Young Tableaux and Homotopy Commutative Algebras

Abstract

A homotopy commutative algebra, or C ∞ -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 {λ : λ = λ ′ }. We prove that this C ∞ -algebra is generated in degree 1 by the binary and the ternary operations.

Homotopy transfer and self-dual Schur modules

“…Due to the isomorphisms H n (g, K) ∼ =H * n (g, K)(i.e., Tor (|λ| + r(λ)). We were able to show in [2] that with the use of the explicit expressions [5] for the operations m 2 (x, y) := pµ(i(x), i(y)) and m 3 (x, y, z) = pµ(i(x), hµ(i(y), i(z))) − pµ(hµ(i(x), i(y)), i(z)) one can generate all the elements in H…”

“…• (g, K) is generated by m 2 and m 3 we will need convenient choice of the homotopy h, the projection p and the inclusion i in the deformation retract (2).…”