1,285 research outputs found
Non-commutative Pieri operators on posets
We consider graded representations of the algebra NC of noncommutative
symmetric functions on the Z-linear span of a graded poset P. The matrix
coefficients of such a representation give a Hopf morphism from a Hopf algebra
HP generated by the intervals of P to the Hopf algebra of quasi-symmetric
functions. This provides a unified construction of quasi-symmetric generating
functions from different branches of algebraic combinatorics, and this
construction is useful for transferring techniques and ideas between these
branches. In particular we show that the (Hopf) algebra of Billera and Liu
related to Eulerian posets is dual to the peak (Hopf) algebra of Stembridge
related to enriched P-partitions, and connect this to the combinatorics of the
Schubert calculus for isotropic flag manifolds.Comment: LaTeX 2e, 22 pages Minor corrections, updated references. Complete
and final version, to appear in issue of J. Combin. Th. Ser. A dedicated to
G.-C. Rot
Analogs of Schur functions for rank two Weyl groups obtained from grid-like posets
In prior work, the authors, along with M. McClard, R. A. Proctor, and N. J.
Wildberger, studied certain distributive lattice models for the "Weyl
bialternants" (aka "Weyl characters") associated with the rank two root
systems/Weyl groups. These distributive lattices were uniformly described as
lattices of order ideals taken from certain grid-like posets, although the
arguments connecting the lattices to Weyl bialternants were case-by-case
depending on the type of the rank two root system. Using this connection with
Weyl bialternants, these lattices were shown to be rank symmetric and rank
unimodal, and their rank generating functions were shown to have beautiful
quotient-of-products expressions. Here, these results are re-derived from
scratch using completely uniform and elementary combinatorial reasoning in
conjunction with some new combinatorial methodology developed elsewhere by the
second listed author.Comment: 15 page
Template iterations with non-definable ccc forcing notions
We present a version with non-definable forcing notions of Shelah's theory of
iterated forcing along a template. Our main result, as an application, is that,
if is a measurable cardinal and are
uncountable regular cardinals, then there is a ccc poset forcing
. Another
application is to get models with large continuum where the groupwise-density
number assumes an arbitrary regular value.Comment: To appear in the Annals of Pure and Applied Logic, 45 pages, 2
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