a signed sum ${\rm g}(\lambda,\mu,\nu)=\sum_{\gamma \succeq \nu} K_{\gamma\nu}^{(-1)} \# {\rm CR}(\lambda,\mu;\gamma)$, enabling computation with Barvinok’s algorithm on the associated polytopes. The authors introduce column-row cones, establish dimension bounds for both CR polytopes and their cones, and develop reduced-dimension strategies by counting points on faces of CR, significantly improving efficiency for Barvinok-based methods. They also provide a detailed, self-contained development of the combinatorial machinery (insertion of words, RSK correspondences) linking 3D contingency tables to LR data, plus a suite of inequalities (column/row) and dimension-reduction techniques that refine the computational workflow. The work yields both new proofs of known results and practical avenues for faster Kronecker coefficient computation, with potential extensions to higher ranks and quasi-polynomial stability phenomena.

Abstract

We present a way of computing Kronecker coefficients that uses a new family of rational convex polytopes, called column-row polytopes. We give several different formulas for the computation. They are alternating sums of numbers of integer points of either column-row polytopes or faces of column-row polytopes. We also compute the maximal dimension of these polytopes and give new proofs of some known results of more theoretical nature.

Theorems & Definitions (38)

• Lemma 3.1
• Corollary 3.3
• Theorem 4.1
• Proposition 5.1
• Proposition 5.2
• Definition 5.3
• Theorem 5.4
• Theorem 5.5
• Remark 5.6
• Corollary 5.7