Topics related to symmetric functions
Here is an overview of various topics which are related to symmetric functions, but do not fit together with a specific family. Some of the pages give a brief overview of the definitions and notation that is used.
We suggest that the reader is somewhat familiar with algebraic combinatorics, although a few topics is covered under preliminaries below. Furthermore, if you are a PhD student or researcher interested in symmetric functions, join our Facebook group!
- Partitions and tableaux
- Permutation matrices and patterns
- Plane partitions
- Unimodality, realrootedness and stable polynomials
- Representation theory
- Root systems
- Schur positivity
- Models for Littlewood–Richardson coefficients
- Border-strip tableaux and ribbon tableaux
- Diagonal harmonics
- Vertex models and the Yang–Baxter equation
- Lagrange inversion
Operations on tableaux
- The Robinson–Schensted–Knuth correspondence
- Operations on Young tableaux (Bender–Knuth, promotion, evacuation, jeu-de-taquin)
- Gelfand–Tsetlin polytopes
- The Kostant partition function
- Kostka–Foulkes polynomials
- Rigged configurations
- Acyclic orientations and Touchard–Riordan polynomials
- Minor problems related to my research
- Research story: R. Stanley — How the upper bound conjecture was proved
- Research story: J. Haglund — The genesis of the Macdonald polynomial statistic
- R. Stanley -- Enumerative and Algebraic combinatorics in the 1960's and 1970's.
- Bridget Tanner's permutation pattern database.
- Mathematica packages (Per Alexandersson) Packages for Catalan objects, Symmetric functions, posets, tableaux and unicellular chromatic calculations.
- Doi 2 bib. This tool is extremely useful.
- Maria Gillespie's Mathematical gemstones.
- Overview of generalized Kostka Polynomials.
- Grammar and math by D. B. West.
- A primer of mathematical writing by S. G. Krantz.
For an introduction to the topic of symmetric functions and algebraic combinatorics, I suggest the following books. Richard Stanley, Enumerative Combinatorics II [Sta01], Lynne Butler, Subgroup Lattices and Symmetric Functions [But94], Bruce Sagan, The Symmetric Group [Sag01], William Fulton, Young Tableaux [Ful97] and Ian Macdonald, Symmetric functions and Hall polynomials [Mac95].
Other related books: David Bressoud, Proofs and confirmations: the story of the alternating sign matrix conjecture [Bre99], which is a very nice reading.
These files are rather old — I intend to update them some time in the future when the catalog include most of the quasisymmetric families.
- [Bre99] David Bressoud. Proofs and confirmations: the story of the alternating sign matrix conjecture. Cambridge University Press, 1999.
- [But94] Lynne M. Butler. Subgroup lattices and symmetric functions. American Mathematical Society, 1994.
- [Ful97] William Fulton. Young tableaux: with applications to representation theory and geometry. London Mathematical Society Student Texts (Book 35), Cambridge University Press, 1997.
- [Mac95] Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications
- [Sag01] Bruce E. Sagan. The symmetric group. Springer New York, 2001.
- [Sta01] Richard P. Stanley. Enumerative Combinatorics: Volume 2. Cambridge University Press, First edition, 2001.