$k$-Schur functions and affine Schubert calculus

Abstract

This book is an exposition of the current state of research of affine Schubert calculus and $k$-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 $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculus

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-0652641

References in corpus (18)

• Perverse sheaves on a Loop group and Langlands' duality
• The Verlinde Algebra And The Cohomology Of The Grassmannian
• Integrality of two variable Kostka functions
• Positroid varieties I: juggling and geometry
• Double Schubert polynomials for the classical groups
• su(3)k fusion coefficients
• Berenstein-Zelevinsky triangles, elementary couplings and fusion rules
• Equivariant K-theory of affine flag manifolds and affine Grothendieck polynomials
• Combinatorial $B_n$-analogues of Schubert polynomials
• The Murnaghan-Nakayama rule for k-Schur functions
• Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions
• Affine Schubert classes, Schur positivity, and combinatorial Hopf algebras
• From double quantum Schubert polynomials to k-double Schur functions via the Toda lattice
• Expansion of $k$-Schur functions for maximal $k$-rectangles within the affine nilCoxeter algebra
• k-shape poset and branching of k-Schur functions
• Mixed Bruhat operators and Yang-Baxter equations for Weyl groups
• The Center of the Nilcoxeter and 0-Hecke Algebras
• $q$-Difference raising operators for Macdonald polynomials and the integrality of transition coefficients

Cited by in corpus (8)

• Back stable Schubert calculus
• Crystal approach to affine Schubert calculus
• Global Brill--Noether Theory over the Hurwitz Space
• The role of residue and quotient tables in the theory of k-Schur functions
• Affine Pieri rule for periodic Macdonald spherical functions and fusion rings
• Filtering cohomology of ordinary and Lagrangian Grassmannians
• A combinatorial formula for the nabla operator
• Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells