66 research outputs found

    Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables

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    We analyze the structure of the algebra N of symmetric polynomials in non-commuting variables in so far as it relates to its commutative counterpart. Using the "place-action" of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of N analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.Comment: 14 page

    The primitives and antipode in the Hopf algebra of symmetric functions in noncommuting variables

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    We identify a collection of primitive elements generating the Hopf algebra NCSym of symmetric functions in noncommuting variables and give a combinatorial formula for the antipode.Comment: 8 pages; footnote added; references added; further remarks adde

    The characteristic polynomial of the Adams operators on graded connected Hopf algebras

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    The Adams operators Ψn\Psi_n on a Hopf algebra HH are the convolution powers of the identity of HH. We study the Adams operators when HH is graded connected. They are also called Hopf powers or Sweedler powers. The main result is a complete description of the characteristic polynomial (both eigenvalues and their multiplicities) for the action of the operator Ψn\Psi_n on each homogeneous component of HH. The eigenvalues are powers of nn. The multiplicities are independent of nn, and in fact only depend on the dimension sequence of HH. These results apply in particular to the antipode of HH (the case n=−1n=-1). We obtain closed forms for the generating function of the sequence of traces of the Adams operators. In the case of the antipode, the generating function bears a particularly simple relationship to the one for the dimension sequence. In case H is cofree, we give an alternative description for the characteristic polynomial and the trace of the antipode in terms of certain palindromic words. We discuss parallel results that hold for Hopf monoids in species and qq-Hopf algebras.Comment: 36 pages; two appendice

    QSym over Sym has a stable basis

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    We prove that the subset of quasisymmetric polynomials conjectured by Bergeron and Reutenauer to be a basis for the coinvariant space of quasisymmetric polynomials is indeed a basis. This provides the first constructive proof of the Garsia-Wallach result stating that quasisymmetric polynomials form a free module over symmetric polynomials and that the dimension of this module is n!.Comment: 12 page

    Hopf structures on the multiplihedra

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    We investigate algebraic structures that can be placed on vertices of the multiplihedra, a family of polytopes originating in the study of higher categories and homotopy theory. Most compelling among these are two distinct structures of a Hopf module over the Loday-Ronco Hopf algebra.Comment: 24 pages, 112 .eps file
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