Symmetry in Tur\'an Sums of Squares Polynomials from Flag Algebras

TL;DR

This paper explores how symmetry properties in flag algebra sums of squares can be exploited using representation theory, leading to more efficient semidefinite programming approaches for Turán problems in extremal combinatorics.

Contribution

It demonstrates that sums of squares from flag algebras can be derived from symmetry-adapted semidefinite programs, revealing structural properties and improving computational methods.

Findings

Sums of squares can be obtained from symmetry-adapted semidefinite programs.

Representation theory simplifies the expression of invariant polynomials.

New tools are provided for Turán and related combinatorial problems.

Abstract

Tur\'an problems in extremal combinatorics ask to find asymptotic bounds on the edge densities of graphs and hypergraphs that avoid specified subgraphs. The theory of flag algebras proposed by Razborov provides powerful methods based on semidefinite programming to find sums of squares that establish edge density inequalities in Tur\'an problems. Working with polynomial analogs of the flag algebra entities, we prove that such sums of squares created by flag algebras can be retrieved from a restricted version of the symmetry-adapted semidefinite program proposed by Gatermann and Parrilo. This involves using the representation theory of the symmetric group for finding succinct sums of squares expressions for invariant polynomials. The connection reveals several combinatorial and structural properties of flag algebra sums of squares, and offers new tools for Tur\'an and other related problems.