Young tableaux and homotopy commutative algebras

TL;DR

This paper introduces a new approach to homotopy commutative algebras using Young tableaux with self-conjugated diagrams, showing they are generated by binary and ternary operations in degree 1.

Contribution

It defines a $C_{ abla}$-algebra based on Young tableaux and proves its generation by binary and ternary operations, advancing the understanding of algebraic structures via combinatorial objects.

Findings

The $C_{ abla}$-algebra is generated in degree 1.

Binary and ternary operations suffice to generate the algebra.

The approach links Young tableaux to homotopy algebra structures.

Abstract

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.