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Positivity of the symmetric group characters is as hard as the polynomial time hierarchy

Christian Ikenmeyer, Igor Pak, Greta Panova

TL;DR

This work establishes a deep connection between the positivity of symmetric group characters and foundational complexity classes. By proving that the vanishing problem for $\chi^\lambda(\pi)$ is $\textup{C$_=$P}$-complete, it follows that the square of a character, $\chi^\lambda(\pi)^2$, cannot lie in $\#P$ unless the polynomial hierarchy collapses to its second level, thereby ruling out any unsigned combinatorial interpretation for these squares under standard complexity assumptions. The authors develop a chain of reductions from $\#\CircuitSAT$ to $3$D- and $4$D-matchings and then to ordered set partitions, translating character values into partition-counting problems via the Frobenius character framework. Additional results show that computing $\chi^\lambda(\mu)$ is GapP-complete (and thus $\textup{PP}$-hard under certain reductions), highlighting tight relationships between representation theory and counting complexity. Overall, the paper clarifies why natural positive combinatorial models for character squares are unlikely and connects these phenomena to the structure of the polynomial hierarchy, with implications for algebraic combinatorics and complexity theory.

Abstract

We prove that deciding the vanishing of the character of the symmetric group is $C_=P$-complete. We use this hardness result to prove that the the square of the character is not contained in $\#P$, unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof we conclude that deciding positivity of the character is $PP$-complete under many-one reductions, and hence $PH$-hard under Turing-reductions.

Positivity of the symmetric group characters is as hard as the polynomial time hierarchy

TL;DR

This work establishes a deep connection between the positivity of symmetric group characters and foundational complexity classes. By proving that the vanishing problem for is _=-complete, it follows that the square of a character, , cannot lie in unless the polynomial hierarchy collapses to its second level, thereby ruling out any unsigned combinatorial interpretation for these squares under standard complexity assumptions. The authors develop a chain of reductions from to D- and D-matchings and then to ordered set partitions, translating character values into partition-counting problems via the Frobenius character framework. Additional results show that computing is GapP-complete (and thus -hard under certain reductions), highlighting tight relationships between representation theory and counting complexity. Overall, the paper clarifies why natural positive combinatorial models for character squares are unlikely and connects these phenomena to the structure of the polynomial hierarchy, with implications for algebraic combinatorics and complexity theory.

Abstract

We prove that deciding the vanishing of the character of the symmetric group is -complete. We use this hardness result to prove that the the square of the character is not contained in , unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof we conclude that deciding positivity of the character is -complete under many-one reductions, and hence -hard under Turing-reductions.
Paper Structure (23 sections, 13 theorems, 27 equations, 1 figure)

This paper contains 23 sections, 13 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. $\textup{\#P}$, $\textup{GapP}$ and combinatorial interpretations
  4. Related work
  5. Preliminaries
  6. Notation
  7. Representation Theory
  8. Computational Complexity
  9. $\textup{C$_=$P}$ and the Collapse of the Polynomial Hierarchy
  10. 3D- and 4D-matchings
  11. Ordered set partitions
  12. Main result
  13. Characters and set partitions
  14. The join of two 3D-matchings
  15. An auxiliary SetPartition instance
  16. ...and 8 more sections

Key Result

Theorem 1.3

Let $\chi^2 : (\lambda,\pi) \mapsto (\chi^\lambda(\pi))^2$, where $\lambda\vdash n$ and $\pi \in \mathfrak{S}_n$. If the function $\chi^2$ is contained in the complexity class $\textup{\#P}$, then $\textup{coNP}=\textup{C$_=$P}$. ConsequentlyIndeed, Tarui (Tar91, see also Gre93) proves that $\textup

Figures (1)

  • Figure 1: On the top left: The #3DM instance $E=\{(1,2,2),(2,1,1),(2,2,1)\}$, where $V_1$ are the red circles in the first column, $V_2$ are the green circles in the second column, and $V_3$ are the blue circles in the third column. For example, $(1,2,2)$ is depicted as a hyperedge containing the red vertex in column 1, row 1, and the green vertex in column 2, row 2, and the blue vertex in column 3, row 2. On the top right: The #3DM instance $\{(1,1,2),(1,2,3),(2,3,1),(3,1,3)\}$. The join of these two instances is obtained by first padding them to the same number of rows $u$ (here $u=4$), and then adding another dimension to each hyperedge (which adds a column of points at the front) and taking the union of both hypergraphs. The two special hyperedges $H=(4,4,4,4)$ and $H'=(1,4,4,4)$ are the ones containing the bottom right vertex. The different shades of gray for the hyperedges are just for illustration.

Theorems & Definitions (26)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Lemma 4.3
  • ...and 16 more