Schur Polynomials do not have small formulas if the Determinant doesn't!

TL;DR

This paper proves that Schur polynomials cannot be computed by small algebraic formulas unless a major complexity class collapse occurs, linking their complexity to the hardness of the determinant and algebraic circuit complexity.

Contribution

It establishes the first known lower bounds for the algebraic formula complexity of Schur polynomials, connecting their hardness to fundamental complexity class separations.

Findings

Schur polynomials lack small algebraic formulas unless VBP equals VF.

Computing determinants of certain generalized Vandermonde matrices is as hard as symbolic determinants.

The study of polynomial composition with algebraically independent polynomials offers new insights into algebraic complexity.

Abstract

Schur Polynomials are families of symmetric polynomials that have been classically studied in Combinatorics and Algebra alike. They play a central role in the study of Symmetric functions, in Representation theory [Sta99], in Schubert calculus [LM10] as well as in Enumerative combinatorics [Gas96, Sta84, Sta99]. In recent years, they have also shown up in various incarnations in Computer Science, e.g, Quantum computation [HRTS00, OW15] and Geometric complexity theory [IP17]. However, unlike some other families of symmetric polynomials like the Elementary Symmetric polynomials, the Power Symmetric polynomials and the Complete Homogeneous Symmetric polynomials, the computational complexity of syntactically computing Schur polynomials has not been studied much. In particular, it is not known whether Schur polynomials can be computed efficiently by algebraic formulas. In this work, we address this question, and show that unless \emph{every} polynomial with a small algebraic branching program (ABP) has a small algebraic formula, there are Schur polynomials that cannot be computed by algebraic formula of polynomial size. In other words, unless the algebraic complexity class $\mathrm{VBP}$ is equal to the complexity class $\mathrm{VF}$, there exist Schur polynomials which do not have polynomial size algebraic formulas. As a consequence of our proof, we also show that computing the determinant of certain \emph{generalized} Vandermonde matrices is essentially as hard as computing the general symbolic determinant. To the best of our knowledge, these are one of the first hardness results of this kind for families of polynomials which are not \emph{multilinear}. A key ingredient of our proof is the study of composition of \emph{well behaved} algebraically independent polynomials with a homogeneous polynomial, and might be of independent interest.