2022-12-13

## 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!

### Preliminaries

- Partitions and tableaux
- Permutation matrices and patterns
- $q$-analogs
- Plethysm
- 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
- Crystals
- Operations on Young tableaux (Bender–Knuth, promotion, evacuation, jeu-de-taquin)

### Kostka coefficients

- Gelfand–Tsetlin polytopes
- The Kostant partition function
- Kostka–Foulkes polynomials
- Rigged configurations

### Cyclic sieving

### Miscellaneous

- 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.

## Software resources

- Bridget Tanner's permutation pattern database.
- Mathematica packages (Per Alexandersson) Packages for Catalan objects, Symmetric functions, posets, tableaux and unicellular chromatic calculations.

### Useful links

- 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.

## Suggested books

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.

## Files

These files are rather old — I intend to update them some time in the future when the catalog include most of the quasisymmetric families.

## References

- [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.