126 research outputs found

### Skew Schubert functions and the Pieri formula for flag manifolds

We show the equivalence of the Pieri formula for flag manifolds and certain
identities among the structure constants, giving new proofs of both the Pieri
formula and of these identities. A key step is the association of a symmetric
function to a finite poset with labeled Hasse diagram satisfying a symmetry
condition. This gives a unified definition of skew Schur functions, Stanley
symmetric function, and skew Schubert functions (defined here). We also use
algebraic geometry to show the coefficient of a monomial in a Schubert
polynomial counts certain chains in the Bruhat order, obtaining a new
combinatorial construction of Schubert polynomials.Comment: 24 pages, LaTeX 2e, with epsf.st

### A Monoid for the Universal K-Bruhat Order

Structure constants for the multiplication of Schubert polynomials by Schur
symmetric polynomials are known to be related to the enumeration of chains in a
new partial order on S_\infty, which we call the universal k-Bruhat order. Here
we present a monoid M for this order and show that $M$ is analogous to the
nil-Coxeter monoid for the weak order on S_\infty. For this, we develop a
theory of reduced sequences for M. We use these sequences to give a
combinatorial description of the structure constants above. We also give
combinatorial proofs of some of the symmetry relations satisfied by these
structure constants.Comment: LaTeX-2e, 21 pages, 3 figures, uses epsf.st

### q and q,t-Analogs of Non-commutative Symmetric Functions

We introduce two families of non-commutative symmetric functions that have
analogous properties to the Hall-Littlewood and Macdonald symmetric functions.Comment: Different from analogues in math.CO/0106191 - v2: 26 pages - added a
definition in terms of triangularity/scalar product relations - to be
submitted FPSAC'0

### Fomin-Greene monoids and Pieri operations

We explore monoids generated by operators on certain infinite partial orders.
Our starting point is the work of Fomin and Greene on monoids satisfying the
relations $(\u{r}+\u{r+1})\u{r+1}\u{r}=\u{r+1}\u{r}(\u{r}+\u{r+1})$ and
$\u{r}\u{t}=\u{s}\u{r}$ if $|r-t|>1.$ Given such a monoid, the non-commutative
functions in the variables $\u{}$ are shown to commute. Symmetric functions in
these operators often encode interesting structure constants. Our aim is to
introduce similar results for more general monoids not satisfying the relations
of Fomin and Greene. This paper is an extension of a talk by the second author
at the workshop on algebraic monoids, group embeddings and algebraic
combinatorics at The Fields Institute in 2012.Comment: 33 pages, this is a paper expanding on a talk given at Fields
Institute in 201

### A combinatorial basis for the free Lie algebra of the labelled rooted trees

The pre-Lie operad can be realized as a space T of labelled rooted trees. A
result of F. Chapoton shows that the pre-Lie operad is a free twisted Lie
algebra. That is, the S-module T is obtained as the plethysm of the S-module
Lie with an S-module F. In the context of species, we construct an explicit
basis of F. This allows us to give a new proof of Chapoton's results. Moreover
it permits us to show that F forms a sub nonsymmetric operad of the pre-Lie
operad T.Comment: 12 pages, uses xypi

### A Pieri-type formula for isotropic flag manifolds

We give the formula for multiplying a Schubert class on an odd orthogonal or
symplectic flag manifold by a special Schubert class pulled back from a
Grassmannian of maximal isotropic subspaces. This is also the formula for
multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the
Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$).
Geometric constructions and intermediate results allow us to ultimately deduce
this from formulas for the classical flag manifold. These intermediate results
are concerned with the Bruhat order of the Coxeter group ${\mathcal B}_\infty$,
identities of the structure constants for the Schubert basis of cohomology, and
intersections of Schubert varieties. We show these identities follow from the
Pieri-type formula, except some `hidden symmetries' of the structure constants.
Our analysis leads to a new partial order on the Coxeter group ${\mathcal
B}_\infty$ and formulas for many of these structure constants

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