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The Newton polytope of the Kronecker product

Greta Panova, Chenchen Zhao

TL;DR

The paper addresses when the Kronecker product $s_\lambda * s_\mu$ has a saturated Newton polytope, linking this property to positivity of Kronecker coefficients. It develops a framework based on multi-Littlewood-Richardson coefficients and Horn inequalities to characterize monomial support via polytopes $\mathcal P(\lambda,\mu;{\mathbf a})$. It proves SNP for key two-/three-row cases and shows that every nonempty polytope in the $k=3$ scenario contains an integral point, yielding SNP in those instances and providing necessary conditions for positivity. It also discusses computational hardness (KronPos) and broader implications for positivity phenomena in Kronecker products and related operations like plethysm, highlighting both constructive results and open challenges.

Abstract

We study the Kronecker product of two Schur functions $s_λ\ast s_μ$, defined as the image of the characteristic map of the product of two $S_n$ irreducible characters. We prove special cases of a conjecture of Monical--Tokcan--Yong that its monomial expansion has a saturated Newton polytope. Our proofs employ the Horn inequalities for positivity of Littlewood-Richardson coefficients and imply necessary conditions for the positivity of Kronecker coefficients.

The Newton polytope of the Kronecker product

TL;DR

The paper addresses when the Kronecker product $s_\lambda * s_\mu$ has a saturated Newton polytope, linking this property to positivity of Kronecker coefficients. It develops a framework based on multi-Littlewood-Richardson coefficients and Horn inequalities to characterize monomial support via polytopes $\mathcal P(\lambda,\mu;{\mathbf a})$. It proves SNP for key two-/three-row cases and shows that every nonempty polytope in the $k=3$ scenario contains an integral point, yielding SNP in those instances and providing necessary conditions for positivity. It also discusses computational hardness (KronPos) and broader implications for positivity phenomena in Kronecker products and related operations like plethysm, highlighting both constructive results and open challenges.

Abstract

We study the Kronecker product of two Schur functions , defined as the image of the characteristic map of the product of two irreducible characters. We prove special cases of a conjecture of Monical--Tokcan--Yong that its monomial expansion has a saturated Newton polytope. Our proofs employ the Horn inequalities for positivity of Littlewood-Richardson coefficients and imply necessary conditions for the positivity of Kronecker coefficients.
Paper Structure (19 sections, 26 theorems, 71 equations)

This paper contains 19 sections, 26 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. SNP for the Kronecker product
  3. Positivity implications
  4. Paper outline
  5. Definitions and tools
  6. Basic notions from algebraic combinatorics
  7. The Kronecker product
  8. Newton polytopes
  9. Two and three-row partitions
  10. Multi-LR coefficients and Horn inequalities
  11. Monomial expansion via multi-LR coefficients
  12. Horn inequalities for multi-LR's
  13. The case $k=3$.
  14. The polytope $P(\mu;{\text{\bf a}}) \cap P(\nu;{\text{\bf a}})$
  15. Integer points in $\mathcal{P}(\mu,\nu,{\text{\bf a}})$
  16. ...and 4 more sections

Key Result

Theorem 1.4

Let $\lambda,\mu \vdash n$ with $\ell(\lambda)\leq 2, \ell(\mu)\leq 3$, and $\mu_1 \geq \lambda_1$ then $s_\lambda*s_\mu(x_1,\ldots,x_k)$ has a saturated Newton polytope for every $k\in \mathbb{N}$.

Theorems & Definitions (56)

  • Definition 1.1
  • Definition 1.2
  • Conjecture 1.3: Yong
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Proposition 1.9: Corollary \ref{['cor:kron_pos_2']}
  • Lemma 2.1: CHM
  • ...and 46 more