263 research outputs found
Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables
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
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
The Adams operators on a Hopf algebra are the convolution powers
of the identity of . We study the Adams operators when 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 on each
homogeneous component of . The eigenvalues are powers of . The
multiplicities are independent of , and in fact only depend on the dimension
sequence of . These results apply in particular to the antipode of (the
case ). 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 -Hopf algebras.Comment: 36 pages; two appendice
QSym over Sym has a stable basis
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
A note on the Markoff condition and central words
We define Markoff words as certain factors appearing in bi-infinite words satisfying the Markoff condition. We prove that these words coincide with central words, yielding a new characterization of Christoffel words
Hopf structures on the multiplihedra
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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