Homotopy transfer and self-dual Schur modules

We consider the free 2-nilpotent graded Lie algebra $\mathfrak{g}$ generated in degree one by a finite dimensional vector space V. We recall the beautiful result that the cohomology $H^ \cdot \left( {\mathfrak{g},\mathbb{K}} \right)$ of $\mathfrak{g}$ with trivial coefficients carries a GL(V)-representation having only the Schur modules V with self-dual Young diagrams {λ: λ = λ′} in its decomposition into GL(V)-irreducibles (each with multiplicity one). The homotopy transfer theorem due to Tornike Kadeishvili allows to equip the cohomology of the Lie algebra g with a structure of homotopy commutative algebra.