Grothendieck bialgebras, Partition lattices, and symmetric functions in noncommutative variables

We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis for this bialgebra.Canada Research ChairsNatural Sciences and Engineering Research Council of Canad