Signed combinatorial interpretations in algebraic combinatorics

TL;DR

This work establishes that a broad array of integral structure constants arising in symmetric, quasisymmetric, and polynomial bases, as well as in Schubert theory and plethysm, admit signed combinatorial interpretations. The authors develop an effective Möbius inversion framework that, together with unitriangular basis changes, places these constants in complexity classes such as $\textup{GapP}$ (and $\textup{GapP}/\textup{FP}$ in some deformations), while identifying many with $\textup{\#P}$ under suitable conditions. By systematically analyzing standard bases (e.g., Schur, monomial, power-sum, elementary, complete homogeneous), deformations (Macdonald, Hall–Littlewood, Jack), and a range of quasisymmetric and polynomial bases, they derive signed interpretations for inverses like Kostka numbers and LR coefficients and extend these results to plethysm coefficients. The approach applies uniformly to a host of combinatorial models (Young tableaux, pipe dreams, Kohnert/Lascoux diagrams) and yields new signed formulas, facilitating bounds, asymptotics, and algorithmic insights, while clarifying which coefficients lack straightforward unsigned combinatorial interpretations. Overall, the paper significantly broadens the toolkit for understanding structure constants via signed combinatorics and Möbius inversion, with implications for complexity and representation theory.

Abstract

We prove the existence of signed combinatorial interpretations for several large families of structure constants. These families include standard bases of symmetric and quasisymmetric polynomials, as well as various bases in Schubert theory. The results are stated in the language of computational complexity, while the proofs are based on the effective Möbius inversion.

Theorems & Definitions (42)

• Theorem 1.1: classic structure constants
• Theorem 1.2
• Theorem 1.3: quasisymmetric structure constants
• Theorem 1.4: polynomial structure constants
• Theorem 1.5: plethysm coefficients
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['t:main-classic']}