Signed combinatorial interpretations in algebraic combinatorics

TL;DR

This paper establishes signed combinatorial interpretations for key structure constants in algebraic combinatorics, connecting computational complexity and M"obius inversion techniques.

Contribution

It introduces new signed combinatorial models for structure constants across multiple algebraic bases, expanding understanding in algebraic combinatorics.

Findings

Signed interpretations for symmetric polynomial bases

Signed interpretations for quasisymmetric polynomial bases

Applications to Schubert calculus

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 M\"obius inversion.