Generalized Kostka Polynomials

Page last updated: 5 September 2005

The goal of this web page is to serve as a starting point for new researchers interested in the subject of Kostka polynomials and their generalizations. This page contains notes for introductory lectures on different aspects of Kostka polynomials, an annotated bibliography of relevant papers from the mathematical literature, a list of conjectures and open problems, and information about some useful computer algebra software.

Lecture Notes

  1. Loehr 1: Introduction to Macdonald Polynomials
  2. Loehr 2: Quick Definition of Macdonald Polynomials
  3. Lam: Lascoux-Leclerc-Thibon (LLT) Polynomials
  4. Haiman: Macdonald Polynomials and the Geometry of Hilbert Schemes
  5. Zabrocki: Creation Operators
  6. Morse: k-Schur Functions
  7. Kedem: Kostka Polynomials and Fusion Products
  8. Schwer: Galleries
  9. Stembridge: Kostka-Foulkes Polynomials in Other Root Systems
  10. Descouens: LLT Polynomials, Ribbon Tableaux, and the Affine Quantum Lie Algebra
  11. Shimozono 1: Generalized Kostka Polynomials as Parabolic Lusztig $q$-analogues
  12. Shimozono 2: One-dimensional Sums for the Impatient
  13. Shimozono 3: Crystals for DUMMIES

Annotated Bibliography

Papers are grouped by subject. Some papers may appear under more than one heading.
  1. Background
  2. Macdonald polynomials
  3. Generalized Kostka polynomials

Conjectures and Open Problems

  1. Schur Positivity Conjectures
  2. K=LLT Conjecture
  3. Problems related to Macdonald Polynomials [from first problem session].
  4. Problems related to k-Schur Functions and Representation Theory [from second problem session].
  5. Problems related to Fusion Products [from third problem session].
  6. Kostka Polynomials and Root Partition Functions
  7. Generalized Quasisymmetric Invariants

Computer Algebra Packages, Tables, etc.

  1. Installing ACE (Algebraic Combinatorics Environment for Maple) in unix .
  2. SF, posets, coxeter, and weyl (John Stembridge's Maple packages for symmetric functions, posets, root systems, and finite Coxeter groups).
  3. Maple code for computing generalized Kostka polynomials
  4. Mupad programs for computing generalized Kostka polynomials
  5. Tables of q,t-Kostka polynomials
  6. Information on a 'Q-function' analogue of Kostka polynomials
  7. Mathematica code for computing generalized Kostka polynomials
  8. Documentation for MuPAD-Combinat