Expansion of k-Schur functions for maximal rectangles within the affine nilCoxeter algebra

@article{Berg2012ExpansionOK,
  title={Expansion of k-Schur functions for maximal rectangles within the affine nilCoxeter algebra},
  author={Chris Berg and Nantel Bergeron and Hugh Thomas and Mike Zabrocki},
  journal={The Journal of Combinatorics},
  year={2012},
  volume={3},
  pages={563-589}
}
We give several explicit combinatorial formulas for the expansion of k-Schur functions indexed by maximal rectangles in terms of the standard basis of the affine nilCoxeter algebra. Using our result, we also show a commutation relation of k-Schur functions corresponding to rectangles with the generators of the affine nilCoxeter algebra. 

Figures from this paper

Expansions of k-Schur Functions in the Affine nilCoxeter Algebra
We give a type free formula for the expansion of $k$-Schur functions indexed by fundamental coweights within the affine nilCoxeter algebra. Explicit combinatorics are developed in affine type $C$.
Canonical Decompositions of Affine Permutations, Affine Codes, and Split k-Schur Functions
TLDR
It is shown that the affine code readily encodes a number of basic combinatorial properties of an affine permutation, and proves a new special case of the Littlewood-Richardson Rule for $k-Schur functions.
Alcove random walks, k-Schur functions and the minimal boundary of the k-bounded partition poset
We use k-Schur functions to get the minimal boundary of the k-bounded partition poset. This permits to describe the central random walks on affine Grassmannian elements of type A and yields a
Primer on k-Schur Functions
The purpose of this chapter is to outline some of the results and open problems related to k-Schur functions, mostly in the setting of symmetric function theory. This chapter roughly follows the
k-Schur expansions of Catalan functions
k-Schur Functions and Affine Schubert Calculus
Author(s): Lam, T; Morse, J; Shimozono, M; Lapointe, L; Schilling, A; Zabrocki, M | Abstract: This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur
Symmetries on the Lattice of k-Bounded Partitions
In 2002, Suter [25] identified a dihedral symmetry on certain order ideals in Young’s lattice and gave a combinatorial action on the partitions in these order ideals. Viewing this result
Schubert Polynomials and $k$-Schur Functions
TLDR
It is shown that the multiplication of a SchUbert polynomial of finite type $A$ by a Schur function, which is referred to as Schubert vs. Schur problem, can be understood from the multiplication in the space of dual $k$-Schur functions.
...
...

References

SHOWING 1-10 OF 18 REFERENCES
Noncommutative schur functions and their applications
The Murnaghan-Nakayama rule for k-Schur functions
A k-tableau characterization of k-Schur functions
Upper Bounds in Affine Weyl Groups under the Weak Order
TLDR
It is determined that the question of which pairs of elements of W have upper bounds can be reduced to the analogous question within a particular finite subposet within an affine Weyl group W0.
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),
Schur function analogs for a filtration of the symmetric function space
Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions
Schubert polynomials for the affine Grassmannian
Let G be a complex simply connected simple group and K a maximal compact subgroup. Let F = C((t)) denote the field of formal Laurent series and O = C[[t]] the ring of formal power series. The
Reflection groups and coxeter groups
Part I. Finite and Affine Reflection Groups: 1. Finite reflection groups 2. Classification of finite reflection groups 3. Polynomial invariants of finite reflection groups 4. Affine reflection groups
Combinatorics of Coxeter Groups
I.- The basics.- Bruhat order.- Weak order and reduced words.- Roots, games, and automata.- II.- Kazhdan-Lusztig and R-polynomials.- Kazhdan-Lusztig representations.- Enumeration.- Combinatorial
...
...