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A combinatorial interpretation for certain plethysm and Kronecker coefficients

Igor Pak, Greta Panova, Joshua P. Swanson

Abstract

We give explicit positive combinatorial interpretations for the plethysm coefficients $\langle s_μ[s_ν], s_λ\rangle$, when $λ$ has at most two rows, as counting certain marked trees. In the special case $μ=(n)$, this also yields a combinatorial interpretation for the corresponding rectangular Kronecker coefficient $g(λ, (n^k), (n^k))$. While it is easy to express these quantities as differences of counting problems in the complexity class $\mathrm{FP}$, putting the problem in $\#\mathrm{P}$, our interpretations give a positive counting formula over explicit marked trees.

A combinatorial interpretation for certain plethysm and Kronecker coefficients

Abstract

We give explicit positive combinatorial interpretations for the plethysm coefficients , when has at most two rows, as counting certain marked trees. In the special case , this also yields a combinatorial interpretation for the corresponding rectangular Kronecker coefficient . While it is easy to express these quantities as differences of counting problems in the complexity class , putting the problem in , our interpretations give a positive counting formula over explicit marked trees.

Paper Structure

This paper contains 11 sections, 12 theorems, 44 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions and background
  3. Kronecker and plethysm coefficients via symmetric functions
  4. Computational complexity
  5. Binomial identities and trees
  6. The KOH identity and KOH trees
  7. The GOH identity and GOH trees
  8. The Kronecker case
  9. Products of symmetric, unimodal polynomials
  10. Marked KOH trees and Kronecker coefficients
  11. The plethysm case

Key Result

Theorem 1.1

The Kronecker coefficient $g(\lambda,(n^k),(n^k))$ for $\lambda = (nk-r,r)$ is equal to the number of marked KOH trees $\mathcal{T}(n,k,r)$.

Figures (4)

  • Figure 1: (Left) A KOH tree $T$ of type $(8, 9)$. Edges are labeled with the distinct row lengths in the parent's partition. The leaf multiset is $\mathcal{L}(T) = \{0, 14, 22\}$ and the corresponding term of \ref{['eq:KOH-trees']} contributing to $\binom{8+9}{9}_q$ is $q^{(72-0-14-22)/2}[0+1]_q [14+1]_q [22+1]_q$. (Right) A KOH tree of type $(12,17)$ with $\mu=(4,4,3,2,2,2)$ and leaf multiset $\mathcal{L}(T)=\{2,6,12,44\}$
  • Figure 2: A GOH tree of type $((3, 3, 2, 1), 6)$.
  • Figure 3: (Left) A marked KOH tree $T$ of type $(8, 9)$, with abstract marks circled. Leaves $((1), a, 1)$ are abbreviated as $a$. (Right) A marked KOH tree of type $(12,17)$, with valid marks $(k_1,k_2,k_3,k_4)=(0,2,6,11)$ for $r=81$.
  • Figure 4: A marked GOH tree $T$ of type $((3, 3, 2, 1), 6)$. Leaves $((1), a, 1)$ are abbreviated as $a$. Marks are circled.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3
  • proof : Proof of \ref{['lem:stab-diff']}
  • Lemma 2.1
  • proof
  • Theorem 3.1: The KOH identity
  • Definition 3.2
  • Remark 3.3
  • Proposition 3.4
  • ...and 18 more