Hopf Structures on Standard Young Tableaux

We review the Poirier-Reutenauer Hopf structure on Standard Young Tableaux and show that it is a distinguished member of a family of Hopf structures. The family in question is related to deformed parastatistics. In this paper K is a field of characteristic zero and all vector spaces are over K. A K[S]-module is a collection of K[Sr]-modules of the symmetric groups Sr. A H(q)-module is a collection of Hr(q)-modules of the Hecke algebras Hr(q). 1 Malvenuto-Reutenauer Hopf structure on S. Malvenuto and Reutenauer [7] introduced two Hopf structures on K[S] in duality, one related to the Solomon descent algebra and the other to the algebras of the quasi-symmetric functions. The product ∗ and the coproduct ∆ of words α ∈ Sr and β ∈ Sp given by α ∗ β = ∑ st(u)=α, st(v)=β uv ∈ Sr+p (1)