Young tableaux and homotopy commutative algebras

A homotopy commutative algebra, or $C_{\infty}$-algebra, is defined via the Tornike Kadeishvili homotopy transfer theorem on the vector space generated by the set of Young tableaux with self-conjugated Young diagrams. We prove that this $C_{\infty}$-algebra is generated in degree 1 by the binary and the ternary operations.
Comment: Talk given by Todor Popov at the International Workshop "Lie Theory-IX", Varna, 2011, (11 pages)