90 research outputs found
A generating algorithm for ribbon tableaux and spin polynomials
We describe a general algorithm for generating various families of ribbon
tableaux and computing their spin polynomials. This algorithm is derived from a
new matricial coding. An advantage of this new notation lies in the fact that
it permits one to generate ribbon tableaux with skew shapes. This algorithm
permits us to compute quickly big LLT polynomials in MuPAD-Combinat
An algorithm for generating ribbon tableaux and spin polynomials
arXiv:math.CO/0611824International audienceWe describe a general algorithm for generating various families of ribbon tableaux and computing their spin polynomials. This algorithm is derived from a new matricial coding. An advantage of this new notation lies in the fact that it permits one to generate ribbon tableaux with skew shapes
q-Supernomial coefficients: From riggings to ribbons
q-Supernomial coefficients are generalizations of the q-binomial
coefficients. They can be defined as the coefficients of the Hall-Littlewood
symmetric function in a product of the complete symmetric functions or the
elementary symmetric functions. Hatayama et al. give explicit expressions for
these q-supernomial coefficients. A combinatorial expression as the generating
function of ribbon tableaux with (co)spin statistic follows from the work of
Lascoux, Leclerc and Thibon. In this paper we interpret the formulas by
Hatayama et al. in terms of rigged configurations and provide an explicit
statistic preserving bijection between rigged configurations and ribbon
tableaux thereby establishing a new direct link between these combinatorial
objects.Comment: 19 pages, svcon2e.sty file require
Ribbon tableaux, ribbon rigged configurations and Hall-Littlewood functions at roots of unity
Hall-Littlewood functions indexed by rectangular partitions, specialized at
primitive roots of unity, can be expressed as plethysms. We propose a
combinatorial proof of this formula using A. Schilling's bijection between
ribbon tableaux and ribbon rigged configurations
Affine insertion and Pieri rules for the affine Grassmannian
We study combinatorial aspects of the Schubert calculus of the affine
Grassmannian Gr associated with SL(n,C). Our main results are: 1) Pieri rules
for the Schubert bases of H^*(Gr) and H_*(Gr), which expresses the product of a
special Schubert class and an arbitrary Schubert class in terms of Schubert
classes. 2) A new combinatorial definition for k-Schur functions, which
represent the Schubert basis of H_*(Gr). 3) A combinatorial interpretation of
the pairing between homology and cohomology of the affine Grassmannian.
These results are obtained by interpreting the Schubert bases of Gr
combinatorially as generating functions of objects we call strong and weak
tableaux, which are respectively defined using the strong and weak orders on
the affine symmetric group. We define a bijection called affine insertion,
generalizing the Robinson-Schensted Knuth correspondence, which sends certain
biwords to pairs of tableaux of the same shape, one strong and one weak. Affine
insertion offers a duality between the weak and strong orders which does not
seem to have been noticed previously.
Our cohomology Pieri rule conjecturally extends to the affine flag manifold,
and we give a series of related combinatorial conjectures.Comment: 98 page
-Schur functions and affine Schubert calculus
This book is an exposition of the current state of research of affine
Schubert calculus and -Schur functions. This text is based on a series of
lectures given at a workshop titled "Affine Schubert Calculus" that took place
in July 2010 at the Fields Institute in Toronto, Ontario. The story of this
research is told in three parts: 1. Primer on -Schur Functions 2. Stanley
symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website:
http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates
and corrections since v1. This material is based upon work supported by the
National Science Foundation under Grant No. DMS-065264
A generalization of the Kostka-Foulkes polynomials
Combinatorial objects called rigged configurations give rise to q-analogues
of certain Littlewood-Richardson coefficients. The Kostka-Foulkes polynomials
and two-column Macdonald-Kostka polynomials occur as special cases.
Conjecturally these polynomials coincide with the Poincare polynomials of
isotypic components of certain graded GL(n)-modules supported in a nilpotent
conjugacy class closure in gl(n).Comment: 37 page
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