Symmetry in Turán Sums of Squares Polynomials from Flag Algebras

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 semidefinite programming based methods 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 the sums of squares created by flag algebras for Tur\’an problems 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. This connection reveals several combinatorial properties of flag algebra sums of squares and offers new tools for Tur\’an problems.