The structure of sl₂-modules on the Young diagrams

This paper establishes the explicit form of derivatives (D̂–, D̂0, D̂+) that define the action of the Lie algebra sl2 on the vector space ℚyn of Young diagrams corresponding to partitions λ of length n: D–(λ) = –(nξ–(λ) + ∇–(λ)),D0(λ) = 2|λ|λ,D+(λ) = ∇+(λ), where ξ–(λ) and ∇±(λ) are sums over all possible Young diagrams obtained by adding or removing a cell □ from the corresponding Young diagram. The proof involves introducing an algebra structure on ℚyn isomorphic to the algebra of symmetric polynomials in n variables, defining the sl2 action on Schur polynomials, and transferring this action to ℚyn.