2023-09-20

## Schur positivity

There are several different techniques to prove that a symmetric function is Schur positive.

### Robinson–Schensted–Knuth (RSK)

Examples of results and papers that use Robinson–Schensted–Knuth correspondence:

- One can show that the skew Schur functions are Schur positive by using RSK.
- In [HNY20], the authors use RSK to get the Schur expansion of certain unicellular LLT polynomials.
- In [AS17a], we use RSK to get the Schur-expansion of certain non-symmetric Macdonald polynomials evaluated at $t=0.$ This is later extended to a skew generalization in [AU20].

### Dual equivalence

The idea behind dual equivalence is to start with the Gessel fundamental expansion of a symmetric function (as a sum over some set of combinatorial objects), and define a graph structure on these objects such that connected components sum to Schur functions.

Austin Roberts give a modified list of axioms, allowing for a *local*
characterization of dual equivalence graphs [Rob13].
It is then enough to verify all graphs with at
most six vertices, which usually can be done on a computer.

The following are extensions and analogues of dual equivalence:

- Type $B$ version of dual equivalence for shifted tableaux, [Def. 4.5.1, Rob14].
- Schur-P and Schur-Q dual equivalence.
- Dual equivalence can be generalized to quasi-symmetric Schur functions, see A. Roberts [Rob16].
- Permutation classes that give rise to Schur positive expressions, see [ER16a].

### Crystal graphs

The idea behind crystal graphs is to define a graph structure on the (combinatorial) objects that generate the monomial expansion. By showing that the graph satisfies a set of axioms, it follows that each connected component sum to a Schur function. For example, one can define a crystal graph on skew SSYT in order to prove that skew Schur functions are Schur positive.

A crystal graph also comes with an $\symS_n$-action on the combinatorial objects, so that one obtains an $\symS_n$-module (a representation), whose Frobenius image is exactly the original symmetric function.

For example, one can define a crystal graph on words of length $n,$ and the $\symS_n$-action is generated by the Lascoux–Schutzenberger involutions $s_i,$ acting on the words. This proves that $(x_1+x_2+x_3+\dotsb)^n$ is Schur-positive.

Notable examples include non-symmetric Macdonald polynomials, [AG18] and Stanley symmetric functions, [MS15c], and Dual stable Grothendieck polynomials, [Gal17].

A flagged version of skew dual stable Grothendieck polynomials are shown to be key positive using crystals in [Kun23]. hyperref[kohnert]{Kohnert polynomials} are key positive with crystal structure defined in [Ass21].

### Representation theory

For a background, see the page on representation theory.

Examples: Modified Macdonald polynomials, LLT polynomials, Eulerian symmetric functions.

### Edelman–Greene

The Edelman–Greene bijection is used to show that the Stanley symmetric functions are Schur positive, see [EG87].

### The slinky rule

The idea is to use the slinky rule to convert an expansion in the Gessel fundamental quasisymmetric basis into a (signed) Schur expansion, and then one must invent some sign-reversing involution. An example of a paper that uses this strategy is [Ser17].

### Positive classes of permutations

In [ER16a], the authors characterize several classes of permutations $A \subseteq \symS_n,$ such that

\[ \sum_{\sigma \in A} \gessel_{n,\DES(\sigma)}(\xvec) \]is Schur-positive. We say that $A$ is Schur-positive if the above sum is Schur-positive.

In [AR15], it is proved that $A \subseteq \symS_n$ is Schur-positive if and only if there are non-negative integers $a_\lambda$ such that

\[ \sum_{\sigma \in A} \yvec_{\DES(\sigma)} = \sum_{\lambda \vdash n} a_\lambda \sum_{T \in \SYT(\lambda)} \yvec_{\DES(T)}. \]Here, $\yvec_{S} = y_{i_1} \dotsm y_{i_\ell}$ where $S = \{i_1,i_2,\dotsc,i_\ell\}.$

## References

- [AG18] Sami Assaf and Nicolle S. González. Crystal graphs, key tabloids, and nonsymmetric Macdonald polynomials. 30th International Conference on Formal Power Series and Algebraic Combinatorics. Séminaire Lotharingien de Combinatoire, 80B(90) 2018. 12 pages
- [AR15] Ron M. Adin and Yuval Roichman. Matrices, characters and descents. Linear Algebra and its Applications, 469:381–418, March 2015.
- [AS17a] Per Alexandersson and Mehtaab Sawhney. A major-index preserving map on fillings. Electronic Journal of Combinatorics, 24(4):1–30, 2017.
- [Ass21] Sami Assaf. Demazure crystals for Kohnert polynomials. Transactions of the American Mathematical Society, 375(3):2147–2186, December 2021.
- [AU20] Per Alexandersson and Joakim Uhlin. Cyclic sieving, skew Macdonald polynomials and Schur positivity. Algebraic Combinatorics, 3(4):913-939, 2020.
- [EG87] Paul Edelman and Curtis Greene. Balanced tableaux. Advances in Mathematics, 63(1):42–99, January 1987.
- [ER16a] Sergi Elizalde and Yuval Roichman. Schur-positive sets of permutations via products and grid classes. Journal of Algebraic Combinatorics, 45(2):363–405, September 2016.
- [ER16b] Sergi Elizalde and Yuval Roichman. Schur-positive sets of permutations via products and grid classes. Journal of Algebraic Combinatorics, 45(2):363–405, September 2016.
- [Gal17] Pavel Galashin. A Littlewood–Richardson rule for dual stable Grothendieck polynomials. Journal of Combinatorial Theory, Series A, 151:23–35, 2017.
- [HNY20] JiSun Huh, Sun-Young Nam and Meesue Yoo. Melting lollipop chromatic quasisymmetric functions and Schur expansion of unicellular LLT polynomials. Discrete Mathematics, 343(3):111728, March 2020.
- [Kun23] Siddheswar Kundu. Demazure crystal structure for flagged reverse plane partitions. arXiv e-prints, 2023.
- [MS15c] Jennifer Morse and Anne Schilling. Crystal approach to affine Schubert calculus. International Mathematics Research Notices, 2016(8):2239–2294, July 2015.
- [Rob13] Austin Roberts. Dual equivalence graphs revisited and the explicit Schur expansion of a family of LLT polynomials. Journal of Algebraic Combinatorics, 39(2):389–428, May 2013.
- [Rob14] Austin Roberts. Dual equivalence graphs and their applications. University of Washington. 2014.
- [Rob16] Austin Roberts. Symmetric fundamental expansions to Schur positivity. 28th International Conference on Formal Power Series and Algebraic Combinatorics. Discrete Mathematics and Theoretical Computer Science, 2016.
- [Ser17] Emily Sergel. A proof of the square paths conjecture. Journal of Combinatorial Theory, Series A, 152:363–379, November 2017.