Abstract
The ring of quasisymmetric functions is free over the ring of symmetric functions. This result was previously proved by M. Hazewinkel combinatorially through constructing a polynomial basis for quasisymmetric functions. The recent work by A. Savage and O. Yacobi on representation theory provides a new proof to this result. In this paper, we proved that under certain conditions, the positive part of a Heisenberg double is free over the positive part of the corresponding projective Heisenberg double. Examples satisfying the above conditions are discussed.
Highlights
Symmetric functions are formal power series which are invariant under every permutation of the indeterminates ([14])
We proved that under certain conditions, the positive part of a Heisenberg double is free over the positive part of the corresponding projective Heisenberg double
It’s well-known that as Hopf algebras Sym is isomorphic to the Grothendieck group of the abelian category C[Sn]-mod, where Sn is the n-th symmetric group
Summary
Symmetric functions are formal power series which are invariant under every permutation of the indeterminates ([14]). The elementary symmetric functions form a polynomial basis of Sym. The existence of comultiplication and counit gives Sym a Hopf algebra structure. The ring of quasisymmetric functions QSym ⊂ Z[[x1, x2, · · · ]] consists of shift invariant formal power series of bounded degrees. As Hopf algebras, NSym and QSym are dual to each other under the bilinear form hα, Mβ = δα,β. Hazewinkel contains the elementary symmetric functions, QSym is free over Sym. From the representation theory point of view, A. Yacobi [12] provide a new proof to the freeness of QSym over Sym. For each dual pair of Hopf algebras (H+, H−), one can construct the Heisenberg double h = h(H+, H−) of H+. Examples satisfying the conditions of Theorem 3.2 are discussed
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