17 research outputs found
Richard Stanley through a crystal lens and from a random angle
We review Stanley's seminal work on the number of reduced words of the
longest element of the symmetric group and his Stanley symmetric functions. We
shed new light on this by giving a crystal theoretic interpretation in terms of
decreasing factorizations of permutations. Whereas crystal operators on
tableaux are coplactic operators, the crystal operators on decreasing
factorization intertwine with the Edelman-Greene insertion. We also view this
from a random perspective and study a Markov chain on reduced words of the
longest element in a finite Coxeter group, in particular the symmetric group,
and mention a generalization to a poset setting.Comment: 11 pages; 3 figures; v2 updated references and added discussion on
Coxeter-Knuth grap
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
Crystal analysis of type Stanley symmetric functions
Combining results of T.K. Lam and J. Stembridge, the type Stanley
symmetric function , indexed by an element in the type
Coxeter group, has a nonnegative integer expansion in terms of Schur
functions. We provide a crystal theoretic explanation of this fact and give an
explicit combinatorial description of the coefficients in the Schur expansion
in terms of highest weight crystal elements.Comment: 39 page
Coxeter-Knuth graphs and a signed Little map for type B reduced words
We define an analog of David Little's algorithm for reduced words in type B,
and investigate its main properties. In particular, we show that our algorithm
preserves the recording tableau of Kra\'{s}kiewicz insertion, and that it
provides a bijective realization of the Type B transition equations in Schubert
calculus. Many other aspects of type A theory carry over to this new setting.
Our primary tool is a shifted version of the dual equivalence graphs defined by
Assaf and further developed by Roberts. We provide an axiomatic
characterization of shifted dual equivalence graphs, and use them to prove a
structure theorem for the graph of Type B Coxeter-Knuth relations.Comment: 41 pages, 10 figures, many improvements from version 1, substantively
the same as the version in Electronic Journal of Combinatorics, Vol 21, Issue